Notes for Algebraic Geometry II

William A. Stein May 4, 1996

Contents

1 Preface 1 2 Ample Invertible Sheaves 4 3 Introduction to Cohomology 5 4 Cohomology in Algebraic Geometry 6 5 Review of Derived Functors 6

5.1 Examples of Abelian Categories . . . . . . . . . . . . . . . . . . . . . . . . . 7 5.2 Exactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5.3 Injective and Projective Objects . . . . . . . . . . . . . . . . . . . . . . . . . 8

6 Derived Functors and Homological Algebra 8 6.1 Construction of R^{i}F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 6.2 Properties of Derived Functors . . . . . . . . . . . . . . . . . . . . . . . . . . 9

7 Long Exact Sequence of Cohomology and Other Wonders 10 8 Basic Properties of Cohomology 10

8.1 Cohomology of Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 8.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

9 Flasque Sheaves 11 10 Examples 12 11 First Vanishing Theorem 12 12 Čech Cohomology 13

12.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 12.2 Sheafify . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

13 Čech Cohomology and Derived Functor Cohomology 16

13.1 History of this Module E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 14 Cohomology of P_{k}^{n} 17 15 Serre’s Finite Generation Theorem 18

15.1 Application: The Arithmetic Genus . . . . . . . . . . . . . . . . . . . . . . . 19 16 Euler Characteristic 19 17 Correspondence between Analytic and Algebraic Cohomology 21 18 Arithmetic Genus 21

18.1 The Genus of Plane Curve of Degree d . . . . . . . . . . . . . . . . . . . . . 22 19 Not Enough Projectives 22 20 Some Special Cases of Serre Duality 23

20.1 Example: O_{X} on Projective Space . . . . . . . . . . . . . . . . . . . . . . . . 23 20.2 Example: Coherent sheaf on Projective Space . . . . . . . . . . . . . . . . . 24 20.3 Example: Serre Duality on P_{k}^{n} . . . . . . . . . . . . . . . . . . . . . . . . . . 25

21 The Functor Ext 25 21.1 Sheaf Ext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 21.2 Locally Free Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

22 More Technical Results on Ext 27 23 Serre Duality 28 24 Serre Duality for Arbitrary Projective Schemes 29 25 Existence of the Dualizing Sheaf on a Projective Scheme 32

25.1 Relative Gamma and Twiddle . . . . . . . . . . . . . . . . . . . . . . . . . . 32 26 Generalized Grothendieck Duality Theory 34

27  35

28 Review of Differentials 37

28.1 The Sheaf of Differentials on a Scheme . . . . . . . . . . . . . . . . . . . . . 39 29 Differentials on P^{n} 40 30 Sheaf of Differentials and Canonical Divisor 42 31 Definitions 44 32 Genus 45 33 Riemann-Roch Theorem 46 34 Serre Duality 47 35 A Bird’s Eye View of Curves 47 36 Moduli Space 47 37 Embeddings in Projective Space 48 38 Elementary Curve Theory 48

38.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 38.2 Maps to Projective Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

39 Low Genus Projective Embeddings 48 39.1 Genus 0 curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 39.2 Genus 1 curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 39.3 Moduli Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

40 Curves of Genus 3 50 41 Curves of Genus 4 53

41.1 Aside: existence of g_{d}^{1}’s in general . . . . . . . . . . . . . . . . . . . . . . . . 53 41.2 Classifying curves of genus 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

42 Curves of Genus 5 55 42.1 The space of quadrics containing an embedded genus 5 curve. . . . . . . . . . 55 42.2 Genus 5 curves with very ample canonical divisor . . . . . . . . . . . . . . . 55

43 Homework Assignment 57 44 Curves of genus 6 57 45 Oral Report Topics 59 46 Curves of general genus 60 47 Halphen’s Theorem 62 48 Hurwitz’s Theorem 63 49 Elliptic Curves 65 50 Automorphisms of Elliptic Curves 67 51 Moduli Spaces 70 52 The Jacobian Variety 71

52.1 Consequence of existence of the Jacobian . . . . . . . . . . . . . . . . . . . . 73 53 The Jacobian 73

53.1 Consequences of existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 53.1.1 Group structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 53.1.2 Natural fibration, dimension . . . . . . . . . . . . . . . . . . . . . . . 75 53.1.3 The Zariski tangent space . . . . . . . . . . . . . . . . . . . . . . . . 76

54 Flatness 76

54.1 Technical definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

54.1.1 General nonsense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 54.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 54.3 Algebraic geometry definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 78 54.4 Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

55 Theorem about flat families 80

56 Examples of Flat Families 80

57 Homework problems 82 57.1 Exercise on basic cohomology and abstract nonsense . . . . . . . . . . . . . . 82 57.2 Chapter III, 4.8, 4.9, 5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

57.2.1 Exercise III.4.8: Cohomological Dimension . . . . . . . . . . . . . . . 83 57.2.2 Exercise III.4.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 57.2.3 Exercise III.5.6: Curves on a nonsingular quadric surface . . . . . . . 86

57.3 IV, 3.6, 3.13, 5.4, Extra Problems . . . . . . . . . . . . . . . . . . . . . . . . 91 57.3.1 Exercise IV.3.6: Curves of Degree 4 . . . . . . . . . . . . . . . . . . . 91 57.3.2 Exercise IV.3.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 57.3.3 Exercise IV.5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 57.3.4 Extra Problem 3, by William Stein . . . . . . . . . . . . . . . . . . . 95 57.3.5 Extra Problem 4, by Nghi Nguyen . . . . . . . . . . . . . . . . . . . . 96

1 Preface

Read at your own risk!

These are my very rough, error prone notes of a second course on algebraic geometry offered at U.C. Berkeley in the Spring of 1996. The instructor was Robin Hartshorne and the students were Wayne Whitney, William Stein, Matt Baker, Janos Csirik, Nghi Nguyen, and Amod. I wish to thank Robin Hartshorne for giving this course and to Nghi Nguyen for his insightful suggestions and corrections. Of course all of the errors are solely my responsibility.

The remarks in brackets [[like this]] are notes that I wrote to myself. They are meant as a warning or as a reminder of something I should have checked but did not have time for. You may wish to view them as exercises.

If you have suggestions, questions, or comments feel free to write to me. My email address

is was@math.berkeley.edu.

2 Ample Invertible Sheaves

Let k be an algebraically closed field and let X be a scheme over k. Let ϕ : X → P_{k}^{n} be a morphism. Then to give ϕ is equivalent to giving an invertible sheaf L on X and sections s_{0}, …, s_{n} ∈ Γ(X, L) which generate L. If X is projective (that is, if there is some immersion of X into some P_{k}^{m}) then ϕ is a closed immersion iff s_{0}, …, s_{n} separate points and tangent vectors.

Definition 2.1. Let X be a scheme and L an invertible sheaf on X. Then we say L is very ample if there is an immersion i : X ↪ P_{k}^{n} such that L ≅ i^{∗}O(1).

Theorem 2.2. Let X be a closed subscheme of P_{k}^{n} and F a coherent sheaf on X, then F(n) is generated by global sections for all n ≫ 0.

Corollary 2.3. Let X be any scheme and L a very ample coherent sheaf on X, then for all n ≫ 0, F ⊗ L^{⊗n} is generated by global sections.

Definition 2.4. Let X be a Noetherian scheme and L be an invertible sheaf. We say that L is ample if for every coherent sheaf F on X, there is n_{0} such that for all n ≥ n_{0}, F ⊗ L^{⊗n} is generated by its global sections.

Thus the previous corollary says that a very ample invertible sheaf is ample.

Proposition 2.5. Let X and L be as above. Then the following are equivalent. 1) L is ample,

2) L^{n} is ample for all n > 0,

3) L^{n} is ample for some n > 0.

Theorem 2.6. Let X be of finite type over a Noetherian ring A and suppose L is an invertible sheaf on A. Then L is ample iff there exists n such that L^{n} is very ample over Spec A.

Example 2.7. Let X = P^{1}, L = O(ℓ), some ℓ ∈ Z. If ℓ < 0 then Γ(L) = 0. If ℓ = 0 then L = O_{X} which is not ample since O_{X}(−1)^{n} ⊗ O_{X} ≅ O_{X}(−1)^{n} is not generated by global sections for any n. Note that O_{X} itself is generated by global sections. Finally, if ell > 0 then L = O_{X}(ℓ) is very ample hence ample.

Example 2.8. Let C ⊆ P^{2} be a nonsingular cubic curve and L an invertible sheaf on C defined by L = L(D), where D = ∑ n_{i}P_{i} is a divisor on C. If deg D < 0 then L has no global sections so it can’t be ample.

3 Introduction to Cohomology

We first ask, what is cohomology and where does it arise in nature? Cohomology occurs in commutative algebra, for example in the Ext and Tor functors, it occurs in group theory, topology, differential geometry, and of course in algebraic geometry. There are several flavors of cohomology which are studied by algebraic geometers. Serre’s coherent sheaf cohomology has the advantage of being easy to define, but has the property that the cohomology groups are vector spaces. Grothendieck introduced ètale cohomology and ℓ-adic cohomology. See, for example, Milne’s Ètale Cohomology and SGA 4\frac{1}{2}, 5 and 6. This cohomology theory arose from the study of the Weil Conjectures (1949) which deal with a deep relationship between the number of points on a variety over a finite field and the geometry of the complex analytic variety cut out by the same equations in complex projective space. Deligne was finally able to resolve these conjectures in the affirmative in 1974.

What is cohomology good for? Cohomology allows one to get numerical invariants of an algebraic variety. For example, if X is a projective scheme defined over an algebraically closed field k then H^{i}(X, F) is a finite dimensional k-vector space. Thus the h_{i} = dim_{k} H^{i}(X, F) are a set of numbers associated to X. “Numbers are useful in all branches of mathematics.”

Example 3.1. Arithmetic Genus Let X be a nonsingular projective curve. Then dim H^{1}(X, O_{X}) is the arithmetic genus of X. If X ⊆ P^{n} is a projective variety of dimension r then, if p_{a} = dim H^{1}(X, O_{X}), then 1 + (−1)^{r}p_{a} = the constant term of the Hilbert polynomial of X.

Example 3.2. Let X be a nonsingular projective surface, then

1 + p_{a} = h^{0}(O_{X}) − h^{1}(O_{X}) + h^{2}(O_{x})

and 1 + (−1)^{r}p_{a} = χ(O_{X}), the Euler characteristic of X.

Example 3.3. Let X be an algebraic variety and Pic X the group of Cartier divisors modular linear equivalence (which is isomorphic to the group of invertible sheaves under tensor product modulo isomorphism). Then Pic X ≅ H^{1}(X, O_{X}^{∗}).

Example 3.4 (Deformation Theory). Let X_{0} be a nonsingular projective variety. Then the first order infinitesimal deformations are classified by H^{1}(X_{0}, T_{X_{0}}) where T_{X_{0}} is the tangent bundle of X_{0}. The obstructions are classified by H^{2}(X_{0}, T_{X_{0}}).

One can define Cohen-Macaulay rings in terms of cohomology. Let (A, 𝔪) be a local Noetherian ring of dimension n, let X = Spec A, and let P = 𝔪 ∈ X, then we have the following.

Proposition 3.5. Let A be as above. Then A is Cohen-Macaulay iff 1) H^{0}(X − P, O_{X−P}) = A and

2) H^{i}(X − P, O_{X−P}) = 0 for 0 < i < n − 1.

A good place to get the necessary background for the cohomology we will study is in Appendices 3 and 4 from Eisenbud’s Commutative Algebra.

4 Cohomology in Algebraic Geometry

For any scheme X and any sheaf F of O_{X}-modules we want to define the groups H^{i}(X, F). We can either define cohomology by listing its properties, then later prove that we can construct the H^{i}(X, F) or we can skip the definition and just construct the H^{i}(X, F). The first method is more esthetically pleasing, but we will choose the second.

We first forget the scheme structure of X and regard X as a topological space and F as a sheaf of abelian groups (by ignoring the ring multiplication). Let Ab(X) be the category of sheaves of abelian groups on X. Let Γ = Γ(X, ·) be the global section functor from Ab(X) into Ab, where Ab is the category of abelian groups. Recall that Γ is left exact so if

0 → F′ → F → F′′ → 0

is an exact sequence in Ab(X) then the following sequence is exact

0 → Γ(F′) → Γ(F) → Γ(F′′)

in Ab.

Definition 4.1. We define the cohomology groups H^{i}(X, F) to be the right derived functors of Γ.

5 Review of Derived Functors

The situation will often be as follows. Let A and B be abelian categories and

A ⟶^{F} B

a functor. Derived functors are the measure of the non-exactness of a functor. Let X be a topological space, Ab(X) the category of sheaves of abelian groups on X and Ab the category of abelian groups. Then Γ(X, ·) : Ab(X) → Ab is a left exact functor. Our cohomology theory will turn out to be the right derived function of Γ(X, ·).

5.1 Examples of Abelian Categories

Although we will not define an abelian category we will give several examples and note that an abelian category is a category which has the same basic properties as these examples. Example 5.1 (A-Modules). Let A be a fixed commutative ring and consider the category Mod(A) of A-modules. Then if M, N are any two modules one has

1) Hom(M, N) is an abelian group,

2) Hom(M, N) × Hom(N, L) → Hom(M, L) is a homomorphism of abelian groups.

3) there are kernels, cokernels, etc.

Mod(A) is an abelian category. Example 5.2. Let A be a Noetherian ring and let our category be the collection of all finitely generated A-modules. Then this category is abelian. Note that if the condition that A be Noetherian is relaxed we may no longer have an abelian category because the kernel of a morphism of finitely generated modules over an arbitrary ring need not be finitely generated (for example, take the map from a ring to its quotient by an ideal which cannot be finitely generated). Example 5.3. Let X be a topological space, then Ab(X) is an abelian category. If (X, O_{X}) is a ringed space then the category Mod(O_{X}) is abelian. If X is a scheme then the category of quasi-coherent O_{X}-modules is abelian, and if X is also Noetherian then the sub-category of coherent O_{X}-modules is abelian. Example 5.4. The category of abelian varieties is not an abelian category since the kernel of a morphism of abelian varieties might be reducible (for example an isogeny of degree n of elliptic curves has kernel n points which is reducible). It may be the case that the category of abelian group schemes is abelian but I don’t know at the moment. Example 5.5. The category of compact Hausdorff abelian topological groups is an abelian category.

5.2 Exactness

Definition 5.6. A functor F : A → B is additive if for all X, Y ∈ A, the map

F : Hom_{A}(X, Y) → Hom_{B}(F X, F Y)

is a homomorphism of abelian groups.

Definition 5.7. A sequence

A ⟶^{f} B ⟶^{g}C

is exact if Im(f) = ker(g).

Definition 5.8. Let F : A → B be a functor and

0 → M′ → M → M′′ → 0

be an exact sequence and consider the sequence

0 → F M′ → F M → F M′′ → 0.

If the second sequence is exact in the middle, then F is a called half exact functor. If the second sequence is exact on the left and the middle then F is called a left exact functor. If the second sequence is exact on the right and in the middle then we call F a right exact functor.

Example 5.9. Let A be a commutative ring and N an A-module. Then N ⊗ − is a right exact functor on the category of A-modules. To see that N ⊗ − is not exact, suppose A, 𝔪 is a local ring and N = k = A/𝔪. Then the sequence

0 → 𝔪 → A → k → 0

is exact, but

0 → k ⊗ 𝔪 → k ⊗ A → k ⊗ k → 0

is right exact but not exact. Example 5.10. The functor Tor_{1}(N, ·) is neither left nor right exact. Example 5.11. The contravarient hom functor, Hom(·, N) is left exact.

Often the following is useful in work.

Theorem 5.12. If

0 → M′ → M → M′′

is exact and F is left exact, then

0 → F M′ → F M → F M′′

is exact.

5.3 Injective and Projective Objects

Let A be an abelian category. Then Hom_{A}(P, −) : A → Ab is left exact.

Definition 5.13. An A module P is said to be projective if the functor Hom_{A}(P, −) is exact. An A module I is said to be injective if the functor Hom_{A}(−, I) is exact.

Definition 5.14. We say that an abelian category A has enough projectives if every X in A is the surjective image of a projective P in A. A category is said to have enough injectives if every X in A injects into an injective objective of A.

Example 5.15. Let A be a commutative ring, then Mod(A) has enough injectives because every module is the quotient of a free module and every free module is projective. If X is a topological space then Ab(X) has enough injectives. If X is a Noetherian scheme, then the category of quasi-coherent sheaves has enough injectives (hard theorem). The category of O_{X}-modules has enough injectives but the category of coherent sheaves on X doesn’t have enough injectives or projectives.

6 Derived Functors and Homological Algebra

Let F : A → B be an additive covariant left-exact functor between abelian categories, for example F = Γ : Ab(X) → Ab. Assume A has enough injectives, i.e., for all X in A there is an injective object I in A such that 0 → X ↪ I. We construct the right derived functors of F. If

0 → M′ → M → M′′ → 0

is exact in A then

0 → F (M′) → F (M) → F (M′′) → R^{1}F (M^{1}) → R^{1}F (M) → ⋯

is exact in B where R^{i}F is the right derived functor of F.

6.1 Construction of R^{i}F

Take any M in A, then since A has enough injectives we can construct an exact sequence

0 → M → I^{0} → I^{1} → I^{2} → ⋯

where each I^{i} is an injective object. (This isn’t totally obvious, but is a straightforward argument by putting together short exact sequences and composing maps.) The right part of the above sequence I^{0} → I^{1} → ⋯ is called an injective resolution of M. Applying F we get a complex

F (I^{0}) ⟶^{d_{0}} F (I^{1}) ⟶^{d_{1}} F (I^{2}) ⟶^{d_{2}} ⋯

in B which may not be exact. The objects H^{i} = ker(d_{2})/ Im(d_{1}) measure the deviation of this sequence from being exact. H^{i} is called the ith cohomology object of the complex.

Definition 6.1. For each object in A fix an injective resolution. The ith right derived functor of F is the functor which assigns to an object M the ith cohomology of the complex F (I^{·}) where I^{·} is the injective resolution of M.

6.2 Properties of Derived Functors

We should now prove the following:

1. If we fix different injective resolutions for all of our objects then the corresponding derived functors are, in a suitable sense, isomorphic.

2. The R^{i}F can also be defined on morphisms in such a way that they are really functors.

3. If 0 → M′ → M → M′′ is a short exact sequence then there is a long exact sequence of cohomology:

0 → F M′ → F M → F M′′ →
R^{1}F M′ → R^{1}F M → R^{1}F M′′ →
R^{2}F M′ → · · · .

4. If we have two short exact sequences then the induced maps on long exact sequences are “δ-compatible”.

5. R^{0}F ≅ F.

6. If I is injective, then for any i > 0 one has that R^{i}F (I) = 0.

Theorem 6.2. The R^{i}F and etc. are uniquely determined by properties 1-6 above.

Definition 6.3. A δ-functor is a collection of functors {R^{i}F } which satisfy 3 and 4 above. An augmented δ-functor is a δ-functor along with a natural transformation F → R^{0}F. A universal augmented δ-functor is an augmented δ-functor with some universal property which I didn’t quite catch.

Theorem 6.4. If A has enough injectives then the collection of derived functors of F is a universal augmented δ-functor.

To construct the R^{i}F choose once and for all, for each object M in A an injective resolution, then prove the above properties hold.

7 Long Exact Sequence of Cohomology and Other Wonders

“Today I sat in awe as Hartshorne effortless drew hundreds of arrows and objects everywhere, chased some elements and proved that there is a long exact sequence of cohomology in 30 seconds. Then he whipped out his colored chalk and things really got crazy. Vojta tried to erase Hartshorne’s diagrams during the next class but only partially succeeded joking that the functor was not ‘effaceable’. (The diagrams are still not quite gone 4 days later!) Needless to say, I don’t feel like texing diagrams and element chases... it’s all trivial anyways, right?”

8 Basic Properties of Cohomology

Let X be a topological space, Ab(X) the category of sheaves of abelian groups on X and

Γ(X, ·) : Ab(X) → Ab

the covariant, left exact global sections functor. Then we have constructed the derived functors H^{i}(X, ·).

8.1 Cohomology of Schemes

Let (X, O_{X}) be a scheme and F a sheaf of O_{X}-modules. To compute H^{i}(X, F) forget all extra structure and use the above definitions. We may get some extra structure anyways.

Proposition 8.1. Let X and F be as above, then the groups H^{i}(X, F) are naturally modules over the ring A = Γ(X, O_{X}).

Proof. Let A = H^{0}(X, F) = Γ(X, O_{X}) and let a ∈ A. Then because of the functoriality of H^{i}(X, ·) the map F → F induced by left multiplication by a induces a homomorphism

a : H^{i}(X, F) → H^{i}(X, F).

8.2 Objective

Our objective is to compute H^{i}(P_{k}^{n}, O(ℓ)) for all i, n, ℓ. This is enough for most applications because if one knows these groups one can, in principle at least, computer the cohomology of any projective scheme. If X is any projective variety, we embed X in some P_{k}^{n} and push forward the sheaf F on X. Then we construct a resolution of F by sheaves of the form O(−ℓ)^{n}. Using Hilbert’s syzigy theorem one sees that the resolution so constructed is finite and so we can put together our knowledge to get the cohomology of X.

Our plan of attack is as follows.

1. Define flasque sheaves which are acyclic for cohomology, i.e., the cohomology vanishes for i > 0.

2. If X = Spec A, A Noetherian, and F is quasi-coherent, show that H^{i}(X, F) = 0 for i > 0.

3. If X is any Noetherian scheme and 𝔘 = (U_{i}) is an open affine cover, find a relationship between the cohomology of X and that of each U_{i}. (The “Čech process”.)

4. Apply number 3 to P_{k}^{n} with U_{i} = {x_{i} ≠ 0}.

9 Flasque Sheaves

Definition 9.1. A flasque sheaf (also called flabby sheaf) is a sheaf F on X such that whenever V ⊂ U are open sets then ρ_{U,V} : F(U) → F(V) is surjective.

Thus in a flasque sheaf, “every section extends”. Example 9.2. Let X be a topological space, p ∈ X a point, not necessarily closed, and M an abelian group. Let j : {P } ↪ X be the inclusion, then F = j_{∗}(M) is flasque. This follows since

j_{∗}(M)(U) = \begin{cases}M & if p ∈ U \\ 0 & if p ∉ U \end{cases} .

Note that j_{∗}(M) is none other than the skyscraper sheaf at p with sections M. Example 9.3. If F is a flasque sheaf on Y and f : Y → X is a morphism then f_{∗}F is a flasque sheaf on X. Example 9.4. If F_{i} are flasque then ⨁_{i} F_{i} is flasque.

Lemma 9.5. If

0 → F′ → F → F′′ → 0

is exact and F′ is flasque then

Γ(F) → Γ(F′′) → 0

is exact.

Lemma 9.6. If

0 → F′ → F → F′′ → 0

is exact and F′ and F are both flasque then F′′ is flasque.

Proof. Suppose V ⊂ U are open subsets of X. Since F′ is flasque and the restriction of a flasque sheaf is flasque and restriction is exact, lemma 1 implies that the sequence

F(V) → F′′(V) → 0

is exact. We thus have a commuting diagram

F(U) ⟶ F′′(U)
↓  ↓
F(V) ⟶ F′′(V) ⟶ 0

which, since F(U) → F(V) is surjective, implies F′′(U) → F′′(V) is surjective.

Lemma 9.7. Injective sheaves (in the category of abelian sheaves) are flasque.

Proof. Let I be an injective sheaf of abelian groups on X and let V ⊂ U be open subsets. Let s ∈ I(V), then we must find s′ ∈ I(U) which maps to s under the map I(U) → I(V). Let Z_{V} be the constant sheaf Z on V extended by 0 outside V (thus Z_{V} (W) = 0 if W ⊄ V). Define a map Z_{V} → I by sending the section 1 ∈ Z_{V} (V) to s ∈ I(V). Then since Z_{V} ↪ Z_{U} and I is injective there is a map Z_{U} → I which sends the section 1 ∈ Z_{U} to a section s′ ∈ I(U) whose restriction to V must be s.

Remark 9.8. The same proof also shows that injective sheaves in the category of O_{X}-modules are flasque.

Corollary 9.9. If F is flasque then H^{i}(X, F) = 0 for all i > 0.

Proof. Page 208 of [Hartshorne].

Corollary 9.10. Let (X, O_{X}) be a ringed space, then the derived functors of Γ : Mod O_{X} → Ab are equal to H^{i}(X, F).

Proof. If

0 → F → I^{0} → I^{1} → ⋯

is an injective resolution of F in Mod O_{X} then, by the above remark, it is a flasque resolution in the category Ab(X) hence we get the regular cohomology.

Remark 9.11. Warning! If (X, O_{X}) is a scheme and we choose an injective resolution in the category of quasi-coherent O_{X}-modules then we are only guaranteed to get the right answer if X is Noetherian.

10 Examples

Example 10.1. Suppose C is a nonsingular projective curve over an algebraically closed field k. Let K = K(C) be the function field of C and let K_{C} denote the constant sheaf K. Then we have an exact sequence

0 → O_{C} → K_{C} → ⨁_{\begin{matrix}P ∈C \\ P closed \end{matrix}} K/O_{P} → 0,

where the map K_{C} → ⨁ K/O_{P} has only finitely many components nonzero since a function f ∈ K has only finitely many poles. Since C is irreducible K_{C} is flasque and since K/O_{P} is a skyscraper sheaf it is flasque so since direct of flasque sheaves are flasque, ⨁ K/O_{P} is flasque. One checks that the sequence is exact and so this is a flasque resolution of O_{C}. Taking global sections and applying the exact sequence of cohomology gives an exact sequence

K → ⨁_{P closed} K/O_{P} → H^{1}(X, O_{C}) → 0,

and H^{i}(X, O_{C}) = 0 for i ≥ 2. Thus the only interesting information is dim_{k} H^{1}(X, O_{C}) which is the geometric genus of C.

11 First Vanishing Theorem

“Anyone who studies algebraic geometry must read French... looking up the more general version of this proof in EGA would be a good exercise.”

Theorem 11.1. Let A be a Noetherian ring, X = Spec A and F a quasi-coherent sheaf on X, then H^{i}(X, F) = 0 for i > 0.

Remark 11.2. The theorem is true without the Noetherian hypothesis on A, but the proof uses spectral sequences. Remark 11.3. The assumption that F is quasi-coherent is essential. For example, let X be an affine algebraic curve over an infinite field k. Then X is homeomorphic as a topological space to P_{k}^{1} so the sheaf O(−2) on P_{k}^{1} induces a sheaf F of abelian groups on X such that

H^{1}(X, F) ≅ H^{1}(P_{k}^{1}, O(−2)) ≠ 0.

Remark 11.4. If I is an injective A-module then Ĩ need not be injective in Mod(O_{X}) or Ab(X). For example, let A = k = F_{p} and X = Spec A, then I = k is an injective A-module but Ĩ is the constant sheaf k. But k is a finite group hence not divisible so Ĩ is not injective. (See Proposition A3.5 in Eisenbud’s Commutative Algebra.)

Proposition 11.5. Suppose A is Noetherian and I is an injective A-module, then Ĩ is flasque on Spec A.

The proposition implies the theorem since if F is quasi-coherent then F = M̃ for some A-module M. There is an injective resolution

0 → M → I^{•}

which, upon applying the exact functor˜ , gives a flasque resolution

0 → M̃ = F → Ĩ^{•}.

Now applying Γ gives us back the original resolution

Γ : 0 → M → I^{•}

which is exact so the cohomology groups vanish for i > 0.

Proof. Let A be a Noetherian ring and I an injective A, then Ĩ is a quasi-coherent sheaf on X = Spec A. We must show that it is flasque. It is sufficient to show that for any open set U, Γ(X) → Γ(U) is surjective.

Case 1, special open affine: Suppose U = X_{f} is a special open affine. Then we have a commutative diagram

Γ(X, Ĩ) ⟶ Γ(X_{f}, Ĩ)
=↓  ^{=}↓
I ⟶^{surjective?} I_{f}

To see that the top map is surjective it is equivalent to show that I → I_{f} is surjective. This is a tricky algebraic lemma (see Hartshorne for proof).

Case 2, any open set: Let U be any open set. See Hartshorne for the rest.

12 Čech Cohomology

Let X be a topological space, 𝔘 = (U_{i})_{i∈I} an open cover and F a sheaf of abelian groups. We will define groups Ȟ^{i}(𝔘, F) called Čech cohomology groups.

Warning: Ȟ^{i}(𝔘, ·) is a functor in F, but it is not a δ-functor.

Theorem 12.1. Let X be a Noetherian scheme, 𝔘 an open cover and F a quasi-coherent sheaf, then Ȟ^{i}(𝔘, F) = H^{i}(X, F) for all i.

12.1 Construction

Totally order the index set I. Let

U_{i_{0}···i_{p}} = ∩_{j=0}^{p}U_{i_{j}}.

For any p ≥ 0 define

C^{p}(𝔘, F) = ∏_{i_{0}<i_{1}<···<i_{p}} F(U_{i_{0}···i_{p}}).

Then we get a complex

C^{0}(𝔘, F) → C^{1}(𝔘, F) → ⋯ → C^{p}(𝔘, F) → ⋯

by defining a map

d : C^{p}(𝔘, F) → C^{p+1}(𝔘, F)

by, for α ∈ C^{p}(𝔘, F),

(dα)_{i_{0}···i_{p+1}} := ∑_{0}^{p+1}(−1)^{j}α_{i_{0}···î_{j}···i_{p+1}}|_{U_{i_{0}···i_{p+1}}}.

One checks that d^{2} = 0. Lemma 12.2. Ȟ^{0}(𝔘, F) = Γ(X, F) Proof. Applying the sheaf axioms to the exact sequence

0 → Γ(X, F) → C^{0} = ∏_{i∈I} F(U_{i}) ⟶^{d}C^{1} = ∏_{i<j} F(U_{ij})

we see that Ȟ^{0}(𝔘, F) = ker d = Γ(X, F).

12.2 Sheafify

Let X be a topological space, 𝔘 an open cover and F a sheaf of abelian groups. Then we define

C^{p}(𝔘, F) = ∏_{i_{0}<···<i_{p}} j_{∗}(F|_{U_{i_{0}···ip}})

and define

d : C^{p}(𝔘, F) → C^{p+1}(𝔘, F)

in terms of the d defined above by, for V an open set,

C^{p}(𝔘, F)(V) = C^{p}(𝔘|_{V} , F|_{V}) ⟶^{d}C^{p+1}(𝔘|_{V} , F|_{V}) = C^{p+1}(𝔘, F)(V).

Remark 12.3. C^{p}(𝔘, F) = Γ(X, C^{p}(𝔘, F) Lemma 12.4. The sequence

0 → F → C^{0}(𝔘, F) → C^{1}(𝔘, F) → ⋯

is a resolution of F, i.e., it is exact.

Proof. We define the map F → C^{0} by taking the product of the natural maps F → f_{∗}(F|_{U_{i}}), exactness then follows from the sheaf axioms.

To show the rest of the sequence is exact it suffices to show exactness at the stalks. So let x ∈ X, and suppose x ∈ U_{j}. Given α_{x} ∈ C_{x}^{p} it is represented by a section α ∈ Γ(V, C^{p}(𝔘, F)), over a neighborhood V of x, which we may choose so small that V ⊂ U_{j}. Now for any p-tuple i_{0} < … < i_{p−1}, we set

(kα)_{i_{0},... ,i_{p−1}} = α_{j,i_{0},... ,i_{p−1}}.

This makes sense because

V ∩ U_{i_{0},... ,i_{p−1}} = V ∩ U_{j,i_{0},... ,i_{p−1}}.

Then take the stalk of kα at x to get the required map k.

Now we check that for any p ≥ 1 and α ∈ C_{x}^{p},

(dk + kd)(α) = α.

First note that

(dkα)_{i_{0},... ,i_{p}} = ∑_{ℓ=0}^{p}(−1)^{ℓ}(kα)_{i_{0},... ,î_{ℓ},... ,i_{p}}
= ∑(−1)^{ℓ}α_{j,i_{0},... ,î_{ℓ},... ,i_{p}}

Whereas, on the other hand,

(kdα)_{i_{0},... ,i_{p}} = (dα)_{j,i_{0},... ,i_{p}}

= (−1)^{0}α_{i_{0},... ,i_{p}} + ∑_{ℓ=1}^{p}(−1)^{ℓ+1}α_{j,i_{0},... ,î_{ℓ},... ,i_{p}}

Adding these two expressions yields α_{i_{0},... ,i_{p}} as claimed.

Thus k is a homotopy operator for the complex C_{x}^{•}, showing that the identity map is homotopic to the zero map. It follows that the cohomology groups H^{p}(C_{x}^{•}) of this complex are 0 for p ≥ 1. Lemma 12.5. If F is flasque then C^{p}(𝔘, F) is also flasque. Proof. If F is flasque then F|_{U_{i_{0},... ,ip}} is flasque so j_{∗}(F|_{U_{i_{0},... ,ip}}) is flasque so ∏ j_{∗}(F|_{U_{i_{0},... ,ip}}) is flasque. Proposition 12.6. If F is flasque then Ȟ^{p}(𝔘, F) = 0. Proof. Consider the resolution

0 → F → C^{•}(𝔘, F).

By the above lemma it is flasque, so we can use it to compute the usual cohomology groups of F. But F is flasque, so H^{p}(X, F) = 0 for p > 0. On the other hand, the answer given by this resolution is

H^{p}(Γ(X, C^{•}(𝔘, F))) = Ȟ^{p}(𝔘, F).

So we conclude that Ȟ^{p}(𝔘, F) = 0 for p > 0. Lemma 12.7. Let X be a topological space, and 𝔘 an open covering. Then for each p ≥ 0 there is a natural map, functorial in F,

Ȟ^{p}(𝔘, F) → H^{p}(X, F).

Theorem 12.8. Let X be a Noetherian separated scheme, let 𝔘 be an open affine cover of X, and let F be a quasi-coherent sheaf on X. Then for all p ≥ 0 the natural maps give isomorphisms

Ȟ^{p}(𝔘, F) ≅ H^{p}(X, F).

13 Čech Cohomology and Derived Functor Cohomology

Today we prove

Theorem 13.1. Let X be a Noetherian, separated scheme, 𝔘 an open cover and F a quasicoherent sheaf on X. Then

Ȟ^{i}(𝔘, F) = H^{i}(X, F).

To do this we introduce a condition (*):

Condition *: Let F be a sheaf of abelian groups and 𝔘 = (U_{i})_{i∈I} an open cover. Then the pair F and 𝔘 satisfy condition (*) if for all i_{0}, …, i_{p} ∈ I,

H^{(}U_{i_{0},... ,i_{p}}, F) = 0, alli > 0.

Lemma 13.2. If 0 → F′ → F → F′′ → 0 is an exact sequence in Ab(X) and F′ satisfies (*) then there is a long exact sequence for Ȟ^{i}(𝔘, ·).

Proof. Since the global sections functor is left exact and cohomology commutes with products, we have an exact sequence

0 → C^{p}(𝔘, F′) = ∏_{i_{0}<···<i_{p}} F′(U_{i_{0},... ,i_{p}}) → C^{p}(𝔘, F) = ∏_{i_{0}<···<i_{p}} F(U_{i_{0},... ,i_{p}})

→ C^{p}(𝔘, F′′) = ∏_{i_{0}<···<i_{p}} F′′(U_{i_{0},... ,i_{p}}) → ∏_{i_{0}<···<i_{p}} H^{1}(𝔘_{i_{0},... ,i_{p}}, F′) = 0

where the last term is 0 because F′ satisfies condition (*). Replacing p by · gives an exact sequence of complexes. Applying Ȟ^{i}(𝔘, ·) then gives the desired result.

Theorem 13.3. Let X be a topological space, 𝔘 an open cover and F ∈ Ab(X). Suppose F and 𝔘 satisfy (*). Then the maps

φ^{i} : Ȟ^{i}(𝔘, F) → H^{i}(X, F)

are isomorphisms.

Proof. The proof is a clever induction.

Lemma 13.4. If 0 → F′ → F → F′′ → 0 is exact and F′ and F satisfy (*) then F′′ satisfies (*).

To prove the main theorem of the section use the fact that X separated implies any finite intersection of affines is affine and then use the vanishing theorem for cohomology of a quasicoherent sheaf on an affine scheme. The above theorem then implies the main result. From now on we will always assume our schemes are separated unless otherwise stated.

Corollary 13.5. If X is a (separated) Noetherian scheme and X can be covered by n + 1 open affines for some n > 0 then H^{i}(X, F) = 0 for i > n.

Example 13.6. Let X = P_{k}^{n}, then the existence of the standard affine cover U_{0}, …, U_{n} implies that H^{i}(X, F) = 0 for i > n. Example 13.7. Let X be a projective curve embedded in P_{n}^{k}. Let U_{0} ⊂ X be open affine, then X − U_{0} is finite. Thus U_{0} ⊂ X ⊂ P^{n} and X − U_{0} = {P_{1}, …, P_{r}}. In P^{n} there is a hyperplane H such that P_{1}, …, P_{r} ∉ H. Then P_{1}, …, P_{r} ∈ P^{n} − H = A^{n} = V. Then U_{1} = V ∩ X is closed in the affine set V, hence affine. Then X = U_{0} ∪ U_{1} with U_{0} and U_{1} both affine. Thus H^{i}(X, F) = 0 for all i ≥ 2.

Exercise 13.8. If X is any projective scheme of dimension n then X can be covered by n + 1 open affines so

H^{i}(X, F) = 0 for all i > n.

[Hint: Use induction.] Hartshorne was unaware of the answer to the following question today.

Question 13.9. If X is a Noetherian scheme of dimension n do there exist n + 1 open affines covering X.

Theorem 13.10 (Grothendieck). If F ∈ Ab(X) then H^{i}(X, F) = 0 for all i > n = dim X.

Example 13.11. Let k be an algebraically closed field. Then X = A_{k}^{2} − {(0, 0)} is not affine since it has global sections k[x, y]. We compute H^{1}(X, O_{X}) by Čechcohomology. Write X = U_{1} ∪ U_{2} where U_{1} = {x ≠ 0} and U_{2} = {y ≠ 0}. Then the Čechcomplex is

C^{·}(𝔘, O_{X}) : k[x, y, x^{−1}] ⊕ k[x, y, y^{−1}] ⟶^{d}k[x, x^{−1}, y, y^{−1}].

Thus one sees with a little thought that H^{0} = ker d = k[x, y] and H^{1} = {∑_{i,j<0} a_{ij}x^{i}x^{j} : a_{ij} ∈ k} = E as k-vector spaces (all sums are finite).

13.1 History of this Module E

E = {∑_{i,j<0} a_{ij}x^{i}x^{j} : a_{ij} ∈ k}

1. Macaulay’s “Inverse System” (1921?)

2. E is an injective A-module, in fact, the indecomposable injective associated to the prime (x, y)

3. E is the dualizing module of A, thus D = Hom_{A}(·, E) is a dualizing functor for finite length modules (so doing D twice gives you back what you started with).

4. Local duality theorem: this is the module you “hom into”.

14 Cohomology of P_{k}^{n}

Today we begin to compute H^{i}(X, O_{X}(ℓ)) for all i and all ℓ.

a) H^{0}(X, O_{X}(ℓ)) is the vector space of forms of degree ℓ in S = k[x_{0}, …, x_{n}], thus

⊕_{ℓ∈Z}H^{0}(O_{X}(ℓ)) = H_{∗}^{0}(O_{X}) = Γ_{∗}(O_{X}) = S.

Proposition 14.1. There is a natural map

H^{0}(O_{X}(ℓ)) × H^{i}(O_{X}(m)) → H^{i}(O_{X}(ℓ + m)).

Proof. α ∈ H^{0}(O_{X}(ℓ)) defines a map O_{X} → O_{X}(ℓ) given by 1 ↦ α. This defines a map

O_{X} ⊗ O_{X}(m) ⟶^{α(m)} O_{X}(ℓ) ⊗ O_{X}(m)

which gives a map O_{X}(m) → O_{X}(ℓ + m). This induces the desired map H^{i}(O_{X}(m)) → H^{i}(O_{X}(ℓ + m)).

b) H^{i}(O_{X}(ℓ)) = 0 when 0 < i < n and for all ℓ. (This doesn’t hold for arbitrary quasi-coherent sheaves!)

c) H_{∗}^{n}(X, O_{X}) is a graded S-module which is 0 in degrees ≥ −n, but is nonzero in degrees ≤ n − 1. As a k-vector space it is equal to

{∑_{i_{j}<0} a_{i_{0},... ,i_{n}}x_{0}^{i_{0}} ⋯ x_{n}^{i_{n}} : sum is finite}.

d) For ℓ ≥ 0 the map

H^{0}(O_{X}(ℓ)) × H^{n}(O_{X}(−ℓ − n − 1)) → H^{n}(O_{X}(−n − 1)) ≅ k

is a perfect pairing so we have a duality (which is in fact a special case of Serre Duality).

15 Serre’s Finite Generation Theorem

We relax the hypothesis from the last lecture and claim that the same results are still true.

Theorem 15.1. Let A be a Noetherian ring and X = P_{A}^{n}. Then

1. H_{∗}^{0}(O_{X}) = ⊕_{ℓ}H_{ℓ}^{0}(O_{X}(ℓ)) = S = A[x_{0}, …, x_{n}]

2. H_{∗}^{i}(O_{X}) = 0 for all 0 < i < n

3. H_{∗}^{n}(O_{X}) = {∑_{I=i_{0},... ,i_{n}} a_{I}x_{0}^{i_{0}} ⋯ x_{n}^{i_{n}} : a_{I} ∈ A}

4. H^{0}(O_{X}(ℓ)) × H^{n}(O_{X}(−ℓ − n − 1)) → H^{n}(O_{X}(−n − 1)) is a perfect pairing of free Amodules. Notice that H^{n}(O_{X}(−n − 1)) is a free A-module of rank 1 so it is isomorphic to A, but not in a canonical way!

Although pairing is in general not functorial as a map into A, there is a special situation in which it is. Let Ω_{X/k}^{1} be the sheaf of differentials on X = P_{k}^{n}. Let ω = Ω_{X/k}^{n} = Λ^{n}Ω^{1} be the top level differentials (or “dualizing module”). Then some map is functorial (??)

“Is ω more important than Ω?” – Janos Csirik

“That’s a value judgment... you can make your own decision on that... I won’t.”

– Hartshorne

Theorem 15.2 (Serre). Let X be a projective scheme over a Noetherian ring A. Let F be any coherent sheaf on X. Then

1. H^{i}(X, F) is a finitely generated A-module for all i

2. for all F there exists n_{0} such that for all i > 0 and for all n ≥ n_{0}, H^{i}(X, F(n)) = 0.

The following was difficult to prove last semester and we were only able to prove it under somewhat restrictive hypothesis on A (namely, that A is a finitely generated k-algebra).

Corollary 15.3. Γ(X, F) is a finitely generated A-module.

Proof. Set i = 0 in 1.

Proof. (of theorem)

I. Reduce to the case X = P_{A}^{r}. Use the fact that the push forward of a closed subscheme has the same cohomology to replace F by i_{∗}(F).

II. Special case, F = O_{P^{r}}(ℓ) any ℓ ∈ Z. 1. and 2. both follow immediately from the previous theorem. This is where we have done the work in explicit calculations.

III. Cranking the Machine of Cohomology

15.1 Application: The Arithmetic Genus

Let k be an algebraically closed field and V ⊂ X = P_{k}^{n} a projective variety. The arithmetic genus of V is

p_{a} = (−1)^{dim V} (p_{V} (0) − 1)

where p_{V} is the Hilbert polynomial of V, thus p_{V} (ℓ) = dim_{k}(S/I_{V})_{ℓ} for all ℓ ≫ 0. The Hilbert polynomial depends on the projective embedding of V.

Proposition 15.4. p_{V} (ℓ) = ∑_{i=0}^{∞}(−1)^{i} dim_{k} H^{i}(O_{V} (ℓ)) for all ℓ ∈ Z.

This redefines the Hilbert polynomial. Furthermore,

p_{a} = (−1)^{dim V} (p_{V} (0) − 1) = (−1)^{dim V} ∑_{i=0}^{∞}(−1)^{i} dim_{k}(H^{i}(O_{V}))

which shows that p_{a} is intrinsic, i.e., it doesn’t depend on the embedding of V in projective space.

16 Euler Characteristic

Fix an algebraically closed field k, let X = P_{k}^{n}. Suppose F is a coherent sheaf on X. Then by Serre’s theorem H^{i}(X, F) is a finite dimensional k-vector space. Let

h^{i}(X, F) = dim_{k} H^{i}(X, F).

Definition 16.1. The Euler characteristic of F is

χ(F) = ∑_{i=0}^{n}(−1)^{i}h^{i}(X, F).

Thus χ is a function Coh(X) → Z.

Lemma 16.2. If k is a field and

0 → V_{1} → V_{2} → ⋯ → V_{N} → 0

is an exact sequence of finite dimensional vector spaces, then ∑_{i=1}^{N}(−1)^{i} dim V_{i} = 0.

Proof. Since every short exact sequence of vector spaces splits, the statement is true when N = 3. If the statement is true for an exact sequence of length N − 1 then, applying it to the exact sequence

0 → V_{2}/V_{1} → V_{3} → ⋯ → V_{N} → 0,

shows that dim V_{2}/V_{1} − dim V_{3} + ⋯ ± dim V_{n} = 0 from which the result follows.

Lemma 16.3. If 0 → F′ → F → F′′ → 0 is an exact sequence of coherent sheaves on X, then

χ(F) = χ(F′) + χ(F′′).

Proof. Apply the above lemma to the long exact sequence of cohomology taking into account that H^{n}(F′′) = 0 by Serre’s vanishing theorem.

More generally, any map χ from an abelian category to Z is called additive if, whenever

0 → F^{0} → F^{1} → ⋯ → F^{n} → 0

is exact, then

∑_{i=0}^{n}(−1)^{i}χ(F^{i}) = 0.

Question. Given an abelian category A find an abelian group A and a map X : A → A such that every additive function χ : A → G factor through A ⟶^{X} A. In the category of coherent sheaves the Grothendieck group solves this problem.

Let X = P_{k}^{n} and suppose F is a coherent sheaf on X. The Euler characteristic induces a map

Z → Z : n ↦ χ(F(n)).

Theorem 16.4. There is a polynomial p_{F} ∈ Q[z] such that p_{F}(n) = χ(F(n)) for all n ∈ Z.

The polynomial p_{F}(n) is called the Hilbert polynomial of F. Last semester we defined the Hilbert polynomial of a graded module M over the ring S = k[x_{0}, …, x_{n}]. Define φ_{M} : Z → Z by φ_{M}(n) = dim_{k} M_{n}. Then we showed that there is a unique polynomial p_{M} such that p_{M}(n) = φ_{M}(n) for all n ≫ 0.

Proof. We induct on dim(supp F). If dim(supp F) = 0 then supp F is a union of closed points so F = ⊕_{i=1}^{k}F_{p_{i}}. Since each F_{p_{i}} is a finite dimensional k-vector space and O_{X}(n) is locally free, there is a non-canonical isomorphism F(n) = F ⊗ O_{X}(n) ≅ F. Thus

χ_{F}(n) = h^{0}(F(n)) = h^{0}(F) = ∑_{i=1}^{k} dim_{k} F_{p_{i}}

which is a constant function, hence a polynomial.

Next suppose dim(supp F) = s. Let x ∈ S_{1} = H^{0}(O_{X}(1)) be such that the hyperplane H := {x = 0} doesn’t contain any irreducible component of supp F. Multiplication by x defines a map O_{X}(−1) ⟶^{x} O_{X} which is an isomorphism outside of H. Tensoring with F gives a map F(−1) → F. Let R be the kernel and Q be the cokernel, then there is an exact sequence

0 → R → F(−1) ⟶^{x} F → Q → 0.

Now supp R∪supp Q ⊂ supp F ∩H so dim(supp R) ≤ dim(supp F)∩H < dim(supp F) and dim(supp Q) ≤ dim(supp F) ∩ H < dim(supp F) so by our induction hypothesis χ(Q(n)) and χ(R(n)) are polynomials. Twisting the above exact sequence by n and applying χ yields

χ(F(n)) − χ(F(n − 1)) = χ(Q(n)) − χ(R(n)) = P_{Q}(n) − P_{R}(N).

Thus the first difference function of χ(F(n)) is a polynomial so χ(F(n)) is a polynomial.

Example 16.5. Let X = P^{1} and F = O_{X}. Then S = k[x_{0}, x_{1}], M = S and dim S_{n} = n + 1. Thus p_{M}(z) = z + 1 and p_{M}(n) = φ(n) for n ≥ −1. Computing the Hilbert polynomial in terms of the Euler characteristic gives

χ(F(n)) = h^{0}(O_{X}(n)) − h^{1}(O_{X}(n)) = \begin{cases}(n + 1) − 0 & n ≥ −1 \\ 0 − (−n − 1) = n + 1 & n ≤ −2 \end{cases}

Thus p_{F}(n) = n + 1.

The higher cohomology corrects the failure of the Hilbert polynomial in lower degrees.

17 Correspondence between Analytic and Algebraic Cohomology

Homework. Chapter III, 4.8, 4.9, 5.6.

Look at Serre’s 1956 paper Geometrie Algebraique et Geometrie Analytique (GAGA). “What are the prerequisites?” asks Janos. “French,” answers Nghi. “Is there an English translation” asks the class. “Translation? ... What for? It’s so beautiful in the French,” retorts Hartshorne.

Let F be a coherent sheaf on P_{C}^{n} with its Zariski topology. Then we can associate to F a sheaf Fan on P_{C}^{n} with its analytic topology. F is locally a cokernel of a morphism of free sheaves so we can define Fan by defining Oan_{X}. The map

Coh(P_{C}^{n}) \underrightarrow{an} Cohan(P_{C}^{n})

is an equivalence of categories and

H^{i}(X, F) ⟶^{∼} H^{i}(Xan, Fan)

for all i. If X/C is affine the corresponding object Xan_{C} is a Stein manifold.

18 Arithmetic Genus

Let X ↪ P_{k}^{n} be a projective variety with k algebraically closed and suppose F is a coherent sheaf on X. Then

χ(F) = ∑(−1)^{i}h^{i}(F)

is the Euler characteristic of F,

P_{F}(n) = χ(F(n))

gives the Hilbert polynomial of F on X, and

p_{a}(X) = (−1)^{dim X}(P_{O_{X}}(0) − 1)

is the arithmetic genus of X. The arithmetic genus is independent of the choice of embedding of X into P_{k}^{n}.

If X is a curve then

1 − p_{a}(X) = h^{0}(O_{X}) − h^{1}(O_{X})

. Thus if X is an integral projective curve then h^{0}(O_{X}) = 1 so p_{a}(X) = h^{1}(O_{X}). If X is a nonsingular projective curve then p_{a}(X) = h^{1}(O_{X}) is called the genus of X.

Let V_{1} and V_{2} be varieties, thus they are projective integral schemes over an algebraically closed field k. Then V_{1} and V_{2} are birationally equivalent if and only if K(V_{1}) ≅ K(V_{2}) over k, where K(V_{i}) is the function field of V_{i}. V is rational if V is bironational to P_{k}^{n} for some n. Since a rational map on a nonsingular projective curve always extends, two nonsingular projective curves are birational if and only if they are isomorphic. Thus for nonsingular projective curves the genus g is a birational invariant.

18.1 The Genus of Plane Curve of Degree d

Let C ⊂ P_{k}^{2} be a curve of degree d. Then C is a closed subscheme defined by a single homogeneous polynomial f(x_{0}, x_{1}, x_{2}) of degree d, thus

C = Proj(S/(f)).

Some possibilities when d = 3 are:

• f : Y^{2} − X(X^{2} − 1), a nonsingular elliptic curve

• f : Y^{2} − X^{2}(X − 1), a nodal cubic

• f : Y^{3}, a tripled x-axis

• f : Y (X^{2} + Y^{2} − 1), the union of a circle and the x-axis

Now we compute p_{a}(C). Let I = (f) with deg f = d. Then

1 − p_{a} = h_{0}(O_{C}) − h_{1}(O_{C}) + h_{2}(O_{C}) = χ(O_{C}).

We have an exact sequence

0 → I_{C} → O_{P^{2}} → O_{C} → 0.

Now I_{C} ≅ O_{P^{2}}(−d) since O_{P^{2}}(−d) can be thought of as being generated by 1/f on D_{+}(f) and by something else elsewhere, and then multiplication by f gives an inclusion so_{P^{2}}(−d)|_{D_{+}(f)} → O_{P^{2}}|_{D_{+}(f)}, etc. Therefore

χ(O_{C}) = χ(O_{P^{2}}) − χ(O_{P^{2}}(−d)).

Now

χ(O_{P^{2}}) = h^{0}(O_{P^{2}}) − h^{1}(O_{P^{2}}) + h^{2}(O_{P^{2}}) = 1 + 0 + 0

and

χ(O_{P^{2}}(−d)) = h^{0}(O_{P^{2}}(−d)) − h^{1}(O_{P^{2}}(−d)) + h^{2}(O_{P^{2}}(−d)) = 0 + 0 + \frac{1}{2}(d − 1)(d − 2).

For the last computation we used duality (14.1) to see that

h^{2}(O_{P^{2}}(−d)) = h^{0}(O_{P^{2}}(d − 3) = dim S_{d−3} = \frac{1}{2}(d − 1)(d − 2).

Thus χ(O_{C}) = 1 − \frac{1}{2}(d − 1)(d − 2) so

p_{a}(C) = \frac{1}{2}(d − 1)(d − 2).

19 Not Enough Projectives

Exercise 19.1. Prove that the category of quasi-coherent sheaves on X = P_{k}^{1} doesn’t have enough projectives.

Proof. We show that there is no projective object P ∈ Qco(X) along with a surjection P → O_{X} → 0.

Lemma 19.2. If P ⟶^{φ} O_{X} is surjective and P is quasi-coherent, then there exists ℓ such that H^{0}(P(ℓ)) → H^{0}(O_{X}(ℓ)) is surjective.

The false proof of this lemma is to write down an exact sequence 0 → R → P → O_{X} → 0 then use the “fact” that H^{1}(R(ℓ)) = 0 for sufficiently large ℓ. This doesn’t work because R might not be coherent since it is only the quotient of quasi-coherent sheaves. A valid way to proceed is to use (II, Ex. 5.15) to write P as an ascending union of its coherent subsheaves, P = ∪_{i}P_{i}. Then since φ is surjective, O_{X} = ∪_{i}φ(P_{i}), where φ(P_{i}) is the sheaf image. Using the fact that φ(P_{i}) is the sheaf image, that O_{X} is coherent and that the union is ascending, this implies O_{X} = φ(P_{i}) for some i. We now have an exact sequence

0 → R_{i} → P_{i} → O_{X} → 0

with R_{i} coherent since P_{i} and O_{X} are both coherent. Thus H^{i}(R_{i}(ℓ)) = 0 for l ≫ 0 which, upon computing the long exact sequence of cohomology, gives the lemma.

Now fix such an ℓ. We have a commutative diagram

P  ⟶ O_{X} ⟶ 0
∃↓  ↓
O_{X}(−ℓ − 1) ⟶ k(p) ⟶ 0

Twisting by ℓ gives a commutative diagram

P(ℓ) ⟶ O_{X}(ℓ) ⟶ 0
↓  ↓
O_{X}(−1) ⟶ k(p) ⟶ 0

Let s ∈ Γ(O_{X}(ℓ)) be a global section which is nonzero at p, then there is t ∈ Γ(P(ℓ)) which maps to s. But then by commutativity t must map to some element of Γ(O_{X}(−1)) = 0 which maps to a nonzero element of k(p), which is absurd.

20 Some Special Cases of Serre Duality

20.1 Example: O_{X} on Projective Space

Suppose X = P_{k}^{n}, then there is a perfect pairing

H^{0}(O_{X}(ℓ)) × H^{n}(O_{X}(−ℓ − n − 1)) → H^{n}(O_{X}(−n − 1)) ≅ k.

For this section let

ω_{X} = O_{X}(−n − 1).

Because the pairing is perfect we have a non-canonical but functorial isomorphism

H^{0}(O_{X}(ℓ)) ≅ H^{n}(O_{X}(−ℓ − n − 1))′.

(If V is a vector space then V′ denotes its dual.)

20.2 Example: Coherent sheaf on Projective Space

Suppose F is any coherent sheaf on X = P_{k}^{r}. View Hom(F, ω) as a k-vector space. By functoriality and since H^{n}(ω) = k there is a map

φ : Hom(F, ω) → Hom(H^{n}(F), H^{n}(ω)) = H^{n}(F)′.

Proposition 20.1. φ is an isomorphism for all coherent sheaves F.

Proof. Case 1. If F = O_{X}(ℓ) for some ℓ ∈ Z then this is just a restatement of the previous example.

Case 2. If E = ⊕_{i=1}^{k}O(ℓ_{i}) is a finite direct sum, then the statement follows from the commutativity of the following diagram.

Hom(⊕_{i=1}^{k}O(ℓ_{i}), ω) ⟶ H^{n}(⊕_{i=1}^{k}O(ℓ_{i}))′
↓≅  ↓≅
⊕_{i=1}^{k} Hom(O(ℓ_{i}), ω) ⟶_{∼} ⊕_{i=1}^{k}H^{n}(O(ℓ_{i}))′

Case 3. Now let F be an arbitrary coherent sheaf. View φ as a morphism of functors

Hom(·, ω) → H^{n}(·)′.

The functor Hom(·, ω) is contravarient left exact. H^{n}(·) is covariant right exact since X = P_{k}^{n} so H^{n+1}(F) = 0 for any coherent sheaf F. Thus H^{n}(·)′ is contravarient left exact.

Lemma 20.2. Let F be any coherent sheaf. Then there exists a partial resolution

E_{1} → E_{0} → F → 0

by sheaves of the form ⊕_{i}O_{X}(ℓ_{i}).

By (II, 5.17) for ℓ ≫ 0, F(ℓ) is generated by its global sections. Thus there is a surjection

O_{X}^{m} → F(ℓ) → 0

which upon twisting by −ℓ becomes

E_{0} = O_{X}(−ℓ)^{m} → F → 0.

Let R be the kernel so

0 → R → E → F

is exact. Since R is coherent, we can repeat the argument above to find E_{1} surjecting onto R. This yields the desired exact sequence.

Now we apply the functors Hom(·, ω) and H^{n}(·)′. This results in a commutative diagram

0 ⟶ Hom(F, ω) ⟶ Hom(E_{0}, ω) ⟶ Hom(E_{1}, ω)
↓  ^{φ(F)}↓  ^{φ(E_{0})}↓  ↓^{φ(E_{1})}
0 ⟶ H^{n}(F)′ ⟶ H^{n}(E_{0})′ ⟶ H^{n}(E_{1})′

From cases 1 and 2, the maps φ(E_{0}) and φ(E_{1}) are isomorphisms so φ(F) must also be an isomorphism.

20.3 Example: Serre Duality on P_{k}^{n}

Let X = P_{k}^{n} and F be a coherent sheaf. Then for each i there is an isomorphism

φ^{i} : Ext_{O_{X}}^{i}(F, ω) → H^{n−i}(F)′.

21 The Functor Ext

Let (X, O_{X}) be a scheme and F, G ∈ Mod(O_{X}). Then Hom(F, G) ∈ Ab. View Hom(F, •) as a functor Mod(O_{X}) → Ab. Note that Hom(F, •) is left exact and covariant. Since Mod(O_{X}) has enough injectives we can take derived functors.

Definition 21.1. The Ext functors Ext_{O_{X}}^{i}(F, •) are the right derived functors of Hom_{O_{X}}(F, •) in the category Mod(O_{X}).

Thus to compute Ext_{O_{X}}^{i}(F, G), take an injective resolution

0 → G → I^{0} → I^{1} → ⋯

then

Ext_{O_{X}}^{i}(F, G) = H^{i}(Hom_{O_{X}(F,I^{•})}).

Remark 21.2. Warning! If i : X ↪ P^{n} is a closed subscheme of P^{n} then Ext_{O_{X}}^{i}(F, G) need not equal Ext_{P^{n}}^{i}(i_{∗}(F), i_{∗}(G)). With cohomology these are the same, but not with Ext! Example 21.3. Suppose F = O_{X}, then Hom_{O_{X}}(O_{X}, G) = Γ(X, G). Thus Ext_{O_{X}}^{i}(O_{X}, •) are the derived functors of Γ(X, •) in Mod(O_{X}). Since we can computer cohomology using flasque sheaves this implies Ext_{O_{X}}^{i}(O_{X}, •) = H^{i}(X, •). Thus Ext generalizes H^{i} but we get a lot more besides.

21.1 Sheaf Ext

Now we define a new kind of Ext. The sheaf hom functor

Hom_{O_{X}}(F, •) : Mod(O_{X}) → Mod(O_{X})

is covariant and left exact. Since Mod(O_{X}) has enough injectives we can defined the derived functors Ext_{O_{X}}^{i}(F, •).

Example 21.4. Consider the functor Ext_{O_{X}}^{i}(O_{X}, •). Since Hom_{O_{X}}(O_{X}, G) = G this is the identity functor which is exact so

Ext_{O_{X}}^{i}(O_{X}, G) = \begin{cases}G i = 0 \\ 0 i > 0 \end{cases}

What if we have a short exact sequence in the first variables, do we get a long exact sequence?

Proposition 21.5. The functors Ext^{i} and Ext^{i} are δ-functors in the first variable. Thus if

0 → F′ → F → F′′ → 0

is exact then there is a long exact sequence

0 → Hom(F′′, G) → Hom(F, G) → Hom(F′, G) → Ext^{1}(F′′, G) → Ext^{1}(F, G) → Ext^{1}(F, G) → ⋯

The conclusion of this proposition is not obvious because we the Ext^{i} as derived functors in the second variable, not the first.

Proof. Suppose we are given 0 → F′ → F → F′′ → 0 and G. Choose an injective resolution 0 → G → I^{•} of G. Since Hom(•, I^{n}) is exact (by definition of injective object), the sequence

0 → Hom(F′′, I^{•}) → Hom(F, I^{•}) → Hom(F′, I^{•}) → 0

is exact. By general homological algebra these give rise a long exact sequence of cohomology of these complexes. For Ext^{i} simply scriptify everything!

21.2 Locally Free Sheaves

Proposition 21.6. Suppose E is a locally free O_{X}-module of finite rank. Let E^{∨} = Hom(E, O). For any sheaves F, G,

Ext^{i}(F ⊗ E, G) ≅ Ext^{i}(F, G ⊗ E^{∨})

and

Ext^{i}(F ⊗ E, G) ≅ Ext^{i}(F, G ⊗ E^{∨}) ≅ Ext^{i}(F, G) ⊗ E^{∨}.

Lemma 21.7. If E is locally free of finite rank and I ∈ Mod(O_{X}) is injective then I ⊗ E is injective.

Proof. Suppose 0 → F → G is an injection and there is a map φ : F → I ⊗ E. Tensor everything with E^{∨}. Then we have an injection 0 → F ⊗ E^{∨} → G ⊗ E^{∨} and a map φ′ : F ⊗ E^{∨} → I. Since I is injective there is a map G ⊗ E^{∨} → I which makes the appropriate diagram commute. Tensoring everything with E gives a map making the original diagram commute.

of proposition. Let 0 → G → I^{•} by an injective resolution of G. Since

Hom(F ⊗ E, I^{•}) = Hom(F, I^{•} ⊗ E^{∨}),

we see that

0 → G ⊗ E^{∨} → I^{·} ⊗ E∨

is an injective resolution of G ⊗ E^{∨}. Thus Hom(F, I^{•} ⊗ E^{∨}) computes Ext(F ⊗ E, •).

Proposition 21.8. If F has a locally free resolution E_{·} → F → 0 then

Ext_{O_{X}}^{i}(F, G) = H^{i}(Hom(E_{•}, G)).

Remark 21.9. Notice that when i > 0 and E is locally free,

Ext^{i}(E, G) = Ext^{i}(O_{X}, G ⊗ E^{∨}) = 0.

Proof. Regard both sides as functors in G. The left hand side is a δ-functor and vanishes for G injective. I claim that that right hand side is also a δ-functor and vanishes for G-injective.

Lemma 21.10. If E is locally free, then Hom(E, •) is exact.

22 More Technical Results on Ext

Let (X, O_{X}) be a scheme and F, G be sheaves in the category Mod(O_{X}). Then Ext(F, G) and Ext(F, G) are the derived functors of Hom, resp. Hom, in the second variable.

Lemma 22.1. If F and G are coherent over a Noetherian scheme X, then Ext^{i}(F, G) is coherent.

This lemma would follow immediately from the following fact which we haven’t proved yet.

Fact 22.2. Let X = Spec A with A Noetherian and let M be an A-module. Then

Ext_{O_{X}}^{i}(M̃, Ñ) = Ext_{A}^{i}(M, N)

and

Ext_{O_{X}}^{i}(M̃, Ñ) = (Ext_{A}^{i}(M, N))^{}˜ .

Instead of using the fact we can prove the lemma using a proposition from yesterday.

Proof. Choose a locally free resolution

L_{•} → F → 0

of F. Then

Ext^{i}(F, G) = H^{i}(Hom(L_{•}, G)).

But all of the kernels and cokernels in Hom(L_{•}, G) are coherent, so the cohomology is. (We can’t just choose an injective resolution of F and apply the definitions because there is no guarantee that we can find an injective resolution by coherent sheaves.)

Proposition 22.3. Let X be a Noetherian projective scheme over k and let F and G be coherent on X. Then for each i there exists an n_{0}, depending on i, such that for all n ≥ n_{0},

Ext_{O_{X}}^{i}(F, G(n)) = Γ(Ext_{O_{X}}^{i}(F, G(n))).

Proof. When i = 0 the assertion is that

Hom(F, G(n)) = Γ(Hom(F, G(n)))

which is obvious.

Claim. Both sides are δ-functors in F. We have already showed this for the left hand side. [I don’t understand why the right hand side is, but it is not trivial and it caused much consternation with the audience.]

To show the functors are isomorphic we just need to show both sides are coeffaceable. That is, for every coherent sheaf F there is a coherent sheaf E and a surjection E → F → 0 such that Ext_{O_{X}}^{i}(E) = 0 and similarly for the right hand side. Thus every coherent sheaf is a quotient of an acyclic sheaf.

Suppose F is coherent. Then for ℓ ≫ 0, F(ℓ) is generated by its global sections, so there is a surjection

O_{X}^{a} → F(ℓ) → 0.

Untwisting gives a surjection

O_{X}(−ℓ)^{a} → F → 0.

Let E = O_{X}(−ℓ)^{a}, then I claim that E is acyclic for both sides. First consider the left hand side. Then

Ext^{i}(⊕O(−ℓ), G(n)) = ⊕ Ext^{i}(O_{X}(−ℓ), G(n))
= ⊕ Ext^{i}(O_{X}, G(ℓ + n))
= H^{i}(X, G(ℓ + n))

By Serre (theorem 5.2 of the book) this is zero for n sufficiently large. For the right hand side the statement is just that

Ext^{i}(E, G(n)) = 0

which we have already done since E is a locally free sheaf.

Thus both functors are universal since they are coeffaceable. Since universal functors are completely determined by their zeroth one they must be equal.

Example 22.4. One might ask if Ext^{i} necessarily vanishes for sufficiently large i. The answer is no. Here is an algebraic example which can be converted to a geometric example. Let A = k[ε]/(ε^{2}), then a projective resolution L_{•} of k is

⋯ ⟶^{ε}A ⟶^{ε}A ⟶^{ε}A ⟶^{ε}k → 0.

Then Hom(L_{•}, k) is the complex

k ⟶^{0}k ⟶^{0}k ⟶^{0}k ⟶^{0}⋯

Thus Ext_{A}^{i}(k, k) = k for all i ≥ 0.

23 Serre Duality

We are now done with technical results on Ext’s so we can get back to Serre duality on P^{n}. Let X = P_{k}^{n} and let ω = O_{X}(−n − 1). Note that this is an ad hoc definition of ω which just happens to work since X = P_{k}^{n}. In the more general situation it will be an interesting problem just to show the so called dualizing sheaf ω actually exists. When our variety is nonsingular, ω will be the canonical sheaf. We have shown that for any coherent sheaf F there is a map

Hom(F, ω) ⟶^{∼} H^{n}(F)^{∨}.

The map is constructed by using the fact that H^{n} is a functor:

Hom(F, ω) → Hom_{k}(H^{n}(F), H^{n}(ω)) = Hom_{k}(H^{n}(F), k) = H^{n}(F)^{∨}.

We shall use satellite functors to prove the following theorem.

Theorem 23.1. Let F be a coherent sheaf on P_{k}^{n}. Then there is an isomorphism

Ext^{i}(F, ω) ⟶^{∼} H^{n−i}(F)^{∨}.

Proof. Regard both sides as functors in F.

1. Both sides are δ-functors in F. We have already checked this for Ext^{i}. Since H^{n−i} is a delta functor in F, so is (H^{n−i})^{∨}. Note that both sides are contravarient.

2. They agree for i = 0. This was proved last time.

3. Now we just need to show both sides are coeffaceable. Suppose E → F → 0 with E = O(−ℓ)^{⊕a}. For some reason we can assume ℓ ≫ 0. We just need to show both sides vanish on this E. First computing the left hand side gives

⊕ Ext^{i}(O(−ℓ), ω) = H^{i}(ω(ℓ)) = 0

for ℓ ≫ 0. Next computing the right hand side we get

H^{n−i}(O(−ℓ)) = 0

by the explicit computations of cohomology of projective space (in particular, note that i > 0).

Next time we will generalize Serre duality to an arbitrary projective scheme X of dimension n. We will proceed in two steps. The first is to ask, what is ω_{X}? Although the answer to this question is easy on P_{k}^{n} it is not obvious what the suitable analogy should be for an arbitrary projective variety. Second we will define natural maps

Ext_{O_{X}}^{i}(F, ω) ⟶^{φ^{i}} H^{n−i}(F)^{∨}

where n = dim X. Unlike in the case when X = P_{k}^{n}, these maps are not necessarily isomorphisms unless X is locally Cohen-Macaulay (the local rings at each point are Cohen-Macaulay).

Definition 23.2. Let A be a nonzero Noetherian local ring with residue field k. Then the depth of A is

depth A = inf{i : Ext_{A}^{i}(k, A) ≠ 0}.

A is said to be Cohen-Macaulay if depth A = dim A.

24 Serre Duality for Arbitrary Projective Schemes

Today we will talk about Serre duality for an arbitrary projective scheme. We have already talked about Serre duality in the special case X = P_{k}^{n}. Let F be a locally free sheaf. We showed there is an isomorphism

Ext^{i}(F, ω_{X}) ⟶^{∼} H^{n−i}(F)^{∨}.

This was established by noting that

Ext^{i}(F, ω_{X}) = Ext^{i}(O_{X}, F^{∨} ⊗ ω_{X}) = H^{i}(F^{∨} ⊗ ω_{X}).

Another thing to keep in mind is that locally free sheaves correspond to what, in other branches of mathematics, are vector bundles. They aren’t the same object, but there is a correspondence.

We would like to generalize this to an arbitrary projective scheme X. There are two things we must do.

1. Figure out what ω_{X} is.

2. Prove a suitable duality theorem.

When X = P_{k}^{n} it is easy to find a suitable ω_{X} = O_{P_{k}^{n}}(−n − 1) because of the explicit computations we did before. We now define ω_{X} to be a sheaf which will do what we hope it will do. Of course existence is another matter.

Definition 24.1. Let X be a Noetherian scheme of finite type over a field k and let n = dim X. Then a dualizing sheaf for X is a coherent sheaf ω_{X} along with a map t : H^{n}(ω_{X}) → k, such that for all coherent sheaves F on X, the map Hom_{O_{X}}(F, ω_{X}) → H^{n}(F)^{∨} is an isomorphism. The latter map is defined by the diagram

Hom_{O_{X}}(F, ω_{X})
↓
Hom(H^{n}(F), H^{n}(ω_{X})) ⟶^{t} Hom(H^{n}(F), k) = H^{n}(F)^{∨}

Strictly speaking, a dualizing sheaf is a pair (ω_{X}, t). Note that on P^{n} we had H^{n}(ω_{P^{n}}) ≅ k, but on an arbitrary scheme X we only have a map from H^{n}(ω_{X}) to k which need not be an isomorphism. The definition never mentions existence.

Proposition 24.2. If X admits a dualizing sheaf (ω_{X}, t) then the pair (ω_{X}, t) is unique up to unique isomorphism, i.e., if (η, s) is another dualizing sheaf for X then there is a unique isomorphism φ : ω_{X} → η such that

H^{n}(ω_{X}) ⟶^{H^{n}(φ)} H^{n}(η)
t ↓  ↓ s
k  =  k

commutes.

Before we prove the proposition we make a short digression to introduce representable functors which give a proof of the uniqueness part of the above proposition.

Definition 24.3. Let C be a category and D a category whose objects happen to be sets. Suppose T : C → D is a contravarient functor. Then T is representable if there exists an object ω ∈ Ob(C) and an element t ∈ T (ω) such that for all F ∈ Ob(C) the map Hom_{C}(F, ω) → T (F) is a bijection of sets. The latter map is defined by the diagram

Hom_{C}(F, ω)

bijection of sets

⟶ T (F)

↘  ↗ evaluation at t
Hom_{D}(Γ(ω), T (F))

Thus there is an isomorphism of functors Hom(•, ω) = T (•). The pair (t, ω) is said to represent the functor T. The relevant application of this definition is to the case when C = Coh(X), D = { k-vector spaces}, T is the functor F ↦ H^{n}(F)^{∨}. Then ω = ω_{X} and

t = t ∈ Hom(H^{n}(ω), k) = H^{n}(ω)^{∨} = T (ω).

Proposition 24.4. If T is a representable functor, then the pair (ω, t) representing it is unique.

Proof. Suppose (ω, t) and (η, s) both represent the functor T. Consider the diagram

Hom(η, ω) ⟶^{T} Hom(T (ω), T (η))
↘  ↙ eval. at t
T (η)

By definition the map Hom(η, ω) → T (η) is bijective. Since s ∈ T (η), there is φ ∈ Hom(η, ω) such that φ ↦ s ∈ T (η). Thus φ has the property that T (φ)(t) = s. This argument uses the fact that (ω, t) represents T. Using the fact that (η, s) represents T implies that there exists ψ ∈ Hom(ω, η) such that T (ψ)(s) = t. We have the following pictures

⟶^{ψ}

ω  η

⟵^{ϕ}

⟵^{T (ψ)=ψ^{∗}}

t ∈ T (ω)  T (η) ∋ s

⟶^{T (ϕ)=ϕ^{∗}}

I claim that

ψ ∘ φ = Id ∈ Hom(η, η).

In diagram form we have

η ⟶^{φ} ω ⟶^{ψ} η

which upon applying T gives

T (η) ⟶^{ψ^{∗}} T (ω) ⟶^{φ^{∗}} T (η)
s ↦ t ↦ s

Where does ψ ∘ φ go to under the map Hom(η, η) ⟶^{∼} T (η)? By definition ψ ∘ φ goes to the evaluation of T (ψ ∘ φ) at s ∈ T (η). But, as indicated above, the evaluation of T (ψ ∘ φ) at s is just s again. But the identity morphism 1_{η} ∈ Hom(η, η) also maps to s under the map Hom(η, η) ⟶^{∼} T (η). Since this map is a bijection this implies that ψ ∘ φ = 1_{η}, as desired. Similarly φ ∘ ψ = 1_{ω}. Thus ψ and φ are both isomorphisms.

“When you define something and it is unique up to unique isomorphism, you know it must be good.”

We return to the question of existence.

Proposition 24.5. If X is a projective scheme over a field k then (ω_{X}, t) exists.

Lemma 24.6. If X is an n dimensional projective scheme over a field k, then there is a finite morphism f : X → P_{k}^{n}.

Proof. Embed X in P^{N} then choose a linear projection down to P^{n} which is sufficiently general.

\begin{aligned}X & ↪ & P^{N} \\ f ↘ & & ↓ \\ & & P^{n} \end{aligned}

Let L be a linear space of dimension N − n − 1 not meeting X. Let the map from P^{N} → P^{n} be projection through L. By construction f is quasi-finite, i.e., for all Q ∈ P^{n}, f^{−1}(Q) is finite. It is a standard QUALIFYING EXAM problem to show that if a morphism is quasi-finite and projective then it is finite. This can be done by applying (II, Ex. 4.6) by covering X by subtracting off hyperplanes and noting that the correct things are affine by construction. See also (III, Ex. 11.2) for the more general case when f is quasi-finite and proper, but not necessarily projective.

25 Existence of the Dualizing Sheaf on a Projective Scheme

Let X be a scheme over k. Recall that a dualizing sheaf is a pair (ω, t) where ω is a coherent sheaf on X and

t : H^{n}(X, ω) → k

is a homomorphism such that for all coherent sheaves F the natural map

Hom_{X}(F, ω) → H^{n}(F)^{∨}

is an isomorphism. We know that such a dualizing sheaf exists on P_{k}^{n}.

Theorem 25.1. If X is a projective scheme of dimension n over k, then X has a dualizing sheaf.

The book’s proof takes an embedding j : X ↪ P_{k}^{N} and works on X as a subscheme of P_{k}^{N}. Then the book’s proof shows that

ω_{X} = Ext_{O_{P^{N}_{k}}}^{N−n}(O_{X}, ω_{P_{k}^{N}}).

Today we will use a different method.

Definition 25.2. A finite morphism is a morphism f : X → Y of Noetherian schemes such that for any open affine U = Spec A ⊂ X, the preimage f^{−1}(U) ⊂ Y is affine, say f^{−1}(U) = Spec B, and the natural map A → B turns B into a finitely generated A-module. We call f an affine morphism if we just require that f^{−1}(U) is affine but not that B is a finitely generated A-module. A morphism f : X → Y is quasi-finite if for all y ∈ Y the set f^{−1}(y) is finite.

Example 25.3. Consider the morphism

j : P^{1} − {pt} ↪ P^{1}.

Since P^{1} minus any nonempty finite set of points is affine j is affine. But it is not finite. Indeed, let a be a point different from pt and let U = P^{1} − {a}. Then U = Spec k[x] and

j^{−1}(U) = P^{1} − {pt, a} = Spec k[x, x^{−1}],

but k[x, x^{−1}] is not a finitely generated k[x]-module.

Exercise 25.4. A morphism can be affine but not finite or even quasi-finite. For example, let f be the natural map

f : A^{n+1} − {0} → P^{n}

then show that f is affine. This is the fiber bundle associated to the invertible sheaf O(1) [[or is it O(−1)?]]

25.1 Relative Gamma and Twiddle

We will now define relative versions of global sections and˜ analogous to the absolute versions. It is not a generalization of the absolute notion, but a relativization. Suppose X is a scheme over Y with structure map f : X → Y and assume f is affine. Then the map sending a sheaf F on X to the sheaf f_{∗}F on Y is the analog of taking global sections. Since f is a morphism there is a map O_{Y} → f_{∗}O_{X} so f_{∗}O_{X} is a sheaf of O_{Y} -modules. Note that f_{∗}F is a sheaf of f_{∗}O_{X}-modules. Thus we have set up a map

Qco(X) → { quasicoherent f_{∗}O_{X}-modules on Y }.

The next natural thing to do is define a map analogous to˜ which goes the other direction. Suppose G is a quasi coherent sheaf of f_{∗}O_{X}-modules on Y. Let U ⊂ Y be an affine open subset of Y. Let G = Γ(U, G) and write U = Spec A. Then since f is an affine morphism, f^{−1}(U) = Spec B where B = Γ(f^{−1}(U), O_{X}). Since G is an f_{∗}O_{X}-module, and f_{∗}O_{X} over U is just B thought of as an A-module, we see that G is a B-module. Thus we can form the sheaf G̃ on Spec B = f^{−1}(U). Patching the various sheaves G̃ together as U runs through an affine open cover of Y gives a sheaf G̃ in Qco(X).

Let G be a quasi-coherent sheaf of O_{Y} -modules. We can’t take ˜ of G because G might not be a sheaf of f_{∗}O_{X}-modules. But Hom_{O_{Y}} (f_{∗}O_{X}, G) is a sheaf of f_{∗}O_{X}-modules, so we can form (Hom_{O_{Y}} (f_{∗}O_{X}, G))^{}˜ . This is a quasi-coherent sheaf on X which we denote f^{!}(G).

Proposition 25.5. Suppose f : X → Y is an affine morphism of Noetherian schemes, F is coherent on X, and G is quasi-coherent on Y. Then

f_{∗} Hom_{O_{X}}(F, f^{!}G) ≅ Hom_{O_{Y}} (f_{∗}F, G)

and passing to global sections gives an isomorphism

Hom(F, f^{!}G) ≅ Hom(f_{∗}F, G).

Thus f^{!} is a right adjoint for f_{∗}.

Proof. The natural map is

f_{∗} Hom_{O_{X}}(F, f^{!}G) → Hom_{O_{Y}} (f_{∗}F, f_{∗}f^{!}G)
= Hom_{O_{Y}} (f_{∗}F, Hom_{O_{Y}} (f_{∗}O_{X}, G)) → Hom_{O_{Y}} (f_{∗}F, G)

where the map Hom_{O_{Y}} (f_{∗}O_{X}, G) → G is obtained obtained by evaluation at 1. Since the question is local we may assume Y = Spec A and X = Spec B. Then F corresponds to a finitely generated module M over the Noetherian ring B and G corresponds to a module N over A. We must show that

Hom_{B}(M, Hom_{A}(B, N)) ≅ Hom_{A}(M, N).

When M is free over B so that M = B^{⊕n} the equality holds. As functors in M, both sides are contravarient and left exact. Now suppose M is an arbitrary finitely generated B-module. Write M as a quotient F_{0}/F_{1} where F_{0} and F_{1} are both free of finite rank. Applying each of the contravarient left-exact functors to the exact sequence

F_{1} → F_{0} → M → 0

and using the fact that equality holds for finite free modules yields a diagram

0 → Hom_{B}(M, Hom_{A}(B, N)) → Hom_{B}(F_{0}, Hom_{A}(B, N)) → Hom_{B}(F_{1}, Hom_{A}(B, N))
||  ||
0 →  Hom_{A}(M, N)  →  Hom_{A}(F_{0}, N)  →  Hom_{A}(F_{1}, N)

The 5-lemma then yields an isomorphism

Hom_{B}(M, Hom_{A}(B, N)) ≅ Hom_{A}(M, N).

Lemma 25.6. Suppose f : X → Y is affine and F is a quasi-coherent sheaf on X. Then

H^{i}(X, F) ≅ H^{i}(Y, f_{∗}F).

Proof. The lemma is proved using Čech cohomology. If {U_{i}} is an open affine cover of Y then {f^{−1}(U_{i})} is an open affine cover of X. But

Γ(U_{i}, f_{∗}F) = Γ(f^{−1}(U_{i}), F)

so the Čech cohomology of f_{∗}F on Y is the same as the Čech cohomology of F on X.

Theorem 25.7 (Duality for a finite flat morphism). Suppose f : X → Y is a finite morphism with X and Y Noetherian and assume every coherent sheaf on X is the quotient of a locally free sheaf (this is true for almost every scheme arising naturally in this course). Assume that f_{∗}O_{X} is a locally free O_{Y} -module. Let F be a coherent sheaf on X and G be a quasi-coherent sheaf on Y. Then for all i ≥ 0 there is an isomorphism

Ext_{O_{X}}^{i}(F, f^{!}G) ⟶^{∼} Ext_{O_{Y}}^{i} (f_{∗}F, G).

Proof. The proposition shows that the theorem is true when i = 0. We next show that the statement is true when F = O_{X}. Applying the above lemma we see that

Ext_{O_{X}}^{i}(O_{X}, f^{!}G) = H^{i}(X, f^{!}G) = H^{i}(Y, f_{∗}f^{!}G)
= H^{i}(Y, Hom_{O_{Y}} (f_{∗}O_{X}, G))

Since f_{∗}O_{X} is a locally free O_{Y} -module,

Ext_{O_{Y}}^{i} (f_{∗}O_{X}, G) = Ext_{O_{Y}}^{i} (O_{Y} , (f_{∗}O_{X})^{∨} ⊗ G)
= H^{i}(Y, (f_{∗}O_{X})^{∨} ⊗ G))

Putting these two computations together by using the fact that

(f_{∗}O_{X})^{∨} ⊗ G = Hom_{O_{Y}} (f_{∗}O_{X}, G)

then gives the desired result.

A clever application of the 5-lemma can be used to obtain the general case. This will be done in a subsequent lecture.

26 Generalized Grothendieck Duality Theory

If X is a projective scheme over k, then

Ext^{i}(F, ω_{X}) ⟶^{∼} H^{n−i}(F)^{∨}.

This is a special type of duality. If X is an affine scheme over Y with structure morphism f : X → Y, then

Ext_{X}^{i}(F, f^{!}G) ⟶^{∼} Ext_{Y}^{i} (f_{∗}F, G).

This is another duality.

At Harvard, Hartshorne was the scribe [Lecture Notes in Math Vol ???] for Grothendieck’s seminar on his duality theory. Suppose X is proper over Y with structure morphism f : X → Y. Assume furthermore that f satisfies hypothesis (*):

(*) f_{∗}O_{X} is a locally free O_{Y} -module

Note that hypothesis start implies Hom(f_{∗}O_{X}, G) is an exact functor in G. We have defined functors f_{∗} : Coh(X) → Coh(Y) and f^{!} : Qco(Y) → Qco(X). Grothendieck’s duality compares Hom_{X}(F, f^{!}G) and its derived functors to Hom_{Y} (f_{∗}F, G) and its derived functors. The two families of functors are essentially equal. We are composing functors here. This often gives rise to spectral sequences.

It is possible to obtain the duality mentioned above as a special case of the more general Grothendieck duality. Suppose X is a projective scheme over Y = Spec k with morphism f : X → Y. Then f_{∗}(F) = Γ(X, F) which has derived functors H^{i}(X, •). A coherent sheaf on Y is a finite dimensional k-vector space. Let ω_{X} be the sheaf f^{!}(k). Substituting this into

Ext_{X}^{i}(F, f^{!}G) ⟶^{∼} Ext_{Y}^{i} (f_{∗}F, G)

with G = k gives

Ext_{X}^{i}(F, ω_{X}) ⟶^{∼} Ext_{Y}^{i} (f_{∗}F, k) = H^{n−i}(F)^{∨}

[[Do we obtain H^{n−i} instead of H^{i} since we are considering the derived functors of Hom(f^{∗}(•), •)?]]

27

Theorem 27.1 (Duality for a projective scheme). Suppose X is a projective scheme. Then X has a dualizing sheaf ω_{X} and their is an isomorphism

Ext_{X}^{i}(F, ω_{X}) ≅ H^{n−i}(X, F)^{∨}.

Proof. Using linear projections find a finite morphism f : X → P^{n}. Let ω_{X} = f^{!}ω_{P^{n}} where

f^{!}ω_{P^{n}} = Hom_{P^{n}}(f_{∗}O_{X}, ω_{P^{n}})^{}˜

and ω_{P^{n}} = O_{P^{n}}(−n − 1). Then for any F,

Hom_{X}(F, ω_{X}) = Hom_{P^{n}}(f_{∗}F, ω_{P^{n}})

⟶^{∼} H^{n}(P^{n}, f_{∗}F)^{∨} = H^{n}(X, F)^{∨}

The last equality holds since f is finite and hence affine. The second isomorphism comes from the fact that ω_{X} is a dualizing sheaf for X.

Next we obtain the duality theorem for X. By duality for a finite (*) morphism

Ext_{X}^{i}(F, ω_{X}) ≅ Ext_{P^{n}}^{i}(f_{∗}F, ω_{P^{n}}).

By Serre duality on P^{n}

Ext_{P^{n}}^{i}(f_{∗}F, ω_{P^{n}}) ⟶^{∼} H^{n−i}(P^{n}, f_{∗}F)^{∨}.

But f is affine so

H^{n−i}(P^{n}, f_{∗}F)^{∨} = H^{n−i}(X, F)^{∨}.

Thus

Ext_{X}^{i}(F, ω_{X}) ≅ H^{n−i}(X, F)^{∨}.

Under what conditions does a finite map f : X → P^{n} satisfy (*)?

Definition 27.2. A scheme X is Cohen-Macaulay if for all x ∈ X the local ring O_{X,x} is Cohen-Macaulay.

Definition 27.3. The homological dimension of a module M over a ring A is the minimum possible length of a projective resolution of M in the category of A-modules. We denote this number by hd M.

We will need the following theorem from pure algebra.

Theorem 27.4. If A is a regular local ring and M a finitely generated A-module, then

hd M + depth M = dim A.

Theorem 27.5. Suppose f : X → Y is a finite morphism of Noetherian schemes and assume Y is nonsingular. Then f satisfies (*) iff X is Cohen-Macaulay.

Proof. The question is local on Y. Suppose y ∈ Y, then A = O_{Y,y} is a regular local ring and f^{−1}(y) ⊂ X is a finite set. Let B be the semi local ring of f^{−1}(y). Thus B = \underrightarrow{lim}_{U}F(U) where the injective limit is taken over all open sets U containing f^{−1}(y). B has only finitely many maximal ideals so we call B semi local. Since f is a finite morphism B is a finite Amodule. Now B is free as an A-module iff hd_{A} B = 0, where hd_{A} B denotes the homological dimension of B as an A-module. This is clear because hd_{A} B is the shortest possible length of a free resolution of B. (Since A is local we need only consider free resolutions and not the more general projective resolutions.) Now (f_{∗}O_{X})_{y} = B so condition (*) is that B is a free A-module. Thus if f satisfies (*) then hd B = 0. By the theorem from pure algebra and the fact that f is finite we see that

depth B = dim A = dim B.

This implies B is a Cohen-Macaulay ring and therefore X is Cohen-Macaulay. Conversely, if X is Cohen-Macaulay then B is Cohen-Macaulay so depth B = dim B. The purely algebraic theorem then implies that hd B = 0 so B is a free A-module and hence f satisfies condition (*). [[We are tacitly assuming that B is Cohen-Macaulay iff it’s localizations at maximal ideals are. It would be nice to know this is true.]]

Now we finish up the proof of duality for a finite morphism.

Proof (of duality, continued). Suppose f : X → Y is a finite morphism of Noetherian schemes which satisfies (*), and assume that X has enough locally free sheaves (i.e. every coherent sheaf is a quotient of a locally free sheaf). We showed that

Hom_{X}(F, f^{!}G) ≅ Hom_{Y} (f_{∗}F, G).

This is the i = 0 case of the theorem and is true even if we drop the assumption that f satisfies (*). It can be shown by taking an injective resolution of f^{!}G and computing Ext_{X}^{i} using it that there are natural maps

φ^{i} : Ext_{X}^{i}(F, f^{!}G) → Ext_{Y}^{i} (f_{∗}F, G).

We have already shown that this is an isomorphism when sF = O_{X}. By the same argument one shows that this is an isomorphism when F is just locally free. The key point to notice is that if F is locally free then F is locally free over f_{∗}O_{X} which is locally free over O_{Y} by condition (*). Finally suppose F is an arbitrary coherent sheaf on X. Writing F as a quotient of of a locally free sheaf E gives an exact sequence

0 → R → E → F → 0

with R coherent. The long exact sequence of Ext’s gives a diagram

Hom_{X}(E, f^{!}G) → Hom_{X}(R, f^{!}G) → Ext_{X}^{1}(F, f^{!}G) → Ext^{1}(E, f^{!}G) → Ext_{X}^{1}(R, f^{!}G)
↓≅ ↓≅ ↓ ?? ↓≅ ↓??
Hom_{X}(f_{∗}E, G) → Hom_{X}(f_{∗}R, G) → Ext_{X}^{1}(f_{∗}F, G) → Ext^{1}(f_{∗}E, G) → Ext_{X}^{1}(f_{∗}R, G)

Now apply the subtle 5-lemma to show that the map

φ^{1} : Ext_{X}^{1}(F, f^{!}G) → Ext_{X}^{1}(f_{∗}F, G)

is injective. Another diagram chase shows that he map

Ext_{X}^{1}(R, f^{!}G) → Ext_{X}^{1}(f_{∗}R, G)

is injective. Then [[I guess??]] the 5-lemma shows that φ^{i} is an isomorphism. Climbing the sequence inductively shows that φ^{i} is an isomorphism for all i.

Exercise 27.6. Give an example of a scheme X which does not have enough locally free sheaves. [Hints: By (III, 6.8) X has enough locally free sheaves if X is quasi-projective or nonsingular. Hartshorne intimated that this problem is hard, but suggested one might search for a counterexample by looking for an appropriate non-projective 3-fold with 2 singular points.]

28 Review of Differentials

Definition 28.1. Let B be a ring and M a B-module. Then a map d : B → M is a derivation if

d(b_{1} + b_{2}) = db_{1} + db_{2}
d(b_{1}b_{2}) = b_{1}db_{2} + b_{2}db_{1}

If B is an A-algebra, then d is a derivation over A if in addition da = 0 for all a ∈ A.

Note that a derivation over A is linear since d(ab) = adb + bda = adb + 0 = adb.

Example 28.2. Let k be a field and let B = k[x]. Let d : B → B be the differentian map f(x) ↦ f′(x). Then d is a derivation.

Definition 28.3. Let B be an A-algebra. Then the module of differentials of B over A is a pair (Ω_{B/A}, d : B → Ω_{B/A}), where Ω_{B/A} is a B-module and d is a derivation of B into Ω_{B/A} over A, which satisfies the following universal property: if d′ : B → M is a derivation over A then there exists a unique B-linear map φ : Ω_{B/A} → M such that d′ = φ ∘ d.

If (Ω_{B/A}, d) exists it is unique as a pair up to unique isomorphism.

We first construct Ω_{B/A} by brute force. Let Ω_{B/A} be the free module on symbols db (all b ∈ B) modulo the submodule generated by the relations d(b_{1} + b_{2}) − db_{1} − db_{2}, d(b_{1}b_{2}) − b_{2}db_{1} − b_{1}db_{2}, and da for all b_{1}, b_{2} ∈ B in a ∈ A. This is obviously a module of differentials.

Example 28.4. Suppose A = B, then Ω_{B/A} = 0.

Example 28.5. Suppose k is a field of characteristic p > 0. Let B = k[x] and let A = k[x^{p}]. Then Ω_{B/A} is the free B-module of rank 1 generated by dx.

Example 28.6. Let B = Q[\sqrt{2}] and A = Q. Then 0 = d(2) = d(\sqrt{2}\sqrt{2}) = 2\sqrt{2}d(\sqrt{2}) so d(\sqrt{2}) = 0. Thus Ω_{Q[\sqrt{2}]/Q} = 0.

Corollary 28.7. The module of differentials Ω_{B/A} is generated by {db : b ∈ B, b ∉ A}.

Now we will construct Ω_{B/A} in a more eloquent manner. Suppose B is an A-algebra. Consider the exact sequence of A-modules

0 → I → B ⊗_{A} B ⟶^{Δ} B → 0

where Δ is the diagonal map b_{1} ⊗ b_{2} ↦ b_{1}b_{2} and I is the kernel of Δ. Make I/I^{2} into a B-module by letting B act on the first factor (thus b(x ⊗ y) = (bx) ⊗ y) and define a map d : B → I/I^{2} by db = 1 ⊗ b − b ⊗ 1.

Proposition 28.8. The module I/I^{2} along with the map d is the module of differentials for B over A.

Proof. Suppose b_{1}, b_{2} ∈ B, then

d(b_{1}b_{2}) = 1 ⊗ b_{1}b_{2} − b_{1}b_{2} ⊗ 1.

One the other hand,

b_{1}db_{2} + b_{2}db_{1} = b_{1}(1 ⊗ b_{2} − b_{2} ⊗ 1) + b_{2}(1 ⊗ b_{1} − b_{1} ⊗ 1)
= b_{1} ⊗ b_{2} − b_{1}b_{2} ⊗ 1 + b_{2} ⊗ b_{1} − b_{1}b_{2} ⊗ 1

Now taking the difference gives

1 ⊗ b_{1}b_{2} − b_{1}b_{2} ⊗ 1 − b_{1} ⊗ b_{2} + b_{1}b_{2} ⊗ 1 − b_{2} ⊗ b_{1} + b_{1}b_{2} ⊗ 1
= (1 ⊗ b_{1} − b_{1} ⊗ 1)(1 ⊗ b_{2} − b_{2} ⊗ 1) ∈ I^{2}

For the universal property see Matsumura.

Corollary 28.9. If B is a finitely generated A-algebra and A is Noetherian then Ω_{B/A} is a finitely generated B-module.

Proof. If B is a finitely generated A-algebra then B is a Noetherian ring. Since I is a kernel of a ring homomorphism I is an ideal so I is finitely generated. Thus I/I^{2} is finitely generated as a B-module.

Example 28.10. Let A be any ring and let B = A[x_{1}, …, x_{n}]. Then Ω_{B/A} is the free Bmodule generated by dx_{1}, …, dx_{n}. The derivation d : B → Ω_{B/A} is the map f ↦ ∑ \frac{∂f}{∂x_{i}} dx_{i}. Since B is generated as an A-algebra by the x_{i}, Ω_{B/A} is generated as a B-module by the dx_{i} and there is an epimorphism B^{r} → Ω_{B/A} taking the ith basis vector to dx_{i}.

On the other hand, the partial derivative ∂/∂x_{i} is an A-linear derivation from B to B, and thus induces a B-module map ∂_{i} : Ω_{B/A} → B carrying dx_{i} to 1 and all the other x_{j} to 0. Putting these maps together we get the inverse map. This proof is lifted from Eisenbud’s Commutative Algebra.

There are a few nice exact sequences.

Proposition 28.11. Suppose A, B, and C are three rings and

A → B ⟶^{g}C

is a sequence of maps between them (it needn’t be exact – in fact it wouldn’t make sense to stipulate that it is exact because kernels don’t exist in the category of commutative rings with 1). Then there is an exact sequence of C-modules

Ω_{B/A} ⊗_{B} C → Ω_{C/A} → Ω_{C/B} → 0.

If d : B → Ω_{B/A} is the derivation associated with Ω_{B/A} then the first map is db ⊗_{B} c ↦ cd(g(b)).

We won’t prove this here, but note that exactness in the middle is the most interesting. The functors T^{i} comes next on the left. See the work of Schlesinger and Lichenbaum, or Illusi and André.

Proposition 28.12. Suppose A, B, and C are rings and

A → B → C

is a sequence of maps. Assume furthermore I ⊂ B is an ideal, that C = B/I, and the map from B → C is the natural surjection. Then there is an exact sequence of C-modules

I/I^{2} ⟶^{d}Ω_{B/A} ⊗_{B} C → Ω_{C/A} → Ω_{C/B} = 0

Next we consider what happens for a local ring.

Proposition 28.13. Suppose B, m is a local ring with residue field k = B/m and there is an injection k ↪ B. Then there is an exact sequence

m/m^{2} ⟶^{d}Ω_{B/k} ⊗_{B} k → 0

and d is actually an isomorphism.

The exact sequence is obtained from the previous proposition by letting A = C = k. The fact that d is an isomorphism is supposed to be tricky. The proposition implies that a lower bound on the number of generators of Ω_{B/k} is dim_{k} m/m^{2}. If B is regular and local, then dim B = dim_{k} m/m^{2} which implies Ω_{B/k} can be generated by dim_{k} B elements.

Proposition 28.14. Let B be a localization of an algebra of finite type over a perfect field, let m be the maximal ideal and let k = B/m. Then B is a regular local ring iff Ω_{B/k} is a free B-module of rank equal to the dimension of B over k.

Proof. (←) If Ω_{B/k} is free of rank n = dim_{k} B then the minimum number of generators of Ω_{B/k} is n, so dim Ω_{B/k} ⊗ k = n = dim_{k} m/m^{2}. Thus dim B = dim m/m^{2} whence B is regular. (→)

28.1 The Sheaf of Differentials on a Scheme

Suppose f : X → Y is a morphism of schemes. Let V = Spec B be an open affine subset of X and U = Spec A an open affine subset of Y such that f(V) ⊂ U. Then B is an algebra over A so we may consider the module Ω_{B/A}. Put Ω̃_{B/A} on V and glue to get a sheaf Ω_{X/Y}. We can glue because localization commutes with forming Ω and the universal property of Ω makes gluing isomorphisms canonical and unique.

If X/k is a nonsingular variety of dimension n then Ω_{X/k} is locally free of rank n. If X is a curve, then Ω_{X/k} is locally free of rank 1 so it is a line bundle.

The sheaf Ω_{X/k} is important because it is intrinsically defined and canonically associated

to X ⟶^{f} Y.

29 Differentials on P^{n}

Remark 29.1. Suppose X is a scheme over Y then one could also define Ω_{X/Y} as follows. Let Δ : X → X ×_{Y} X be the diagonal morphism and let I_{Δ} be the ideal sheaf of the image of Δ. Define Ω_{X/Y} = Δ^{∗}(I_{Δ}/I_{Δ}^{2}).

Proposition 29.2. Let X, Y, and Z be schemes along with maps X ⟶^{f} Y ⟶^{g}Z. Then there is an exact sequence of O_{X}-modules

f^{∗}(Ω_{Y/Z}) → Ω_{X/Z} → Ω_{X/Y} → 0.

Proposition 29.3. Let Y be a closed subscheme of a scheme X over S. Let I_{Y} be the ideal sheaf of Y on X. Then there is an exact sequence of sheaves of O_{X}-modules

I_{Y} /I_{Y}^{2} → Ω_{X/S} ⊗ O_{Y} → Ω_{Y/S} → 0.

Example 29.4. Let S be a scheme and let X = A_{S}^{n} be affine n-space over S. Then Ω_{X/S} is the free O_{X}-module generated by dx_{1}, …, dx_{n}.

Projective space is more interesting.

Theorem 29.5. Let X = P_{k}^{n}, then there is an exact sequence

0 → Ω_{X/k} → O_{X}(−1)^{n+1} → O_{X} → 0.

The proof in the book is too computational and nobody understands it therefore we present an “explanation” even though it doesn’t quite have the force of proof.

Explanation. Since U_{i} = {x_{i} ≠ 0} is affine, Ω_{X}|_{U_{i}} = Ω_{U_{i}} is free of rank n thus Ω_{X} is locally free of rank n. Let W = A^{n+1} − {0} and let f : W → X be the natural quotient map. The sequence W ⟶^{f} X → k gives rise to an exact sequence

f^{∗}Ω_{X} → Ω_{W} → Ω_{W/X} → 0.

Since the open subset W ⊂ A^{n+1} is a nonsingular variety, Ω_{W} is free of rank n+1, generated by dx_{0}, …, dx_{n}.

The affine subset U_{0} = {x_{0} ≠ 0} ⊂ X can be represented as

U_{0} = Spec k[\frac{x_{1}}{x_{0}} , …, \frac{x_{n}}{x_{0}} ] = Spec k[y_{1}, …, y_{n}].

The inverse image is

f^{−1}(U_{0}) = A^{n+1} − \{x_{0} = 0\} = Spec(k[x_{0}, …, x_{n}, \frac{1}{x_{0}} ])

= Spec(k[y_{1}, …, y_{n}][x_{0}, \frac{1}{x_{0}} ])

Thus f^{−1}(U_{0}) ≅ U_{0} × (A^{1} − {0}) which is affine. Therefore Ω_{W/X}|_{f^{−1}(U_{0})} is locally free of rank 1 generated by dx_{0}. Consider again the exact sequence

f^{∗}Ω_{X} → Ω_{W} → Ω_{W/X} → 0.

Since Ω_{W} is free of rank n + 1, Ω_{W/X} is locally free of rank 1, and f^{∗}Ω_{X} is locally free of rank n (pullback preserves rank locally), we conclude that the map f^{∗}Ω_{X} → Ω_{W} must be injective. We thus obtain an exact sequence of sheaves on W

0 → f^{∗}Ω_{X} → Ω_{W} → Ω_{W/X} → 0.

Passing to global sections and using the fact that W is affine we obtain an exact sequence of modules over S = k[x_{0}, …, x_{n}]

0 → Γ(f^{∗}Ω_{X}) → S^{n+1} ⟶^{ψ} S.

The last term is S because any invertible sheaf on W = A^{n} − {0} is isomorphic to O_{W}. This follows from the exact sequence (II, 6.5)

0 → Pic A^{n} ⟶^{∼} Pic W → 0

and the fact that Pic A^{n} = 0. Take generators e_{0}, …, e_{n} of S^{n+1}. Then ψ is the map e_{i} ↦ x_{i}, i.e. the multiplication by x map.

To finish the proof we need to know that if F is a coherent sheaf on P^{n}, then Γ(W, f^{∗}F)^{}˜ = F. This assertion is completely natural but doesn’t carry the force of proof, i.e., Hartshorne gave no proof. Taking global sections and applying this to the above sequence yields the exact sequence of sheaves on X

0 → Ω_{X/k} → Ω_{X}(−1)^{n+1} → O_{X} → 0

which is just what we want.

Proposition 29.6. If X is a nonsingular variety over a field k, then Ω_{X} is locally free of rank n = dim X.

The proof can be found in the book.

Example 29.7. Let C be a nonsingular curve in P_{k}^{2} defined by an equation f of degree d. Then Ω_{C/k} is a locally free sheaf of rank 1 so it corresponds to a divisor. Which divisor class will Ω_{C/k} correspond to? Let I be the ideal sheaf of C ⊂ P^{2}. Then we have an exact sequence

0 → I/I^{2} → Ω_{P^{2}} ⊗ O_{C} → Ω_{C/k} → 0.

The left map is injective since I/I^{2} ≅ O_{C}(−d) so I/I^{2} is locally free of rank 1. [[Why is I/I^{2} ≅ O_{C}(−d)?]] Taking the second exterior power gives an isomorphism

Λ^{2}(Ω_{P^{2}} ⊗ O_{C}) ≅ (I/I^{2}) ⊗ Ω_{C/k}.

This is a fact from the general theory of locally free sheaves. We have an exact sequence

0 → Ω_{P^{2}} → O(−1)^{3} → O → 0

thus

Λ^{3}(O(−1)^{3}) ≅ Λ^{2}Ω_{P^{2}} ⊗ Λ^{1}O_{P^{2}}

so O(−3) ≅ Λ^{2}Ω_{P^{2}}. Thus

O(−3) ≅ I/I^{2} ⊗ Ω_{C/k} ≅ O_{C}(−d) ⊗ Ω_{C/k}.

Tensoring with O(3) gives O_{C} ≅ O_{C}(3 − d) ⊗ Ω_{C/k} so Ω_{C/k} ≅ O_{C}(d − 3).

If C is a cubic then

Ω_{C/k} = O_{C}(3 − 3) = O_{C}(0) = O_{C}.

Furthermore

Ω_{P^{1}} = O(−2) ≅ O̸(0) = Ω_{C/k}

so a nonsingular plane cubic is not rational.

Proposition 29.8. Suppose 0 → E′ → E → E′′ → 0 is an exact sequence of locally free sheaves of ranks r′, r, and r′′. Then

Λ^{r}E ≅ Λ^{r′}E′ ⊗ Λ^{r′′}E′′.

30 Sheaf of Differentials and Canonical Divisor

Theorem 30.1. Let X be a nonsingular projective variety of dimension n. Then Ω_{X/k}^{n} ≅ ω_{X} where ω_{X} is the dualizing sheaf on X. Furthermore, Ω_{X/k} is locally free of rank n and so Ω_{X/k}^{n} is locally free of rank 1. Thus Ω_{X/k}^{n} is an invertible sheaf on X.

Note that Ω_{X/k}^{n} is an (abusive) shorthand notation for Λ^{n}Ω_{X/k} and that Ω_{X/k}^{n} is not the direct sum of n copies of Ω_{X/k}.

Recall the construction of the dualizing sheaf ω_{X}. Let f : X → P^{n} be a finite morphism. Let ω_{P^{n}} = O_{P^{n}}(−1). Then

ω_{X} = f^{!}O_{P^{n}}(−1) = f^{!}ω_{P^{n}} = Hom(f_{∗}O_{X}, ω_{P^{n}})^{}˜ .

Since X is Cohen-Macaulay, f_{∗}O_{X} is locally free so f^{!}ω_{P^{n}} is locally free of rank 1.

First we show that the theorem holds when X = P^{n}. From last time we have an exact sequence

0 → Ω_{P^{n}} → O(−1)^{n+1} → O → 0

so taking highest exterior powers gives an isomorphism

Λ^{n+1}(O(−1)^{n+1}) ≅ Λ^{n}Ω_{P^{n}} ⊗ Λ^{1}O ≅ Λ^{n}Ω_{P^{n}}.

For the last isomorphism we used the fact that Λ^{1}O ⟶^{∼} O. But since the highest exterior power changes a direct sum into a tensor product

Λ^{n+1}(O(−1)^{n+1}) = O(−1)^{⊗n+1} = O(−n − 1) = ω_{P^{n}}.

so Combining these shows that Λ^{n}Ω_{P^{n}} = ω_{P^{n}}.

Now suppose X is an arbitrary nonsingular projective variety of dimension n. As we have done before let f : X → P^{n} be a finite morphism (do this using suitable linear projections). To prove the theorem it is enough to show that f^{!}(Ω_{P^{n}/k}^{n}) ≅ Ω_{X/k}^{n} since f^{!}(Ω_{P^{n}/k}^{n}) = f^{!}(ω_{P^{n}}) = ω_{X}. This is just a statement about differentials.

More generally suppose f : X → Y is a finite map and that both X and Y are nonsingular projective varieties of dimension n. Then we have the duality

Hom_{X}(F, f^{!}G) = Hom_{Y} (f_{∗}F, G).

Thus to give a map φ : Ω_{X}^{n} → f^{!}Ω_{Y}^{n} is equivalent to giving a map \overline{φ} : f_{∗}Ω_{X}^{n} → Ω_{Y}^{n}.

It is an open problem to find a natural yet elementary way to define a map f_{∗}Ω_{X}^{n} → Ω_{Y}^{n} which corresponds to an isomorphism Ω_{X}^{n} → f^{!}Ω_{Y}^{n}. Because the map sends differentials “above” to differentials “below” it should be called a “trace” map. In Hartshorne’s book the existence of such a map is proved by embedding X in some large P^{N}, showing that Ω_{X} = Ext^{N−n}(O_{X}, ω_{P^{n}}) and then applying the “fundamental local isomorphism”. The approach is certainly not elementary because it involves the higher Ext groups.

Continue to assume f : X → Y is a finite morphism of nonsingular varieties of dimension n. Assume f is separable so that the field extension K(X)/K(Y) is finite and separable. A theorem proved last time gives an exact sequence

f^{∗}Ω_{Y} → Ω_{X} → Ω_{X/Y} → 0.

Since X is nonsingular of dimension n, Ω_{X} is locally free of rank n. Similarly Ω_{Y} is locally free of rank n so f^{∗}Ω_{Y} is locally free of rank n. (Locally this is just the fact that A^{n} ⊗_{A} B ≅ B^{n}.) Since localization commutes with taking the module of differentials the stalk at the generic point of Ω_{X/Y} is Ω_{K(X)/K(Y)}. This is 0 since K(X)/K(Y) is finite separable, thus Ω_{X/Y} is a torsion sheaf. By general facts about free modules this implies the map f^{∗}Ω_{Y} → Ω_{X} is injective, i.e., the sequence

0 → f^{∗}Ω_{Y} → Ω_{X} → Ω_{X/Y} → 0

is exact.

We pause to consider the simplest illustrative example, namely the case when X is a parabola and Y is the line.

Example 30.2. Let A = k[x] and let B = k[x, y]/(x − y^{2}) ≅ k[y] where k is a field of characteristic not equal to 2. Let X = Spec B and let Y = Spec A. Let f : X → Y be the morphism induced by the inclusion A ↪ B (thus x ↦ y^{2}). Since B ≅ k[y] it follows that Ω_{X} = Bdy, the free B-module generated by dy. Similarly Ω_{Y} = Adx. The exact sequence above becomes

0 → Ω_{Y} ⊗_{A} B → Ω_{X} →  Ω_{X/Y}  → 0
||  ||  ≀|
Bdx → Bdy → Bdy/B(2ydy)

The point is that Ω_{X/Y} = (k[y]/(2y))dy is a torsion sheaf supported on the ramification locus of the map f : X → Y. (The only ramification point is above 0.) Note that Ω_{X/Y} is the quotient of Ω_{X} by the submodule generated by the image of dx in Ω_{X} = Bdy. The image of dx is 2ydy.

More generally we define the ramification divisor as follows.

Definition 30.3. Let f : X → Y be a finite separable morphism of nonsingular varieties of dimension n. Then the ramification divisor of X/Y is

R = ∑_{ζ∈Z⊂X} length_{O_{ζ}}((Ω_{X/Y})_{ζ}) · Z

where the sum is taken over all closed irreducible subsets Z ⊂ X of codimension 1 and ζ is the generic point of Z.

Since the sequence

0 → f^{∗}Ω_{Y} → Ω_{X} → Ω_{X/Y} → 0

is exact it will follow that

Ω_{X}^{n} ≅ f^{∗}Ω_{Y}^{n} ⊗ L(R)

where R is the ramification divisor of X/Y and L(R) denotes the corresponding invertible sheaf. [[This is some linear algebra over modules.]]

The next part of the argument is to study f^{!}. As usual let f : X → Y be a finite morphism of nonsingular varieties of dimension n and assume furthermore that f_{∗}O_{X} is a locally free O_{Y} -module. Define a trace map Tr : f_{∗}O_{X} → O_{Y} locally as follows. Let Spec A be an open affine subset of Y and let Spec B = f^{−1}(Spec A). Then B is a free A-module of rank d = deg f. Choose a basis e_{1}, …, e_{d} for B/A. Let b ∈ B, and suppose be_{i} = ∑_{j} a_{ij}e_{j}. Define Tr(b) = ∑_{i} a_{ii}. Let \overline{Tr} ∈ Hom_{X}(O_{X}, f^{!}O_{Y}) correspond to Tr ∈ Hom_{Y} (f_{∗}O_{X}, O_{Y}) under the isomorphism between these Hom groups.

Claim. f^{!}O_{Y} ≅ L(R).

Once we have proved the claim, the theorem will follow. To see this tensor both sides by f^{∗}Ω_{Y}^{n}. Then

f^{!}Ω_{Y}^{n} = f^{!}O_{Y} ⊗ f^{∗}Ω_{Y}^{n} = L(R) ⊗ f^{∗}Ω_{Y}^{n} = Ω_{X/Y}^{n} .

We look what happens locally. Let Spec A ⊂ Y and Spec B = f^{−1}(Spec A) ⊂ X. We want to show that f^{!}(O_{Y}) = L(R). Since f^{!}O_{Y} = Hom_{O_{Y}} (f_{∗}O_{X}, O_{Y})^{}˜ we look at B^{∗} = Hom_{A}(B, A). First consider the special case of the parabola investigated above. Then B has a basis 1, y over A and B^{∗} is spanned by e_{0} and e_{1} where e_{0}(1) = 1, e_{0}(y) = 0, and e_{1}(1) = 0, e_{1}(y) = 1. Thus ye_{0}(1) = e_{0}(y) = 0, ye_{0}(y) = e_{0}(y^{2}) = e_{0}(x) = xe_{0}(1) = x, and ye_{1}(1) = e_{1}(y) = 1, ye_{1}(y) = e_{1}(y^{2}) = e_{1}(x) = xe_{1}(1) = 0. Therefore ye_{1} = e_{0} so e_{1} generates B^{∗} over A. The trace T r : B → A is an element of B^{∗}. We determine it. We see that T r(1) = 2 and T r(y) = 0 since

1 ↔ (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) and y ↔ (\begin{matrix}0 & x \\ 1 & 0 \end{matrix})

Thus Tr = 2e_{0} = 2ye_{1} since ye_{1} = e_{0} as shown above. We have shown that the map

\overline{Tr} : B → B^{∗} : 1 ↦ Tr

has image generated by 2ye_{1}. More generally suppose B = A[y]/(f(y)) where f(y) = y^{d}+a_{1}y^{d−1}+⋯+a_{d} is a monic polynomial of degree d. Then B has A-basis 1, y, y^{2}, …, y^{d−1} and B^{∗} is spanned by e_{0}, e_{1}, …, e_{d−1}. By a similar argument to the one above, we can show that Tr = f′(y)e_{d−1}. Thus locally f^{!}O_{Y} is a free O_{X}-module of rank 1 generated by f′(y)e_{d−1}. But this is really what we defined L(R) to be (where this derivative is zero is where there is ramification). Thus f^{!}O_{Y} = L(R) and we are done.

Curves

31 Definitions

Definition 31.1. A curve is a connected nonsingular projective 1-dimensional scheme over an algebraically closed field k.

We have proved that if C_{1} and C_{2} are curves then C_{1} ≅ C_{2} iff C_{1} is birational to C_{2} (i.e., they have isomorphic open subsets) iff K(C_{1}) ≅ K(C_{2}). Here K(C_{1}) = O_{C,ζ} where ζ is the generic point of C_{1}. If any of the hypothesis nonsingular, connected, projective, or one-dimensional is removed then these equivalences can fail to hold.

32 Genus

Let X ↪ P^{n} be an embedded projective variety. Let P_{X}(n) be its Hilbert polynomial. The arithmetic genus of X is the quantity p_{a} defined by the equation P_{X}(0) = 1 + (−1)^{dim X}p_{a}. Thus if C is a curve then p_{a} = 1 − P_{X}(0). For a nonsingular projective curve we also define the quantity g = h^{1}(O_{C}) and call it the genus. The geometric genus p_{g} = h^{0}(ω_{X}) is a third notion of genus for a nonsingular projective variety X. For curves all types of genus coincide.

Theorem 32.1. If C is a curve then

p_{a} = g = p_{g}.

Proof. P_{C}(n) = ∑(−1)^{i}h^{i}(O_{X}(n)) so P_{C}(0) = ∑(−1)^{i}h^{i}(O_{X}). But C has dimension 1 so by Grothendieck vanishing

1 − p_{a} = P_{C}(0) = h^{0}(O_{C}) − h^{1}(O_{C}) = 1 − h^{1}(O_{C}) = 1 − g

so p_{a} = g.

Serre duality says that if X is nonsingular of dimension n then Ext_{X}^{i}(F, ω_{X}) is linearly dual to H^{n−i}(X, F). Thus when X = C, F = O_{C}, i = 0, and n = 1 we see that

H^{0}(ω_{C}) = Ext_{C}^{0}(O_{C}, ω_{C})

is dual to H^{1}(O_{C}). Thus p_{g} = h^{1}(O_{C}) = h^{0}(ω_{C}) = g.

Thus we may speak of the genus of a curve. For more general varieties the concepts diverge.

The classification problem is to describe all curves up to isomorphism. The set of curves is a disjoint union C = ∐_{g≥0} C_{g} where C_{g} consists of all curves of genus g.

Theorem 32.2. Any curve of genus 0 is isomorphic to P^{1}.

Proof. Let C be a curve of genus 0 and let P ∈ C be a closed point. There is an exact sequence

0 → O_{C} → L(P) → κ(P) → 0.

AS a divisor P corresponds to the invertible sheaf L(P). Let s ∈ Γ(L(P)) be a global section which generates L(P) as an O_{C}-module. Thus s has a pole of order 1 at P and no other poles. Then s defines a morphism

O_{C} → L(P) : 1 ↦ s

which has cokernel κ(P). Taking cohomology yields

0 → H^{0}(O_{C}) → H^{0}(L(P)) → H^{0}(κ(P)) → H^{1}(O_{C}) = 0.

Since H^{0}(O_{C}) = k and H^{0}(κ(P)) = k it follows that H^{0}(L(P)) = k ⊕k. View O_{C} ⊂ L(P) ⊂ K where K = K(X) is the constant function field sheaf. Then

H^{0}(O_{C}) ↪ H^{0}(L(P)) ↪ K.

Since dim H^{0}(L(P)) > dim H^{0}(O_{C}), there is f ∈ H^{0}(L(P))−H^{0}(O_{C}) and f is non constant. Thus f ∈ K = K(X) and f ∉ k. So f defines a morphism C → P^{1} as follows. If Q ∈ C and f ∈ O_{Q} send Q to \overline{f} ∈ O_{Q}/m_{Q} = k ⊂ P_{k}^{1}. If Q ∉ C so that f has a pole at Q, send Q to ∞ ∈ P_{k}^{1}. Since f lies in H^{0}(L(P)) which has dimension 1 over k = H^{0}(O_{C}), v_{P} (f) = −1. Thus ∞ does not ramify so the degree of the morphism defined by f is the number of points lying over ∞ which is 1. Thus P_{k}^{1} ≅ C.

Remark 32.3. This result is false if k is not algebraically closed. But it is true if C has a rational point, i.e., a point P ∈ C so that κ(P) = O_{P} /m_{P} = k.

Example 32.4. Let k = R and let C be the curve x^{2} + y^{2} + z^{2} = 0 on P_{R}^{2}. C is nonsingular of degree 2 so g = 0. But C is not isomorphic over R to P_{R}^{1} since C has no rational points whereas P_{R}^{1} does. Let P = (1, i, 0) ∈ C_{C} and \overline{P} = (1, −i, 0) ∈ C_{C}. Let P^{∗} ∈ C_{R} be the Galois conjugacy class \{P, \overline{P} \}. Then κ(P^{∗}) = C is of degree 2 over R.

Exercise 32.5. Over R an curve of genus 0 not isomorphic to P_{R}^{1} is isomorphic to

C : x^{2} + y^{2} + z^{2} = 0

.

[[If the curve is planar this follows from diagonalizability of quadratic forms.]]

33 Riemann-Roch Theorem

The Riemann-Roch theorem is the cornerstone of all of curve theory.

Theorem 33.1 (Riemann-Roch). Let C be a curve and D = ∑_{i=1}^{t} n_{i}P_{i} be a divisor on C. Then

h^{0}(L(D)) − h^{1}(L(D)) = deg D + 1 − g

where deg D = ∑_{i=1}^{t} n_{i}.

Proof. When D = 0 the assertion is that

h^{0}(O_{C}) − h^{1}(O_{C}) = 1 − g

which is easily checked. Next suppose D is any divisor and P any point. Compare D and D + P as follows. The statement of the theorem for D is

h^{0}(L(D)) − h^{1}(L(D)) = deg D + 1 − g

and for D + P the theorem is

h^{0}(L(D + P)) − h^{1}(L(D + P)) = deg D + 1 + 1 − g.

Use the exact sequence

0 → L(D) → L(D + P) → κ(P) → 0

to compute the Euler characteristic χ = h^{0} − h^{1} of L(D + P). This yields

χ(L(D + P)) = χ(L(D)) + χ(κ(P))

so

h^{0}(L(D + P)) − h^{1}(L(D + P)) = h^{0}(L(D)) − h^{1}(L(D)) + 1.

The theorem is thus true for D+P iff it is true for D. Since the theorem is true for D = 0 and we can obtain any divisor by adding or subtracting points starting with D = 0 the theorem follows.

34 Serre Duality

By Serre Duality H^{1}(L(D)) is dual to

Ext^{0}(L(D), ω_{C}) = Ext^{0}(O_{C}, ω_{C} ⊗ L(D)^{∨})
= H^{0}(ω_{C} ⊗ L(−D)) = H^{0}(L(K − D))

Thus h^{1}(L(D)) = h^{0}(L(K − D)) where K is some divisor so that L(K) = ω. K is often called the canonical divisor. Thus we can restate the Riemann-Roch theorem as

h^{0}(L(D)) − h^{0}(L(K − D)) = deg D + 1 − g.

Sometimes one abbreviates h^{0}(L(D)) as ℓ(D) and then Riemann-Roch becomes

L(D) − L(K − D) = deg D + 1 − g.

Corollary 34.1. deg K = 2g − 2.

Proof. By Riemann-Roch,

h^{0}(L(K)) − h^{0}(L(0)) = deg K + 1 − g

but h^{0}(L(K)) = g and h^{0}(L(0)) = 1. Thus deg K = 2g − 2.

Homework II, Exercise 8.4; III, Exercise 6.8, 7.1, 7.3

35 A Bird’s Eye View of Curves

3g − 3 + g + 4(d + 1 − g − 4) + 15 = 4d

1. 3g − 3 is the dimension of the moduli space M_{g} of curves of genus g.

2. g is the dimension of Pic^{d} C.

3. 4(d + 1 − g − 4) is the number of ways to choose a linear system W in W^{0}(L).

4. 15 is the dimension of Aut P^{3} ≅ PGL(4, k).

5. 4d is the dimension of the Hilbert scheme H_{d,g}^{0} of nonsingular curves of genus g and degree d.

36 Moduli Space

As a set M_{g} is the set of curves of genus g modulo isomorphism. M_{g} can be made into a variety in a natural way. As a variety M_{g} is irreducible and

dim M_{g} = \begin{cases}0 & if g = 0 \\ 1 & if g = 1 \\ 3g − 3 & if g ≥ 2 \end{cases}

M_{g} is not a projective variety, but its closure \overline{M}_{g} is. The points of \overline{M}_{g} not in M_{g} are called stable curves.

37 Embeddings in Projective Space

38 Elementary Curve Theory

38.1 Definitions

Let C be a curve thus C is a nonsingular connected projective variety of dimension 1 over an algebraically closed field k. Let g be the genus of C. A divisor D is a sum ∑_{i=1}^{k} n_{i}P_{i} where n_{i} ∈ Z and P_{i} is a closed point of C. Let deg D = ∑_{i=1}^{k} n_{i}. A divisor D corresponds to an invertible sheaf L(D) = I_{D}^{∨} where I_{D} is the ideal sheaf of D and I_{D}^{∨} = Hom_{C}(I_{D}, O_{C}) is the dual of I_{D}. Note that L(nD) = L(D)^{⊗n} since ⊗ and ∨ commute.

Notation: O(D) := L(D), M(D) = M ⊗ L(D).

We say a divisor ∑ n_{i}D_{i} is effective if all n_{i} ≥ 0. Let |D| denote the complete linear system associated to D. Thus

|D| = {D′ : D′ is effective and D′ ∼ D}.

Here ∼ denotes linear equivalence. Two divisors D and D′ are linearly equivalent if there exists f ∈ K = K(C) (the function field of C) so that D − D′ = (f) where (f) = ∑_{P ∈C closed} v_{P} (f)P.

Since the condition D + (f) ≥ 0 is exactly the condition that f ∈ H^{0}(C, L(D)), it follows that there is a bijection

|D| ⟶^{∼} H^{0}(C, L(D))/k^{∗}

D′ = D + (f) ↦ f.

Two functions f, g which differ by an element of k^{∗} give the same divisor. Note that |D| may be empty.

If s ∈ H^{0}(L) then there is an injection 0 → O ⟶^{s} L(D) given by multiplication by s. Dualizing we obtain

L(−D) = I_{D} = L^{∨} ⟶^{s^{∨}} O → O_{D} → 0.

Thus D ≠ 0 is effective iff D corresponds to a closed subscheme defined locally by (t_{P_{i}}^{n_{i}} ⊂ O_{P_{i}} where t_{P_{i}} ∈ m_{P_{i}} ⊂ O_{P_{i}} is a uniformizing parameter at m_{P_{i}}.

|D| can be regarded as a projective space and as such

dim |D| = h^{0}(ℓ(D)) − 1 = ℓ(D) − 1,

where ℓ(D) = h^{0}(L(D)). We can generalize the notion of a complete linear system by considering linear subspaces of |D|. A (general) linear system is a linear subspace D ⊂ |D|. Thus D corresponds to a vector subspace W ⊂ H^{0}(C, L(D)).

38.2 Maps to Projective Space

39 Low Genus Projective Embeddings

Let C be a curve of genus g, let K be the canonical divisor (which corresponds to the invertible sheaf Ω = ω of differentials) and let D be a divisor.

Theorem 39.1 (Riemann-Roch). ℓ(D) − ℓ(K − D) = deg(D) + 1 − g

Proposition 39.2. A divisor D is ample iff deg D > 0.

Proposition 39.3. If deg D ≥ 2g + 1 then D is very ample.

39.1 Genus 0 curves

Suppose C is a curve of genus g = 0. If deg D ≥ 1 then D is very ample. Suppose D = P is a point. Then D gives rise (after a choice of basis for the corresponding invertible sheaf) to an embedding C ↪ P^{n} where n = ℓ(D) − 1 = 1. When D = 2P we obtain the 2-uple embedding C ↪ P^{2} which is a conic since g = 0. When D = 3P, C ↪ P^{2} is the twisted cubic. More generally D = dP gives the rational normal curve C ↪ P^{d} of degree d (which is in fact projectively normal).

39.2 Genus 1 curves

If C is a curve of genus g = 1 then deg D ≥ 2g + 1 = 3 iff D is very ample. Thus the converse to proposition 2 holds when g = 1. Suppose D is a very ample divisor of degree 2. (For degree 1 or 0 a similar argument works.) Then D gives rise to an embedding C ↪ P^{1} since ℓ(D) − 1 = 2 − 1 = 1 which is absurd since C has genus 1 but P^{1} has genus 0. To see that ℓ(D) = 2 apply Riemann-Roch to obtain ℓ(D) − ℓ(K − D) = 2 + 1 − 1 = 2. Then since deg K = 2g − 2 = 0 we see that deg(K − D) < 0 and hence ℓ(K − D) = 0 so ℓ(D) = 2 as desired.

Suppose D is a divisor of degree 3 on a genus 1 curve C. Then D gives rise to an embedding C ↪ P^{2} since ℓ(D) − 1 = (deg D + 1 − g) − 1 = 3 − 1 = 2. Thus any curve of genus 1 can be embedded as a nonsingular cubic curve in P^{2}. This embedding also allows us to show that K ∼ 0. We showed before that if C ⊂ P^{2} is of degree d then ω_{C} ≅ O_{C}(d − 3). Thus choosing an embedding arising from a divisor of degree 3 as above we see that ω_{C} ≅ O_{C}(0), thus the canonical sheaf corresponds to the trivial divisor class so K ∼ 0. Alternatively, we can prove this by using Riemann-Roch to see that

ℓ(K) = ℓ(0) + deg(K) + 1 − g = 1 + 0 + 1 − 1 = 1

and thus K is linearly equivalent to an effective divisor of degree 2g − 2 = 0.

39.3 Moduli Space

The moduli spaces of curves of low genus are

M_{0} = { curves of genus 0 } = {P^{1}}
M_{1} = { curves of genus 1 } = A^{1}

A curve C of genus g = 1 can be given by a degree 3 embedding C ↪ P^{2}. We will show [[in nice characteristic only?]] that two embedded curves C_{1} ↪ P^{2} and C_{2} ↪ P^{2} are isomorphic as abstract curves iff there is an automorphism in Aut P^{2} = PGL(3) sending C_{1} to C_{2}. Note that Aut P^{2} is a 3^{2} − 1 = 8 dimensional family. A degree 3 curve C ↪ P^{2} is given by a degree 3 polynomial

F = a_{0}x^{3} + ⋯ ∈ H^{0}(O_{P^{2}}(3)).

Since dim H^{0}(O_{P^{2}}(3)) = 10 we obtain a 10 − 1 = 9 dimensional projective space of such curves (including the singular ones). The nonsingular ones form an open subset Δ ≠ 0. The moduli space M_{g} has dimension 9 − 8 = 1. [[Not clear.]]

How can degree 3 embedded curves C_{1}, C_{2} ⊂ P^{2} be isomorphic? We generalize to arbitrary degree and ask the following question.

Question 39.4. Suppose C_{1}, C_{2} ⊂ P^{2} are both nonsingular curves of degree d and suppose C_{1} ≅ C_{2} as abstract curves. Does is necessarily follow that there is an automorphism g of P^{2} sending C_{1} to C_{2}, i.e., such that g(C_{1}) = C_{2}?

Suppose d = 1. Then C_{1} and C_{2} are both lines in P^{2} so the answer is yes.

Suppose d = 2. Then C_{1} and C_{2} are both conics. If k is algebraically closed and char k ≠ 2 the defining equations C_{1} and C_{2} can be transformed into x^{2} + y^{2} + z^{2} = 0 by an automorphism of P^{2} (by “completing the square”). Thus in this case the answer to the question is yes. When char k = 2 the answer is no. [[give an easy counterexample here.]]

Suppose d = 3. Thus C_{1} and C_{2} are both cubic curves in P^{2} and C_{1} ≅ C_{2} as abstract curves. Equivalently we are given an abstract curve C and two embeddings

φ_{1} : C ↪ P^{2}
φ_{2} : C ↪ P^{2}

The question is then: does there exist an automorphism g of P^{2} such that g(φ_{1}(C)) = φ_{2}(C)? The embedding data giving φ_{1} is a divisor D along with a basis of global sections s_{0}, s_{1}, s_{2} ∈ H^{0}(O_{C}(D)). An automorphism of P^{2} induces a map on C which preserves D but changes the basis s_{0}, s_{1}, s_{2}. Suppose φ_{1} is given by D_{1} and global sections s_{0}, s_{1}, s_{2}, and that φ_{2} is given by D_{2} and t_{0}, t_{1}, t_{2}. The automorphism g induces a map

g′ : C ≅ φ_{1}(C) ⟶^{g}φ_{2}(C) ≅ C.

Then g′(D_{1}) = D_{2} and g′ : s_{0}, s_{1}, s_{2} ↦ t_{0}, t_{1}, t_{2}.

This generalizes so that in degree d ≥ 3 a necessary condition for a yes answer to the above question is that for any two divisors D_{1} and D_{2} of degree d on C there is g ∈ Aut P^{2} such that g′(D_{1}) = D_{2}.

40 Curves of Genus 3

Today we will study curves of genus 3.

Example 40.1. A plane curve C ⊂ P^{2} of degree d = 4 has genus \frac{1}{2}(d − 1)(d − 2) = 3.

Example 40.2. A curve C on the quadric surface Q in P^{3} of type (2,4) has degree 6 and genus 3.

These two examples are qualitatively different. The curves in the first example are “canonical” whereas the curves in the second class are “hyperelliptic”.

We consider the first example in more detail. Let C ⊂ P^{2} be a genus 3 plane curve (so C has degree d = 4). Then

ω_{C} = O_{C}(d − 3) = O_{C}(1)

so ω_{C} is very ample. Thus the canonical embedding arising from the canonical divisor is exactly the given embedding C ↪ P^{2}.

Next we consider example 2 in more detail. Let C be a genus three curve of type (2,4) on the quadric surface Q ≅ P^{1} ×P^{1} ⊂ P^{3}. The projections p_{1}, p_{2} : Q → P^{1} give rise to a degree 2 and a degree 4 morphism of C to P^{1}. Thus there exists a 2-to-1 morphism f : C → P^{1}. f corresponds to a base point free linear system on C of degree 2 and dimension 1. This linear system in turn corresponds to an effective divisor D of degree 2 with ℓ(D) = 2 so |D| = 1. The existence of such a divisor means there exists P, Q such that |P + Q| has dimension 1.

Definition 40.3. A curve C is hyperelliptic if g ≥ 2 and there is a base point free linear system of degree 2 and dimension 1.

It is classical notation that a base point free linear system of degree d and dimension r is called a g_{d}^{r}. To say that a curve is hyperelliptic is to say that it has a g_{2}^{1}.

• If g = 0 then there is always a g_{2}^{2}.

• If g = 1 any divisor of degree 2 gives a g_{2}^{1} by Riemann-Roch. Indeed, if D has degree

2 then

dim |D| − dim |K − D| = 2 + 1 − g = 2

and deg(K − D) = −2 so dim |D| − (−1) = 2 and hence dim |D| ≤ 1.

• If g = 2 every curve is hyperelliptic since |K| is a g_{2}^{1}. Indeed, applying Riemann-Roch

we see that

dim |K| = dim |K| − dim |0| = 2 − 1 = 1.

Lemma 40.4. Let C be any curve and D any divisor of degree d > 0. Then dim |D| ≤ d with equality iff C is a rational curve.

Proof. This is (IV, Ex. 1.5) in Hartshorne. Although one might guess that this lemma follows from Riemann-Roch this is not the case. Riemann-Roch gives a different sort of relationship between the dimension and degree of a divisor. We induct on d.

First suppose d = 1. First note that

dim |P | = ℓ(P) − 1 = h^{0}(O_{C}(P)) − 1.

There is an exact sequence

0 → O_{C} → O_{C}(P) → k(P) → 0.

Now h^{0}(O_{C}) = 1 and h^{0}(k(P)) = 1 therefore h^{0}(O_{C}(P)) ≤ 2 so dim |P | ≤ 1. If dim |P | = 1 then |P | has no base points so we obtain a morphism C → P^{1} of degree deg P = 1 which must be an isomorphism so C is rational.

Next suppose D = P_{1} + ⋯ + P_{d}. Let D′ = P_{1} + ⋯ + P_{d−1}. There is an exact sequence

0 → O_{C}(D′) → O_{C}(D) → k(P_{d}) → 0.

Now h^{0}(O_{C}(D′)) ≤ d by induction and h^{0}(k(P_{d})) = 1 so h^{0}(O_{C}(D)) ≤ d + 1, therefore dim |D| ≤ d with equality iff h^{0}(O_{C}(D′)) = d. By induction h^{0}(O_{C}(D′)) = d iff C is rational.

Theorem 40.5. Suppose C is a curve of genus g ≥ 2. Then ω_{C} is very ample iff C is not hyperelliptic.

Proof. Let K be the canonical divisor. Then by a previous result K is very ample iff for all points P, Q, dim |K − P − Q| = dim |K| − 2. By Riemann-Roch,

dim |P + Q| − dim |K − P − Q| = 2 + 1 − g = 3 − g.

Now K is very ample iff dim |K − P − Q| = dim |K| − 2 = (g − 1) − 2 = g − 3 so when K is very ample the above becomes

dim |P + Q| − (g − 3) = 3 − g.

Thus K is very ample iff for all P and Q, dim |P + Q| = 0. Thus K is not very ample iff there exists P and Q so that dim |P + Q| = 1. But the latter condition occurs precisely when C is hyperelliptic. (We can exclude the case dim |P + Q| ≥ 2 by using the previous lemma and the fact that C is not rational.)

Corollary 40.6. If C is a curve of genus g ≥ 1 then |K| has no base points.

Proof. If g = 1 then K = 0 so |K| = {0} and we are done. If g ≥ 2 then |K| is base point free iff

dim |K − P | = dim |K| − 1

for all P. (If |K| has a base point P then every effective divisor D linearly equivalent to K is such that D − P is effective and linearly equivalent to K − P. If |K| has no base points then the dimension of the space of effective divisors equivalent to K − P must go down for ever P.) Since

dim |P | = dim |K − P | + 1 + 1 − g

and

dim |K| = 2g − 2 + 1 − g = g − 1

we see that

dim |K − P | = dim |K| − 1 + dim |P |

so |K| is base point free iff dim |P | = 1 for all P. But dim |P | ≤ 1 for every P and we have equality iff C is rational (i.e., g=0). Since C is not rational it follows that |K| is base point free.

Suppose given an abstract curve C of genus 3. Then C belongs to one of two disjoint classes. If the canonical sheaf ω_{C} is very ample then we obtain an embedding of C into P^{g−1} = P^{2} as a nonsingular quartic curve. If ω_{C} is not very ample then C is hyperelliptic (since the map induced by ω_{C} is 2-to-1). Does every hyperelliptic curve arise as a curve of type (2,4) on Q ⊂ P^{3}? Hartshorne claims to have three-fourths of a proof.

Lemma 40.7. If C is any curve of genus 3 then there exists a very ample divisor of degree 6 = 2g.

Proof. Let D be a divisor of degree 6. We have shown that D is very ample iff

dim |D − P − Q| = dim |D| − 2

for all P and Q. Since deg K = 2g − 2 = 4 Riemann-Roch asserts that

dim |D| = dim |K − D| + 6 + 1 − g = −1 + 6 + 1 − g = 6 − g = 3
dim |D − P | = dim |K − (D − P)| + 5 + 1 − g = 5 − g = 2
dim |D − P − Q| = dim |K − (D − P − Q)| + 4 + 1 − g.

Thus for D to be very ample it must be the case that

dim |K − (D − P − Q)| = −1.

Since deg(K − (D − P − Q)) = 0 it follows that dim |K − (D − P − Q)| = −1 iff K is not linearly equivalent to D − P − Q. The assertion is thus reduced to showing that among all divisors of degree 6 the set with D − P − Q ∼ K is a proper closed subset.

[[I do not understand Hartshorne’s proof of this. He says D−P −Q ∼ K iff D ∼ K+P +Q. He then claims that the family of D of degree 6 is a 6 dimensional family and that the family of divisor K + P + Q is a 5 dimensional family.]]

What about the converse? If C is hyperelliptic must C then have to lie on a quadric surface?

41 Curves of Genus 4

Recall that curves of genus g ≥ 2 split up into two disjoint classes.

(a) hyperelliptic

(b) ω_{X} is very ample

If g = 3 and C is of type (2, 4) on Q ⊂ P^{3} then C is hyperelliptic. Also, if g = 3 then ω_{X} is very ample iff C is a degree 4 curve in P^{2}. Any curve of genus g = 3 can be embedded as a curve of degree 6 in P^{3}. We do not know whether any such curve can actually be put on a quadric surface. “This would make a great homework problem — I do not know the answer.”

Next we consider curves of genus 4.

Example 41.1. Consider a type (2, 5) curve C on Q ⊂ P^{3}. Then C has degree 7 = 2 + 5 and C is hyperelliptic (because of the degree 2 map coming from projection onto the first fact p_{1} : Q → P^{1}). A type (3, 3) curve on Q is also of genus 4. It is a degree 6 complete intersection of Q and a cubic surface. Curves of type (3, 3) have at least two g_{3}^{1}’s.

41.1 Aside: existence of g_{d}^{1}’s in general

Assume for this discussion that g_{d}^{1}’s are allowed to have base points. Call such g_{d}^{1}’s trivial. Given a g_{d}^{1} adding a point p trivially gives a (trivial) g_{d+1}^{1}.

Question 41.2. Given any curve C what is the least d for which there exists a g_{d}^{1}?

• g = 0 there is a g_{1}^{1} coming from the embedding P^{1} ↪ P^{1},

• g = 1 there are infinitely many g_{2}^{1}’s,

• g = 2 there is a g_{2}^{1} namely ω_{C},

• g ≥ 2] if there is a g_{2}^{1} then C is hyperelliptic,

• g > 2 there exists nonhyperelliptic curves.

If a curve C has a g_{3}^{1} it is called trigonal. If g ≤ 2 then there exists a trivial g_{3}^{1} (just add a point to a g_{2}^{1}). If g = 3 and C is hyperelliptic then it is trivial that there is a g_{3}^{1}. If C is not hyperelliptic it is not trivial. But projection through a point P ∈ C gives a 3-to-1 map to P^{1}. This corresponds to a g_{3}^{1}. Thus a curve of genus 3 has infinitely many g_{3}^{1}’s (one for each point, coming from projection). If g = 4 and C is hyperelliptic then it is trivial that there is a g_{3}^{1}. In general given any genus 4 curve one can always construct a g_{3}^{1}. When g ≥ 5 in general there will not be a g_{3}^{1}. This pattern repeats itself.

41.2 Classifying curves of genus 4

Start with an abstract curve C of genus 4. We do not deal with the case C hyperelliptic now. A related question is the following.

Question 41.3. Does every hyperelliptic curve live on the quadric surface?

We postpone this question or maybe put it on an upcoming homework assignment. [Everyone shudders.]

If C is not hyperelliptic then ω_{C} is very ample. Therefore we have the canonical embedding C ↪ P^{g−1} = P^{3}. The degree of the embedded curve is deg ω_{C} = 2g − 2 = 6. Thus view C as a degree 6 genus 4 curve in P^{3}. What does C lie on? There is an exact sequence

0 → H^{0}(I_{C}(2)) → H^{0}(O_{P^{3}}(2)) → H^{0}(O_{C}(2)) → ⋯

Since Riemann-Roch states that ℓ(D) = deg D + 1 − g + h^{1}(O(D)) we see that h^{0}(O_{C}(2)) = 12 + 1 − 4 + 0 = 9. [[This is not quite clear.]] Since h^{0}(O_{P^{3}}(2)) = 10 it follows that the map H^{0}(O_{P^{3}}(2)) → H^{0}(O_{C}(2)) must have a nontrivial kernel so h^{0}(I_{C}(2)) > 0. Therefore C lies in some surface of degree 2. Can the surface be twice a hyperplane? No. Can the surface be the union of two planes? No. Could the surface by the singular quadric cone Q_{one}? Yes. Could the surface be the nonsingular quadric surface Q_{ns}? Yes.

If C lies on Q_{ns} then it must have a type (a, b) which must satisfy a + b = 6 and (a − 1)(b − 1) = 4. The only solution is a = b = 3.

The other possibility is that C lies on Q_{one}. One way to understand C is to figure out all divisors on Q. Another way is to compute an exact sequence like the one above. We obtain

0 → H^{0}(I_{C}(3)) → H^{0}(O_{P^{3}}(3)) → H^{0}(O_{C}(3)) → ⋯

As before one sees that h^{0}(O_{C}(3)) = 15 and h^{0}(O_{P^{3}}(3)) = 20. Thus h^{0}(I_{C}(3)) ≥ 5. Let q ∈ H^{0}(I_{C}(2)) be the defining equation of Q_{one}. Then xq, yq, zq, wq ∈ H^{0}(I_{C}(3)). But h^{0}(I_{C}(3)) ≥ 5 so there exists an f ∈ H^{0}(I_{C}(3)) so that the global sections xq, yq, zq, wq, f are independent. Thus there is an f not in (q). Since f ∉ (q) we see that F_{3} = Z(f) ⊅ Q so C′ = F_{3} ∩ Q is a degree 6 not necessarily nonsingular or irreducible curve. Since C ⊂ F_{3} and C ⊂ Q it follows that C ⊂ C′. Since deg C = 6 = deg C′ it follows by an easy exercise that C = C′.

Lemma 41.4. Suppose C ⊂ C′ are both closed subschemes of P^{n} with the same Hilbert polynomial. Then C = C′.

Thus in the case that C lies on Q_{one} we see that C is also a complete intersection C = Q_{one} ∩ F_{3}.

Next we comment on the g_{3}^{1} question in this situation. Projection from the cone point to the conic (the base of the cone) induces a g_{3}^{1} on C. So in genus g = 4 there is a g_{3}^{1}. There is one in the case that C lies on Q_{one} (it is not clear that there is just one), there are two in the case that C lies on Q_{ns}, and there is a trivial one in the case that C is hyperelliptic.

Proposition 41.5. Suppose C is a genus 4 nonsingular complete intersection Q ∩ F_{3} ⊂ P^{3} with Q of degree 2 and F_{3} of degree 3. Then ω_{C} ≅ O_{C}(1).

Recall that if C is of degree d in P^{2} then ω_{C} ≅ O_{C}(d − 3). This can be seen from an analysis of the exact sequence

0 → I/I^{2} → Ω_{P^{2}}|C → Ω_{C} → 0

together with the exact sequence

0 → Ω_{P^{2}}| → O_{C}(−1)^{3} → O_{P^{2}}|C → 0.

We could do the same thing for C ⊂ P^{3}.

Proposition 41.6. Suppose C = F_{e} · F_{f} ⊂ P^{3} with F_{e} and F_{f} nonsingular of degree e, f, respectively. Then ω_{C} = O_{C}(e + f − 4).

This was (II, Ex. 8.4 e) and it can be found in my homework solutions.

42 Curves of Genus 5

There are hyperelliptic curves of genus 5. For example a curve of type (2, 6) on the quadric surface Q ⊂ P^{3}. Are there any more curves of genus 5?

If C is a curve of genus 5 which is not hyperelliptic then ω_{C} is very ample so there is a canonical embedding C ↪ P^{4} in which C has degree 8. How many quadric hypersurfaces does such a C lie on?

42.1 The space of quadrics containing an embedded genus 5 curve.

There is an exact sequence

0 → H^{0}(I_{C}(2)) → H^{0}(O_{P^{4}}(2)) → H^{0}(O_{C}(2)) → ⋯

Riemann-Roch implies h^{0}(O_{C}(2)) = deg(O_{C}(2)) + 1 − g = 16 + 1 − 5 = 12. Here O_{C}(2) has degree 16 since the degree is additive on the class group and C has degree 8 so the divisor O_{C}(1) has degree 8. Also O_{C}(2) is superspecial since the canonical divisor has degree 2g − 2 = 8. Combining this with the fact that h^{0}(O_{P^{4}}(2)) = 15 implies h^{0}(I_{C}(2)) ≥ 3. Thus C lies on a 3 dimensional space of quadric hypersurfaces. Let F_{2}, F′_{2}, F′′_{2} be linearly independent quadric hypersurfaces containing C. In the genus 4 case F_{2} ∩ F_{3} was a curve C′ which we were able to show was equal to C. But in our situation it might be possible that

F_{2} ∩ F′_{2} = H ≅ P^{3}.

But in that case C ⊂ F_{2} ∩ F′_{2} would be contained in P^{3}. But this is not true since C is not contained in any linear subspace since the embedding comes from the complete linear system |ω_{C}|. (If C were contained in a linear subspace then there would be a dependence relation between the linearly independent global sections of ω_{C} giving rise to the embedding.)

If F_{2} ∩ F′_{2} is a hypersurface then it is defined by a single polynomial equation f = 0. Then f divides the quadratic defining F_{2} and the quadratic defining F′_{2} so f must be linear. But this case was ruled out above. Thus S = F_{2} ∩ F′_{2} is a surface. We cannot conclude that F_{2} ∩ F′_{2} ∩ F′′_{2} is the curve C. It could happen that S ⊂ F′′_{2} and so S = F_{2} ∩ F′_{2} ∩ F′′_{2}. It could also happen that F_{2} ∩ F′_{2} ∩ F′′_{2} is just a part S_{1} of the surface S. But this gives us a hint as to how to construct curves of genus 5.

42.2 Genus 5 curves with very ample canonical divisor

In P^{4} take F_{2}, F′_{2}, F′′_{2} three quadric hypersurfaces which are sufficiently general so that C = F_{2} ∩ F′_{2} ∩ F′′_{2} is a nonsingular curve. (That this can be done appeals to Bertini’s theorem.) Given such a curve C then deg C = 2^{3} = 8 and ω_{C} = O_{C}(2 + 2 + 2 − 4 − 1) = O_{C}(1). Therefore C is the canonical embedding of the abstract curve hiding in the shadows behind C. The genus of C is 5 since 2g −2 = deg C = 8. This construction gives examples of curves of genus 5 for which ω_{C} is very ample, i.e., curves of genus 5 which are not hyperelliptic.

Does this type of curve have a g_{3}^{1}, i.e., a complete linear system of degree 3 and dimension 1?

Suppose C is an abstract curve of genus 5 which is not hyperelliptic. Embed C in P^{4} by its canonical embedding C ↪ P^{4}. There are two possibilities:

1. C = F_{2} ∩ F′_{2} ∩ F′′_{2}

2. C ⊂ S = F_{2} ∩ F′_{2} ∩ F′′_{2} As seen above case 1 can occur. In case 2, S is a surface of degree 8 [[I missed the proof.]] We do not yet know if case 2 can occur.

Is C trigonal, that is, does C have a g_{3}^{1}? To study this question we introduce a new technique. Suppose C ↪ P^{4} is the canonical embedding of C into P^{4} as a degree 8 curve. Then there is a g_{3}^{1} iff there exists three points P, Q, R ∈ C such that dim |P + Q + R| = 1. Given three points P, Q, R Riemann-Roch implies

dim |P + Q + R| = 3 + 1 − 5 + dim |K − P − Q − R| = −1 + dim |K − P − Q − R|

so dim |P +Q+R| = 1 iff dim |K−P −Q−R| = 2. The condition that dim |K−P −Q−R| = 2 is that there is a 2 dimensional linear system of effective canonical divisors K which contain P, Q, R.

The technique is to translate the condition dim |K − P − Q − R| = 2 into a geometric criterion involving the embedding C ↪ P^{4}. Since the embedding C ↪ P^{4} is canonical every effective divisor in the canonical divisor class is the intersection of C with a hyperplane in P^{4}. We obtain every effective divisor because |K| has dimension 4 and the dimension of the space of lines in P^{4} is 4. Thus an effective canonical divisor contains P, Q, R iff there is a hyperplane in P^{4} containing P, Q, R. Hence there is a 2 dimensional linear system in P^{4} containing P, Q, R iff P, Q, R are collinear in P^{4}. We have thus interpreted dim |P + Q + R| in terms of the geometry of where P, Q, R lie on C in the canonical embedding. The upshot of this is

Proposition 42.1. Suppose C is a not a hyperelliptic curve. Then C has a g_{3}^{1} iff there are 3 points P, Q, R ∈ C which are collinear in the canonical embedding.

Notice that the proposition is even true without the assumption that C has genus 5. We return to our situation. Suppose C ↪ P^{4} is the degree 8 canonical embedding of

some nonhyperelliptic curve C of genus 5. Suppose P, Q, R ∈ C are collinear so they all lie on some line L.

First suppose C is the complete intersection C = F_{2} ∩ F′_{2} ∩ F′′_{2}. Then P, Q, R ∈ F_{2} and P, Q, R ∈ L so L ∩ F_{2} contains at least 3 points. If L is not contained in F_{2} then L ∩ F_{2} has (deg L) · (deg F_{2}) = 2 points so it must be the case that L ⊂ F_{2}. By similar reasoning we conclude that L ⊂ F′_{2} and L ⊂ F′′_{2} so L ⊂ C, a contradiction. So C does not have a g_{3}^{1}.

Next suppose C ⊂ S = F_{2} ∩ F′_{2} ∩ F′′_{2}. Then it is possible that L ⊂ S since this leads to no contradiction. So maybe there could be a g_{3}^{1} on C, we still do not know.

Next we show that there do exist genus 5 trigonal curves. We construct directly a genus 5 curve C with nontrivial g_{3}^{1}. The things to do is look in P^{2} for a curve D of degree 5 which has one node and no other singularities.

As an aside we consider a more general problem. Consider singular irreducible curves C of degree d in P^{2} containing r nodes and k cusps. A node looks locally like xy = 0 and a cusp looks like y^{2} = x^{3}. What are the possible triples (d, r, k) which can occur? The answer for small d is

\begin{matrix}d & (d, r, k) \\ 1 & (1,0,0) \\ 2 & (2,0,0) \\ 3 & (3,1,0), (3,0,1) \\ 4 & (4,3,0), … \end{matrix}

In general this is an open problem. Solving it is a guaranteed thesis, seven years of good luck, and an... academic position! The complete answer is probably known up to degree 10.

One constraint comes from the fact that if C̃ is the normalization of C then

g(C̃) = \frac{1}{2}(d − 1)(d − 2) − r − k ≥ 0

so r + k ≤ \frac{1}{2}(d − 1)(d − 2).

Finding a D as above means finding a (5, 1, 0). Suppose char k ≠ 3, 5. Let

f = xyz^{3} + x^{5} + y^{5}.

The point x = y = 0 is a nodal singularity. There are no other singularities. See this by computing f_{x} = yz^{3} + 5x^{4}, f_{y} = xz^{3} + 5y^{4}, and f_{z} = 3xyz^{2}. For these to all vanish it must be the case that x, y, or z is 0. If x or y is 0 then both x and y are 0 so we recover the nodal singularity. If x and y are nonzero then z = 0 so from f_{x} = 0 and f_{y} = 0 it follows that x = y = 0, but x = y = z = 0 is not a point.

Let C = D̃ be the normalization of C. Then C is a genus 5 nonsingular curve. (That the normalization of a curve is nonsingular is a fact from commutative algebra.) Pick a node P on D. Then lines through P in P^{2} give a map D − {P } → P^{1}. A point Q in P^{1} corresponds to a line L through P. Since P is a double point and D has degree 5, we see that L intersects D in 3 other points. These three points map to Q ∈ P^{1}. This map extends to a map on C which corresponds to a g_{3}^{1}.

We can now say something about the 3g − 3 = 12 dimensional space of genus 5 curves. They can be divided into 3 nonempty classes: trigonal, hyperelliptic and general ones with no g_{3}^{1}. The hyperelliptic curves form a 2g − 1 = 9 dimensional family. [[The trigonal curves might form an 11 dimensional family??]] It might be the case that every hyperelliptic curve is trigonal.

Next time we will do the case of genus 6 since a new phenomenon appears.

43 Homework Assignment

Do the following from the book: IV Ex. 3.6, 3.12, 5.4.

Do any one of the following problems (and try to do one that someone else is not doing!)

1. If C is a hyperelliptic curve of genus g ≥ 2, find the least possible degree of a very ample divisor on C. Is it independent of the particular hyperelliptic curve chosen?

2. Can every hyperelliptic curve of genus g ≥ 2 be embedded in P^{3}, so as to be a curve of bidegree (2, g + 1) on a non-singular quadric surface Q?

3. If C is hyperelliptic of genus g ≥ 3, then C does not have a g_{3}^{1} (without base points).

4. If C is a non-hyperelliptic curve of genus g ≥ 4 show that C has at most a finite number of g_{3}^{1}’s.

5. If C is a non-hyperelliptic curve of genus g ≥ 3, then it admits a very ample divisor of degree d ≤ g + 2.

44 Curves of genus 6

Examples of curves of genus 6.

(a) degree 5 curve in P^{2},

(b) a type (2, 7) curve (of degree 9) on the quadric surface Q in P^{3},

(c) a type (3, 4) curve (of degree 7) on the quadric surface Q in P^{3}.

Claim 44.1. The abstract curves C which can be realized in types (a), (b), and (c) are mutually disjoint.

Since (b) is hyperelliptic it has a g_{2}^{1}. Since (c) is trigonal (i.e. it has a g_{3}^{1}) by exercise 3 it does not have a g_{2}^{1}. Thus (b) and (c) are disjoint classes.

I will next show that (a) has no overlap with (b) and (c). First note that (a) has infinitely many g_{4}^{1}’s, one for each point. This is because projection through a point gives a degree 4 map to P^{1}. I will prove that (a) has no g_{2}^{1} or g_{3}^{1}.

Lemma 44.2. If C is a nonsingular curve of deg 5 in P^{2} then C does not have a g_{2}^{1} or a g_{3}^{1}.

The lemma is a special case of a result of Max Noether:

“On a plane curve, the only linear systems g_{d}^{r} of maximal dimension (so r is maximal with respect to d) are the obvious ones.”

The obvious linear systems are the ones arising in a natural way by intersecting the curve with a straight line, or with a conic section, or a conic section but fixing one point, and so on. Suppose C has degree 5 in P^{2}. Then cutting with a line gives a g_{5}^{2} since the lines in P^{2} form a 2 parameter family and they intersect C in 5 points. Cutting with a conic gives a g_{10}^{5} since the conics form a 5 parameter family. By fixing two points one obtains a g_{9}^{4}, by fixing three points a g_{8}^{3} and similarly a g_{7}^{2} and g_{6}^{1}. All lines with 1 fixed point gives a g_{6}^{2}. More generally, if C has degree n in P^{2} then there exists a g_{n−1}^{1} but Noether’s result implies that there does not exist a g_{d}^{1} for d < n − 1.

Proof. Now we prove the lemma. Let C be a nonsingular curve of degree 5 in P^{2}. Then ω_{C} = O_{C}(d − 3) = O_{C}(2). This curve is subcanonical, i.e., ω_{C} = O_{C}(ℓ) for some ℓ > 0. Thus the canonical embedding is obtained by following C ↪ P^{2} by the 2-uple embedding.

From last time we know that there exists a g_{3}^{1} on C iff there exists points P, Q, R ∈ C such that P, Q, R are collinear in the canonical embedding. This might lead to a proof but I can not think of it right now, so forget it!

C is hyperelliptic iff there exists P, Q such that dim |P + Q| = 1. By Riemann-Roch

dim |P + Q| = 2 + 1 − 6 + dim |K − P − Q|

so in this situation dim |K − P − Q| = 4. Since C has degree 5 and the canonical divisor has degree 10, the canonical divisor is cut by conics. To see this note that |K| = g − 1 = 5 and the dimension of the space of conics in P^{2} is also 5. If dim |K − P − Q| = 4 then there exists P, Q ∈ C such that the family

{ conics containing P, Q }

has dimension 4. But the family of all conics in P^{2} has dimension 5, the family of conics through one fixed point has dimension 4, and the family of conics through two fixed points has dimension 3. Thus C can not be hyperelliptic.

C is trigonal iff there exists points P, Q, R such that dim |P + Q + R| = 1. By Riemann-Roch this latter condition implies that dim |K − P − Q − R| = 3. This would mean that we could find three points P, Q, R in C such that the dimension of the family of conics containing P, Q, R is 3. But the family of conics through P and Q has dimension 3 and there are conics passing through P and Q but not through R so the family of conics through all of P, Q, R must have dimension less than 3. Thus C is not trigonal and the lemma is proved.

Can every curve of genus 6 be realized as one of type (a), (b), or (c)? The answer is no.

g  g_{d}^{r}’s
g = 2  ∃g_{2}^{1}
g = 3  ∃g_{3}^{1}
g = 4  ∃ finitely many g_{3}^{1}
g = 5 ∃ infinitely many g_{4}^{1} (not shown in class)
g = 6 we should expect that the general curve

has only finitely many g_{4}^{1}’s

Kleiman-Lacksov prove that what we expect is actually the case in general. Our examples (a), (b), and (c) all have infinitely many g_{4}^{1}’s so we suspect that our examples do not cover all genus 6 curves.

There exists a curve of a fourth type (d)=none of the above, and this will be the general curve. It is hard to get our hands on a general curve since any time we explicitly make a curve it has special properties and is thus not general.

We want to find a plane curve of degree 6 with 4 nodes and no other singularities. The space of all curves of degree 6 has dimension \frac{1}{2}d(d + 3) = 27. It takes 3 linear conditions to force a node at a particular point, thus it takes 12 linear conditions to get 4 nodes. Since 12 < 27 there exist such curves. The curve could have weird singularities, but there is a way to get the curve we want. This is like homework IV.5.4. Thus suppose C_{0} ⊂ P^{2} is a curve of degree 6 with exactly 4 singularities which are all nodes. Let C = C̃_{0} be the normalization of C_{0}. Then

g = g(C) = \frac{1}{2}(d − 1)(d − 2) − 4 = 10 − 4 = 6.

There are five obvious g_{4}^{1}’s. Four come from projecting away from any of the double points. To get the fifth consider conics passing through all four double points. Thus C has g_{4}^{1}’s but none of our previous curves did and so the class (d) is nonempty.

Another question is the following.

Question 44.3. When g is even there are only finitely many g_{g−1}^{1}’s. For example when g = 4 there are two g_{3}^{1}’s, and when g = 6 there are five g_{4}^{1}’s. In general how many g_{g−1}^{1}’s are there?

We could have also approached showing classes (a), (b), (c), and (d) are disjoint by asking: what is the least degree of a very ample divisor? For (a) the least degree is 5, for (b) it is 9, for (c) it is 7, and for (d) it is 8. We give no proof of this here. This type of classification method should generalize to arbitrary genus.

45 Oral Report Topics

During the week of the 29th of April oral reports will be presented by the students. The suggested topics are

• Curves /k where k is not necessarily algebraically closed and rational points on curves over finite fields.

• The variety of moduli M_{g}.

• Duality for a finite smooth morphism X → Y.

• Jacobian variety of a curve. (“what does it really mean?”)

• Curves on a nonsingular cubic surface in P^{3} (ch 5, sec 4).

• Flat families of curves in P^{3} (ch 3, sec 9).

Your oral reports must be 20 minutes in length. They should contain precise definitions, statements of the main theorems, some examples, and maybe some proofs if there is time. You should consult with me before you begin.

For the rest of the semester we will be studying curves of genus g, g general and elliptic curves.

46 Curves of general genus

Today’s lecture contains hints for some of the homework problems.

Proposition 46.1. If C is hyperelliptic of genus g ≥ 2, then the g_{2}^{1} is unique. Furthermore, K ∼ (g − 1)D for any D ∈ g_{2}^{1}.

Proof. The complete linear system |K| has no base points iff dim |K−P | = dim |K|−1 for any point P on C. By Riemann-Roch dim |K| = 2g−2+1−g = g−1 and dim |P |−dim |K−P | = 2−g so dim |K−P | = dim |P |+g−2. Thus dim |K−P | = dim |K|−1 for all P iff dim |P | = 0 for all P which is true since g ≥ 1. (If there were a one dimensional space of points linearly equivalent to a given point P then C would have genus 0.) Thus K defines a morphism (which is not necessarily an embedding)

C → P^{g−1}.

The map is into P^{g−1} since dim |K| = g − 1. We saw before that if C is not hyperelliptic then this is an embedding. If C is hyperelliptic then the canonical divisor can not be very ample so this map will not be an embedding.

First consider the special case g = 1. Then we have a map C → P^{0} = { point } which is not interesting.

Next suppose g = 2. Then the canonical divisor induces a map

C → P^{1}

and K is the pullback of O(1). Since deg K = 2g − 2 = 2 the map C → P^{1} has degree 2, and the complete linear system |K| is a g_{2}^{1}. In fact, any |D| is a g_{2}^{1} then D ∼ K. To see this suppose P, Q are two points and dim |P + Q| = 1. By Riemann-Roch,

1 = dim |P + Q| = 2 + 1 − 2 + dim |K − P − Q|

so dim |K − P − Q| = 0. Thus there is an effective divisor linearly equivalent to the degree 0 divisor K−P −Q. But the only effective divisor of degree 0 is the 0 divisor. Thus K ∼ P +Q. Thus in genus 2 any g_{2}^{1} is given by K since any effective D giving a g_{2}^{1} is linearly equivalent to K.

Suppose now that g ≥ 3. The complete linear system |K| gives a morphism

φ : C → P^{g−1}.

Since K is not very ample, φ is not a closed immersion. Now assume C is hyperelliptic. Suppose P and Q are two points on C such that P + Q lies in some g_{2}^{1}. Then by Riemann-Roch,

1 = dim |P + Q| = 2 + 1 − g + dim |K − P − Q|.

Thus dim |K − P − Q| = g − 2. But dim |K| = g − 1 and |K| has no base points so dim |K − P | = g − 2. Thus dim |K − P | = dim |K − P − Q| so Q is a base point of |K − P |. This means that any canonical divisor containing P also contains Q. Therefore φ(P) = φ(Q) if we look at this in terms of the morphism. Thus if P + Q is in some g_{2}^{1} then φ(P) = φ(Q).

Thus φ is at least two-to-one. Let C_{0} = φ(C) ⊂ P^{g−1} be the image curve. Pulling back O(1) on P^{g−1} gives O_{C_{0}}(1) on C_{0} which pulls back to the canonical divisor O_{C}(1) = K which has degree 2g − 2. There are two numbers to consider. First the degree of the curve C_{0}, call it d. Let e be the degree of the finite morphism φ. Then

2g − 2 = deg |K| = de.

See this by noting that O_{C_{0}}(1) is obtained by cutting C_{0} with a hyperplane and seeing that it intersects in d points then pulling these points back to their de preimages to obtain the de degree divisor O_{C} = K.

Since e ≥ 2 the equality de = 2(g −1) implies d ≤ g −1. Consider C_{0} which is a (possibly singular) integral curve. Let D_{0} = O_{C_{0}}(1), then D_{0} has degree d and since |D_{0}| gives the embedding of C_{0} into P^{g−1} and C_{0} lies in no linear subspace, dim |D_{0}| ≥ g −1. By a previous proposition (IV, Ex. 1.5), we know that dim |D_{0}| ≤ deg D_{0} with equality iff C_{0} ≅ P^{1}. But

g − 1 ≤ dim |D_{0}| ≤ deg D_{0} = d ≤ g − 1

so we do have equality and C_{0} ≅ P^{1} is nonsingular and is in fact the d − 1-uple embedding of P^{1} into P^{g−1}. Furthermore, since d = g − 1 we also see that e = 2.

The upshot of all this is that if C is hyperelliptic and g ≥ 3 then |K| gives an embedding φ : C → P^{g−1} which factors through the g − 1-uple embedding of P^{1}. Thus φ : C → P^{g−1} can be written as a composition

C ⟶^{g_{2}^{1}} C_{0} = P^{1} ↪ P^{g−1}.

[[One shows that the g_{2}^{1} is uniquely determined, etc. as on page 343 of the book. WRITE THIS UP HERE!]]

I thought of another proof of the uniqueness of g_{2}^{1}’s. It is not in the book. This technique is useful for the exercises. The drawback is that it does not explicitly give K.

Theorem 46.2. If C is a curve of genus g ≥ 2 then C can not have two distinct g_{2}^{1}’s.

Proof. The proof is by contradiction. Suppose C has non-linearly equivalent divisors D and D′ such that |D| and |D′| are g_{2}^{1}’s. Then |D| and |D′| give morphisms p and p′ to P^{1} Taking the product gives a morphism φ = p × p′ : C → P^{1} × P^{1}. Since |D| and |D′| do not contain any of the same divisors the morphisms p and p′ collapse different points and so they are different. Let

C_{0} = φ(C) ⊂ P^{1} × P^{1} = Q ⊂ P^{3}

then C_{0} is a possibly singular but still integral curve on the quadric surface Q. By construction each ruling gives a g_{2}^{1} on C.

Let (a, b) be the bidegree of C_{0} on Q. It is clear that C_{0} is not a point since p and p′ are not constant. Let e be the degree of φ : C → C_{0}. A divisor of type (1, 0) on Q pulls back to a divisor on C_{0} of degree a. Since the morphism C → C_{0} has degree e this divisor of degree a pulls back to a degree ea divisor on C. On the other hand a divisor of type (1, 0) on Q pulls back to a divisor of degree 2 since it gives rise to p. Thus 2 = ae and similarly 2 = be.

Case 1: First suppose e = 2 and hence a = b = 1. Then C_{0} is of type (1, 1) so the normalization of C_{0} is of genus 0. But the arithmetic genus of C_{0} is (1 − 1)(1 − 1) = 0 so, since the genus can only go now upon normalization, we see that C_{0} must already be nonsingular. Thus φ maps C into P^{1}. But p is just the composition of φ with the first projection Q → P^{1} and this projection is the identity C_{0} = P^{1} ⊂ Q → P^{1}. Thus p and p′ collapse the same points, a contradiction.

Case 2: Next suppose e = 1 and hence a = b = 2. Then φ : C → C_{0} induces a birational morphism to the normalization of C_{0}. But the normalization of C_{0} has genus less than or equal to p_{a}(C_{0}) = (2 − 1)(2 − 1) = 1 so the genus of C is less than or equal to 1. In fact, if the genus of C is one then there are two distinct g_{2}^{1}’s. But under our assumption that g ≥ 2 we have the desired contradiction.

The above proof probably can be generalized to show that, for large enough genus, C can not have any g_{d}^{1}’s.

As in case 1 we saw that a type (1,1) curve C_{0} can not be singular since the normalization would then have negative genus. This might help with problems 3 and 4.

Next we give a statement of the theorem which will be proved next time.

Theorem 46.3 (Halphen). Let C be a curve of genus g ≥ 2. Then there exists a very ample nonspecial divisor D of degree d iff d ≥ g + 3.

The existence is actually very strong in the sense that a Zariski open subset has the property.

This generalizes our proof that g = 3 implies that there exists a very ample divisor of degree 6. Note that 2g + 1 = 7.

There can be very ample special divisors of degree less than g + 3.

47 Halphen’s Theorem

Today we will give one more general result about curves.

Theorem 47.1 (Halphen). Suppose C is a curve of genus g ≥ 2. Then there exists a very ample nonspecial divisor D of degree d iff d ≥ g + 3.

Proof. (⇒) Suppose D is a very ample nonspecial divisor of degree d. We will show that d ≥ g + 3. By Riemann-Roch ℓ(D) ≥ d + 1 − g. Since D is very ample D gives rise to an embedding C ↪ P^{n}. If n = 1 then C has genus 0, a contradiction. If n = 2 then somehow [I can not figure this out from my notes], this implies g < 2, a contradiction. Thus D gives an embedding C ↪ P^{n} with n ≥ 3. Thus ℓ(D) ≥ 4 and so d + 1 − g ≥ 4 so d ≥ g + 3 as claimed.

(⇐) This direction is a little harder. Assume d ≥ g + 3 and look for a very ample nonspecial divisor D. Remember our criterion for when a divisor is very ample? A divisor D is very ample iff dim |D − P − Q| = dim |D| − 2 for all points P, Q. By Riemann-Roch this is equivalent to the assertion that for all points P, Q,

dim |K − D| = dim |K − D + P + Q|.

As a “warm-up exercise” we fix d and ask the following. Does there exist a nonspecial effective divisor D of degree d?

Recall:

• For D to be nonspecial means that dim |K − D| = −1.

• For D to be special means that K − D is linearly equivalent to an effective divisor. Thus D is special iff there exists an effective divisor E such that D + E ∼ K, i.e., D + E is a canonical divisor.

The last condition shows that D is special iff D is “inside” some canonical divisor.

Since dim |K| = g − 1 there is a g − 1 dimensional family of effective canonical divisors. From each one there are only finitely many ways to get a special divisor. Thus the dimension of the family of all special divisors of degree d is at most g − 1. This shows that if d ≥ g then there exists nonspecial divisors of degree d. In fact, most are nonspecial.

Next we prove something which is more difficult. We want to show that there exists a nonspecial very ample divisor of any given degree d ≥ g + 3. It is enough to prove that the collection of nonspecial not very ample divisors has dimension ≤ g + 2.

Suppose D is nonspecial. Then for D to be not very ample means that there exists points P and Q so that

dim |D − P − Q| > dim |D| − 2.

A straightforward check using Riemann-Roch shows that this is equivalent to

dim |K − D| < dim |K − D + P + Q|.

Since D is assumed nonspecial dim |K − D| = −1 thus

dim |K − D + P + Q| ≥ 0.

So there exists an effective divisor E which is special and has degree d − 2 such that E ∼ D − P − Q. [[I thought I saw this yesterday but not even this makes any sense today.]]

By the above work the dimension of the set of effective special divisors is ≤ g − 1. Thus the set

{ E + P + Q : E is effective, special, degree d − 2, P, Q any points }

has dimension ≤ g + 1. But there is another wrinkle. We must count all divisors linearly equivalent to any such E + P + Q.

Somehow [[and I haven’t figured out how!!]] Hartshorne counts this and concludes that it has dimension ≤ g + 2. [[I thought sort of hard about this and can not see it, but I got confused at this point in class when I was taking notes so that may be why my notes do not reveal the truth.]]

48 Hurwitz’s Theorem

Suppose X and Y are curves (nonsingular, projective, over an algebraically closed field k). Suppose f : X → Y is a finite morphism. Then f induces a map of function fields K(Y) ↪ K(X) which makes K(X) into a finite extension of K(Y). The degree of f is the degree of the corresponding extension of function fields K(Y)/K(X). We say f is separable (not to be confused with separated) if K(X) is a separable field extension of K(Y).

Suppose P ∈ X maps to Q ∈ Y. Then there is an induced map of local rings f^{#} : O_{Q} ↪ O_{P}. Since X and Y are nonsingular curves this is an extension of discrete valuation rings. Let t ∈ O_{Q} be a uniformizing parameter for O_{Q} and let u ∈ O_{P} be a uniformizing parameter for O_{P}. Then f^{#}(t) ∈ m_{P} ⊂ O_{P}. The ramification index of P over Q is e_{P} := v_{P} (f^{#}(t)) ≥ 1.

We say that a point P lying over a point Q is wildly ramified if char k = p > 0 and p|e. Otherwise the ramification is called tame, i.e., when char k = 0 or char k = p |̸e. Note that we do not, as in some definitions, need to worry about the extension of residue fields being separable because k is algebraically closed.

Theorem 48.1 (Hurwitz). Suppose f : X → Y is a finite separable morphism of curves which is at most tamely ramified. Then

2g(X) − 2 = (degf)(2g(Y) − 2) + ∑_{P ∈X}(e_{P} − 1).

Note that as a consequence of this theorem, a finite separable tamely ramified morphism of curves has at most finitely many points of ramification. “Any good theorem has a counterexample.”

Example 48.2 (Frobenius map). If the morphism f is not separable then there can be infinitely many points of ramification. For example, let k be a field of characteristic p. The divisor corresponding to the invertible sheaf O_{P^{1}} gives a map from P^{1} → P^{1}. It is given explicitly simply by

(x : y) ↦ (x^{p} : y^{p}).

[[Is this true, or do I just want it to be true?]] For any point P ∈ P^{1} we have that e_{P} = p since the map corresponds to the map

k(t) → k(t) : t ↦ t^{p}.

Thus this map is widely ramified everywhere.

Hurwitz’s theorem. [[First part omitted.]] Thus we have the exact sequence

0 → f^{∗}Ω_{Y/k} → Ω_{X/k} → Ω_{X/Y} → 0

and Ω_{X/Y} is a torsion sheaf so it equals ⊕_{P ∈X}(Ω_{X/Y})_{P}. Next we study Ω_{X/Y} locally. Let P be a point in X lying over Q in Y. Then there is an exact sequence

0 → f^{∗}Ω_{Q} → Ω_{P} → Ω_{P/Q} → 0.

Here Ω_{P/Q} is the module of differentials of the local ring O_{P} over O_{Q}. Let t be a uniformizing parameter for O_{Q} and let u be a uniformizing parameter for O_{P}. Then Ω_{Q} is a free O_{Q}- module of rank 1 locally generated by dt.

Next let e = e_{P} = v_{P} (t), thus t = au^{e} in O_{P} (with a a unit in O_{P}). Differentiate to see that

dt = aeu^{e−1}du + u^{e}da.

The term u^{e}da is some mysterious element of Ω_{P}. We do not know anything about what a looks like so da can be very strange. If e = 0 in k this means that dt would be wild in the sense that we have no real control over da. Since we are assuming that the ramification is tame, char k |̸e, so e ≠ 0 in k. Thus aeu^{e−1}du ≠ 0. Define b ∈ O_{Q} by u^{e}da = bdu. Since v_{p}(b) ≥ e and v_{p}(aeu^{e−1}) = e − 1 we see that if dt = Adu then v_{p}(A) = e − 1. This means that u^{e−1}du is zero in Ω_{P/Q} [[but why is no lower power of u times du also 0.]] Thus Ω_{P/Q} is a principal module generated by du of length e − 1. Thus noncanonically

Ω_{P/Q} ≅ O_{P} /u^{e−1}.

We thus have an exact sequence

0 → f^{∗}(Ω_{Y}) → Ω_{X} → R → 0

where R = ⊕_{P} O_{P} /u_{P}^{e_{P} −1}. Let K_{X} denote the canonical divisor on X and K_{Y} be the canonical divisor on Y. Then Ω_{X} = L(K_{X}) and Ω_{Y} = L(K_{Y}).

Example 48.3. Assume char k ≠ 2. Let C be the cubic curve in P^{2} defined by y^{2} = x(x + 1)(x − 1). Let π by the degree 2 projection of C from ∞ onto P^{1} (i.e., the x-axis). There are 4 points of ramification, namely (−1, 0), (0, 0), (1, 0), and ∞. Hurwitz’s theorem is satisfied since

2 · 1 − 2 = 2(2 · 0 − 2) + ∑_{4 points}(2 − 1).

Example 48.4. Let C be a cubic curve in P^{2}. Let O be a point not on C and fix a copy of P^{1} ⊂ P^{2}. Then projection from O onto P^{1} defines a degree 3 map C → P^{1}. This map is ramified at a point P ∈ C exactly when the line from O to P is tangent to C. Assume there are no inflectional tangents so that the ramification degree of ramified points is 2. By Hurwitz’s theorem,

0 = 3(−2) + ∑(e_{P} − 1)

so there are 6 points of ramification 2. This means that from O one can draw 6 tangent lines to C.

Example 48.5. Let C be a cubic curve in P^{2}. Define a map φ : C → P^{1} by sending a point P to the intersection of the tangent space to C at P with P^{1} ⊂ P^{2}. By the above example this map has degree 6. By Hurwitz’s theorem,

0 = 6(−2) + ∑(e_{P} − 1)

so there are 12 points of ramification. [[Of course one must show (or set things up so) that the ramification of a point is at most 2.]] There are 3 ramification points where P^{1} intersects C. The other 9 ramification points are inflection points. Thus Hurwitz’s theorem implies that there are 9 inflections on a cubic.

49 Elliptic Curves

Definition 49.1. An elliptic curve C is a nonsingular projective curve of genus 1.

We know that any divisor of degree d ≥ 3 is very ample. Thus there is an embedding C ↪ P^{2} of C into P^{2} as a degree 3 curve.

Theorem 49.2. Suppose char k ≠ 2 and k is algebraically closed. If C is an elliptic curve then there is an embedding C ↪ P^{2} with equation

y^{2} = x(x − 1)(x − λ)

for some λ ∈ k, λ ≠ 1, 0.

Proof. Fix a point P_{0} ∈ C and let D = 3P_{0}. Then, after choosing section x_{0}, x_{1}, x_{2} ∈ H^{0}(L(3P_{0})) we obtain an embedding C ↪ P^{2}. Construct basis for the global sections of L(nP_{0}) for various n.

O_{C} L(P_{0}) L(2P_{0}) L(3P_{0}) L(4P_{0})  L(5P_{0})  L(6P_{0})
h^{0}  1  1  2  3  4  5  6
basis of H^{0} 1  1  1, x  1, x, y 1, x, y, x^{2} 1, x, y, x^{2}, xy 1, x, y, x^{2}, xy, x^{3}, y^{2}

Since the seven function 1, x, y, x^{2}, xy, x^{3}, y^{2} lie in a six dimensional space there must be a dependence relation

ay^{2} + bx^{3} + cxy + dx^{2} + ey + fx + g = 0

for some a, b, c, d, e, f, g ∈ k. Furthermore, both x^{3} and y^{2} occur with coefficient not equal to zero, because they are the only functions with a 6-fold pole at P_{0}. So replacing y be a suitable scalar multiple we may assume a = 1. Preparing to complete the square we rewrite the relation as

y^{2} + (cxy + ey) + (\frac{1}{2}cx + \frac{1}{2}e)^{2} − (\frac{1}{2}cx + \frac{1}{2}e)^{2} + bx^{3} + dx^{2} + fx + g = 0.

Replacing y by \frac{1}{2}cx + \frac{1}{2}e transforms the equation into

y^{2} = e(x − a)(x − b)(x − c)

where a, b, c, e ∈ k are new constants. Next absorb e to obtain

y^{2} = (x − a)(x − b)(x − c).

Now translate x by a to obtain an equation of the form

y^{2} = x(x − a)(x − b).

Multiply and divide by a^{3} to obtain

y^{2} = a^{3} \frac{x}{a}(\frac{x}{a} − 1)(\frac{x}{a} − \frac{b}{a}).

Replace x by \frac{x}{a} and absorb a^{3} to get

y^{2} = x(x − 1)(x − λ)

where λ ≠ 0, 1 since C is nonsingular.

In this proof we definitely used that k is algebraically closed to absorb constants and that the characteristic is not 2 in order to complete the square. Remark 49.3. Since A = k[x, y]/(y^{2} − x(x − 1)(x − λ)) is not a UFD the class group of C is nontrivial.

How unique is the representation y^{2} = x(x − 1)(x − λ)? The answer is that it is not unique. For example, replace x by x + 1 to obtain

y^{2} = (x + 1)x(x + 1 − λ)

then divide by −(1 − λ) to obtain

y^{2} = −(\frac{x}{1 − λ} + \frac{1}{−λ} \frac{x}{1 − λ}(\frac{x}{1 − λ} + 1).

Next replace x by −\frac{x}{1−λ} and absorb the minus sign to obtain

y^{2} = x(x − 1)(x − \frac{1}{1 − λ}).

By similar methods we get any of six choices:

λ ↦ λ, \frac{1}{λ}, 1 − λ, \frac{1}{1 − λ}, \frac{λ}{λ − 1}, 1 − \frac{1}{λ}.

Proposition 49.4. In the representation C in the form

y^{2} = x(x − 1)(x − λ)

, λ up to the group S_{3} depends only on C.

The first thing to struggle with is the dependency on the choice of P_{0}.

50 Automorphisms of Elliptic Curves

Let C be an elliptic curve so C has genus 1. Suppose char k ≠ 2. Fix a point P_{0} on C. Then the linear system |3P_{0}| gives an embedding

|3P_{0}| : C ↪ P^{2}.

(The divisor 3P_{0} is very ample by (3.3.3) and Riemann-Roch implies dim |3P_{0}| = 2.) For suitable choice of coordinates C is given by

y^{2} = x(x − 1)(x − λ), λ ∈ k, λ ≠ 0, 1.

The choice of λ is not unique since S_{3} acts by

λ ↦ λ, \frac{1}{λ}, 1 − λ, 1 − \frac{1}{λ}, \frac{1}{1 − λ}, \frac{λ}{λ − 1}.

Except for this ambiguity, λ is uniquely determined. Define

j(λ) = \frac{2^{8}(λ^{2} − λ + 1)^{3}}{λ^{2}(1 − λ)^{2}} .

Then Theorem 50.1. The j-invariant has the following properties.

• j ∈ k,

• j depends only on C,

• C_{1} ≅ C_{2} iff j(C_{1}) = j(C_{2}), and

• for all j ∈ k there is a curve C such that j(C) = j.

The theorem implies that there is a bijection of sets

{ isomorphism classes of elliptic curves / k } ⟶^{∼} { elements of k }.

This can be given a moduli theoretic interpretation. Example 50.2. Let E be the elliptic curve defined by

y^{2} = x(x − 1)(x + 1) = x^{3} − x.

Thus λ = −1 so

j = \frac{2^{8}(3)^{3}}{1 · 2^{2}} = 2^{6} · 3^{3} = 12^{3} = 1728.

If C is an elliptic curve then Aut C is transitive. So to study the automorphisms of C we need to cut down the number under consideration. To do this fix a point P_{0} and consider

Aut(C, P_{0}) = { automorphisms of C fixing P_{0} }.

Suppose C is given by the equation

y^{2} = x(x − 1)(x − λ)

and P_{0} = (0 : 1 : 0) is the point at infinity. Then the map σ defined by y ↦ −y and x ↦ x is an element of Aut(C, P_{0}). Note that σ is the covering transformation of some 2-to-1 map C → P^{1}.

Is σ unique? Let P be any point on C such that Q = σ(P) ≠ P. Then σ is a covering transformation of the morphism determined by the linear system |P + Q|. Associated to any degree 2 map to P^{1} there is a covering transformation (the hyperelliptic involution). Thus if R + S is an effective divisor of degree 2 which is not linearly equivalent to P + Q then we obtain a different degree 2 morphism to P^{1} and hence a different automorphism of degree two. The conclusion in class was that σ is not unique because you obtain many different automorphisms of degree 2 in the above manner. But there is no reason any of these should fix P_{0}. Furthermore, if Aut(C, P_{0}) really turns out to be the units in an order in a number field then there σ must be unique because there is only one nontrivial square root of −1.

Let C be as in the above example, so C is defined by

y^{2} = x(x − 1)(x + 1).

Then the map τ defined by

τ : \begin{cases}x ↦ −x \\ y ↦ iy \end{cases}

is an automorphism of C fixing P_{0} = (0 : 1 : 0). Furthermore σ = τ^{2} so Aut(C, P_{0}) contains at least Z/4Z. Is Aut(C, P_{0}) exactly Z/4Z? We will come back to this question later.

The map sending every point to its double under the group law is not an automorphism although it does fix P_{0}. Since k is algebraically closed there are 4 points on C which map to P_{0} under multiplication by 2.

Example 50.3. Let C be the genus 1 curve in P^{3} defined by

x^{3} + y^{3} + z^{3} = 0.

Suppose char k ≠ 2, 3. We want to find an equation for C of the form

y^{2} = x(x − 1)(x − λ).

Since ∞ is an inflectional tangent of y^{2} = x(x − 1)(x − λ) we hunt for an inflectional tangent of C. The line y + z = 0 meets C in an inflectional tangent since

x^{3} + y^{3} + z^{3} = x^{3} + (y + z)(y^{2} − yz + z^{2})

so setting y + z = 0 gives x^{3} = 0.

Next perform the change of variables z = z′ − y. The reason for doing this is so that z′ = 0 will give the point of intersection of C with y + z = 0. After substitution the equation becomes

x^{3} + y^{3} + (z′ − y)^{3} = 0

which is

x^{3} + y^{3} + z′^{3} − 3z′^{2}y + 3z′y^{2} − y^{3} = 0.

Setting z′ = 1 we obtain

x^{3} − 3y + 3y^{2} + 1 = 0

or equivalently, after factoring the -1 into x^{3},

y^{2} − y = x^{3} − \frac{1}{3}.

Next working modulo linear changes of variables gives

y^{2} − y + \frac{1}{4} = x^{3} − \frac{1}{3} + \frac{1}{4},

(y − \frac{1}{2})^{2} = x^{3} − \frac{1}{12},

y^{2} = x^{3} − \frac{1}{12},

y^{2} = x^{3} − 1,

y^{2} = (x − 1)(x − ω)(x − ω^{2}), where ω^{3} = 1.

In general

y^{2} = (x − a)(x − b)(x − c)

is equivalent to y^{2} = x(x − 1)(x − λ) where λ = \frac{a−c}{b−c}. In our situation this means

λ = \frac{1 − ω}{ω^{2} − ω} = \frac{1 − ω}{ω(ω − 1)} = −\frac{1}{ω} = −ω^{2} = \frac{1}{2} + \frac{1}{2}\sqrt{−3}.

Thus j(λ) = 0 and C is defined by

y^{2} = x(x − 1)(x + ω^{2}).

At this point Hartshorne tried to explicitly describe a degree 3 automorphism of C which fixes infinity. But try as he may it did not come out right. Note that x ↦ x and y ↦ −y defines a degree 2 automorphism of C.

To define a degree 3 automorphism look at the Fermat form of C

x^{3} + y^{3} + z^{3} = 0, P_{0} = (0 : 1 : −1).

Then

σ : \begin{cases}x ↦ ωx \\ y ↦ y \\ z ↦ z \end{cases}

is an automorphism of degree 3 fixing P_{0}. Interchanging y and z yields an automorphism of degree 2 fixing P_{0} and which commutes with σ. Thus Aut(C, P_{0}) is at least Z/6Z.

Now we approach the automorphism group abstractly.

Theorem 50.4. If C is an elliptic curve and P_{0} is a fixed point then

Aut(C, P_{0}) = \begin{cases}Z/2Z & if j ≠ 0, 12^{3} \\ Z/4Z & if j = 12^{3} ≠ 0, char k ≠ 3 \\ Z/6Z & if j = 0 ≠ 12^{3}, char k ≠ 3 \\ Z/12Z & if j = 0 = 12^{3}, char k = 3 \\ Z/24Z & if j = 0 = 12^{3}, char k = 2 \end{cases}

We do not prove the theorem now but the idea is to stare at the following square

\begin{aligned}& C & ⟶^{σ} & C \\ |2P_{0}| & ↓ & & ↓ & |2P_{0}| \\ & P^{1} & ⟶^{∃ τ} & P^{1} \end{aligned} .

Example 50.5. Consider

y^{2} = x(x − 1)(x + 1)

in characteristic 3. Define automorphisms

τ : \begin{cases}x ↦ −x \\ y ↦ iy \end{cases} ρ : \begin{cases}x ↦ x + 1 \\ y ↦ y \end{cases}

of orders 4 and 3. Clearly τρ = ρτ … but upon checking this, clearly τρ ≠ ρτ. Thus Aut(C, P_{0}) is obviously not cyclic of order 12, in fact it is an extension of S_{3}.

Let C be an elliptic curve and P_{0} a fixed point on C. Then we associate to the pair C, P_{0} the algebraic objects

• Aut(C, P_{0}),

• a group structure on C with P_{0} as the identity, and

• the endomorphism ring End(C, P_{0}).

The ring End(C, P_{0}) is defined to be the set of morphisms θ : C → C fixing P_{0}. By general facts about abelian varieties any such θ is in fact a group endomorphism. In characteristic 0 this endomorphism ring is either Z or an order in an imaginary quadratic number field. In characteristic p the endomorphism ring can be an order in a rank 4 quaternionic extension of Z. In characteristic p the endomorphism ring can not be just Z because of the Frobenius endomorphism which verifies a certain quadratic equation. [[How does this last statement work?]]

51 Moduli Spaces

For the rest of the semester I am going to talk about Jacobian varieties, variety of moduli and flat families. In each of these situations we parameterize “something or others” by points on a variety then apply algebraic geometry to the variety.

Here are some examples.

Start  Set  Parameter Space
A curve C Pic^{0} C = { divisors of degree Closed points of
0 modulo linear equivalence } Jacobian variety J.
Not just a set but a functor J represents the
Sch → Set  functor.
P_{k}^{n} and set of closed subschemes The Hilbert scheme
P (z) ∈ Q[z] Z ⊂ P_{k}^{n} with Hilbert  Hilb^{P} (P^{n})

polynomial P (z)

fix g > 0 isomorphism classes of curves The variety M_{g} of
of genus g  moduli. Fine if reps. the

functor, coarse if not.

P_{k}^{n}, set of cycles of dim. r and The Chow scheme.
degree d, and degree d in P^{n} modulo  It does not represent
dimension r rational equivalence.  a functor.

A cycle of dimension r is a Z-linear combination of subschemes of dimension r. If Z = ∑ n_{i}Y_{i} is a cycle of dimension r then deg Z = ∑ n_{i} deg(Y_{i}).

52 The Jacobian Variety

Fix a curve C of genus g. Then Pic^{0} C is the group of divisors of degree 0 modulo linear equivalence. The Jacobian is going to be a variety J whose closed points correspond to divisors D ∈ Pic^{0} C.

A closed point of J is a map φ : Spec k → J. Taking the product (over Spec k) of this map with the identity map C → C gives a commuting square

C ⟶^{φ′} C × J
↓  ↓
Spec k ⟶^{φ} J

Given an invertible sheaf D on C × J and a closed point t of J (given by a morphism φ : Spec k → J) define an invertible sheaf D_{t} on C by

D_{t} = φ′^{∗}D.

In this way we obtain a map from the closed points on J to Pic(C).

We first strengthen our requirement for the Jacobian of C by asking for a divisor D on C × J such that the map D ↦ D_{t} gives a correspondence

closed points(J) ⟶^{∼} Pic^{0}(C).

Grothendieck’s genius was to generalize all of this. The point is to replace Spec k-valued points of J with T -valued points of J where T is any scheme over k. Taking the product of T ⟶^{φ} J with C ⟶^{id_{C}} C we obtain a diagram

C × T ⟶^{φ′} C × J
↓  ↓
T ⟶^{φ} J

If L is an invertible sheaf on C × J then associate to L the invertible sheaf M = φ′^{∗}L.

Next we make a stronger requirement for the Jacobian of C. We require that there exist an invertible sheaf with the following universal property. For any scheme T and any invertible sheaf M on C × T which is “of degree 0 along the fibers” there exists a unique morphism φ : T → J such that M = φ′^{∗}L.

The point is that we are parameterizing families of divisor classes. We make the following tentative definition. It will not turn out to be the right one.

Definition 52.1. The Jacobian variety is a pair (J, L) with L an invertible sheaf on C × J such that for all schemes T and for all invertible sheaves M on C × T of degree 0 on the fibers, there exists a unique morphism φ : T → J such that M ≅ φ′^{∗}L.

What does “degree 0 on the fibers” mean? Suppose M is an invertible sheaf on C × T. Let t be a point in T. Thus t can be thought of as a map

ψ : Spec κ(t) → T

where κ(t) is the residue field at t. Tensoring this map with the identity map id : C → C yields a map

ψ′ : C_{t} = C ×_{k} Spec κ(t) → C × T.

If M is an invertible sheaf, define M_{t} to be the pullback ψ′^{∗}M. The degree of M_{t} as a divisor on the curve C_{t} makes sense. Although κ(t) may not be algebraically closed we can still define the degree of a divisor in a natural way.

Definition 52.2. M on C × T is said to be of degree 0 along the fibers if for all t ∈ T, deg M_{t} = 0.

Remark 52.3. One can show that the map t ↦ deg(M_{t}) is a continuous function T → Z.

There is still one subtlety. Our tentative definition of the Jacobian is bad since it obviously cannot exist.

Consider the diagram

C × T ⟶^{ψ′} C = C_{t} ⟶^{φ′} C × J
p↓  ↓  ↓
T ⟶^{ψ} Spec k ⟶^{φ} J

Pullback L on C × J via φ′ to obtain L_{t} = φ′^{∗}L. Then pullback L_{t} to obtain ψ′^{∗}L_{t} on C × T. Let N be any invertible sheaf on T. Then F = p^{∗}N ⊗ ψ′^{∗}L_{t} is an invertible sheaf on C × T. Furthermore p^{∗}N is trivial on the fibers so the fibers of F do not depend on N. [[I can not seem to figure out why this all leads to a contradiction!]] For some reason the map from T → J which corresponds to F must be the constant map sending everything to the point corresponding to φ. Since in general there are many possibilities for p^{∗}N this gives a contradiction. [[No matter what, I can not seem to understand this. I do not see why the map from T to J corresponding to F must be anything in particular.]]

So to the correct definition is that we require M to equal φ′^{∗}L in the quotient group

Pic^{0}(C × T/T) := Pic^{0}(C × T)/p^{∗} Pic T.

Now we have the correct definition of the Jacobian variety.

This says the pair (J, L) represents the functor

(F : Sch /k)^{o} → Set
T ↦ Pic^{0}(C × T/T).

Here (Sch /k)^{o} is the opposite category of the category of schemes over k obtained by reversing all the arrows. If ψ : T′ → T is a morphism then F sends ψ to the group homomorphism

Pic^{0}(C × T/T) → Pic^{0}(C × T′/T′)

defined by M ↦ ψ′^{∗}M. Here ψ′ is the map defined by the diagram

C × T′ ⟶^{ψ′} C × T
↓  ↓
T′  ⟶^{ψ} T

To say that (J, L) represents F is to say that Hom(T, J) = F (T) in the sense that the map φ ↦ φ^{∗}L is a bijection.

Existence of the Jacobian variety in general is difficult. When the genus is 1 we are lucky because the Jacobian is just the curve itself. For higher genus we must do something nontrivial. Let me give some consequences of existence.

52.1 Consequence of existence of the Jacobian

1. J is automatically an abelian group scheme.

Definition 52.4. If C is any category with finite products then a group object is G ∈ Ob(C) given with morphisms

μ : G × G → G (multiplication)
ρ : G → G  (inverse)
ε : {e} → G  (identity)

For any G ∈ Ob C there exists a natural representable functor h_{G} = Hom(·, G). To say that the functor h_{G} factors through the category Grp of groups is equivalent to saying that G is a group object. In particular since J represents a functor to the category of abelian groups it is an abelian group scheme.

2. The Zariski tangent space to J at 0.

Let P_{0} be the zero point of J. Let m_{0} ⊂ O_{P_{0}} be the maximal ideal of the local ring at P_{0}. Then the Zariski tangent space at P_{0} is the dual vector space (m_{0}/m_{0}^{2})′. [[I will figure this out later.]]

53 The Jacobian

In this lecture we reconsider more directly the Jacobian in greater depth. Let C be a curve of genus g. How can we parameterize divisors of degree 0 on C? The following definition gives a notion of a family of divisors of degree 0.

Definition 53.1. If T is any scheme, a family of divisor classes on C of degree 0 parameterized by T is an element of

Pic^{0}(C × T/T) := Pic^{0}(C × T)/p^{∗} Pic^{0} T

where p : C × T → T is projection.

Definition 53.2. If M is an invertible sheaf on C × T then the fiber M_{t} of M at t is the pullback of M by the map C_{t} → C × T. The diagram is

C_{t} ⟶ C × T
↓  ↓^{p}
Spec κ(t) ⟶ T

Definition 53.3. The Jacobian variety is a pair (J, L) where J is a scheme over k and L ∈ Pic^{0}(C × J/J) such that the pair (J, L) represents the functor

T ↦ Pic^{0}(C × T/T).

In other words, for every family M on C × T there exists a unique φ : T → J such that M = φ′^{∗}L.

C × T ⟶^{φ′} C × J
↓  ↓
T ⟶^{φ} J

53.1 Consequences of existence

53.1.1 Group structure

J is automatically an abelian group scheme. In particular,

μ : J × J → J

is a morphism.

Example 53.4. When g = 1 we obtain a group structure on the elliptic curve C.

Let L be any divisor class of degree 1. By Riemann-Roch h^{0}(L) = 1 + 1 − 1 = 1 so there is some s which spans H^{0}(L). This means that there is exactly one effective divisor of degree 1 corresponding to L, namely a point. Thus for any such L of degree 1 there is a unique point P ∈ C such that L ≅ L(P).

Fix a point P_{0} ∈ C. Consider the map C × C → C defined as follows. Send the pair ⟨P, Q⟩ to the point R corresponding to the degree 1 divisor

L(P + Q − P_{0}) ≅ L(R).

The map ⟨P, Q⟩ ↦ R defines a group structure on C. It is not obvious that this rule gives a morphism C × C → C. [[We have only defined a map on closed points. Hmm.]] 53.1.2 Natural fibration, dimension

Fix d > 0. Pick a basepoint P_{0} ∈ C. Consider the product

C^{d} = C × ⋯ × C.

Its closed points are (P_{1}, …, P_{d}) with P_{i} ∈ C.

Proposition 53.5. There exists a natural morphism C^{d} → J. On closed points it is

(P_{1}, …, P_{d}) ↦ ∑_{i=1}^{d} P_{i} − dP_{0}.

Proof. To give a morphism C^{d} → J is equivalent to giving an appropriate family on C ×T/T where T = C^{d}. Let

D = {(R, P_{1}, …, P_{d}) : R is one of the P_{i}}.

Since D is the sum of pullbacks of various diagonals it defines a divisor of degree d. Let p : C × T → C be projection onto C. Let

M = L(D) − d(p^{∗}L(P_{0})).

[[Probably if someone thinks about it for awhile she sees that M gives the desired family.]] Check that M = φ^{∗}L where φ is the map we want on closed points.

The first interesting case to consider is when d > 2g − 2. Then the fiber of a point j ∈ J of the map

\begin{matrix}C^{d} \\ ↓ \\ J \end{matrix}

is

C_{j}^{d} = {(P_{1}, …, P_{d}) : ∑ P_{i} − dP_{0} = j in J } = |j + dP_{0}|.

By Riemann-Roch the size of a fiber is thus

dim |j + dP_{0}| = d + 1 − g − 1 = d − g.

Let C^{(d)} by C^{d} modulo the action of S_{d}. Then C^{(d)} is of dimension d and the above map factors as a surjection C^{(d)} → J with fibers of dimension d − g. Thus J has dimension g.

A second interesting special case is when d = g. If P_{1} + ⋯ + P_{g} is chosen sufficiently generally then |P_{1} + ⋯ + P_{g}| = {P_{1} + ⋯ + P_{g}}. To see this choose each P_{i} so that it is not a basepoint of |K − P_{1} − ⋯ − P_{i−1}|. Then ℓ(K − P_{1} − ⋯ − P_{g}) = 0 so by Riemann-Roch

ℓ(P_{1} + ⋯ + P_{g}) = g + 1 − g + ℓ(K − P_{1} − ⋯ − P_{g}) = 1.

This means that the general fiber of C^{(g)} → J has degree 1. Thus it is a birational map, but it is not necessarily an isomorphism since there can exist special divisors of degree g.

Let U be the open subset of nonspecial divisors on J. This map hints at the construction of the Jacobian since it gives rise to a “germ of the group law” or a “group chunk”

U × U → U
⟨D, E⟩ ↦ D + E − gP_{0}.

The next step is to try to fill out the group law to get a group law on J. This is Weil’s method.

53.1.3 The Zariski tangent space

We compute the Zariski tangent space to J at 0.

Definition 53.6. The Zariski tangent space to 0 ∈ J is

T_{J,0} = (m_{0}/m_{0}^{2})^{∨}.

Let ε be such that ε^{2} = 0. The elements of T_{J,0} are in one-to-one correspondence with morphisms

Spec k[ε] → J

sending the point of Spec k[ε] to 0 ∈ J. The corresponding map O_{0} → k[ε] sends m_{0} to (ε). Since ε^{2} = 0 this map factors through m_{0}/m_{0}^{2} giving an element of T_{J,0}.

Now rewrite T_{J,0} as

T_{J,0} = Hom_{0}(Spec k[ε], J)

where Hom_{0} means all homomorphisms sending the point of Spec k[ε] to 0 ∈ J. The next step is to use the fact that J represents a certain functor. The point is that

Hom_{0}(Spec k[ε], J) = ker(Pic^{0}(C × T) → Pic^{0}(C)).

[[This is a direct check using the fact that 0 ∈ J, by representability, must correspond to O_{C} on C.]]

The last step is to write down some exact sequences.

54 Flatness

[[The orals are in 959 1-3pm on Monday and Wednesday. Monday Janos, William, then Nghi talk. Wednesday Wayne, Amod, then Matt talk.]]

The outline for this lecture is

1. technical definition

2. significance

3. properties

4. examples

54.1 Technical definitions

Let A be a commutative ring with identity.

Definition 54.1. An A-module is flat if the functor M ⊗_{A} · is exact.

For M ⊗ · to be exact means that whenever

0 → N′ → N → N′′ → 0

is an exact sequence of A-modules then

0 → M ⊗ N′ → M ⊗ N → M ⊗ N′′ → 0

is exact. Note that M ⊗ · is right exact even if M is not flat. The salient property of a flat module is that it preserves injectivity.

54.1.1 General nonsense

If A is Noetherian then M is flat if the functor M ⊗· preserves the exactness of any sequence of finitely generated A-modules. Even better, M is flat if for any ideal I ⊂ A the map M ⊗ I → M ⊗ A is injective.

Definition 54.2. Suppose A → B is a morphism of rings. Then B is flat over A if B is flat as an A-module.

54.2 Examples

Example 54.3. Suppose A is a ring and S a multiplicative set. Then S^{−1}A is a flat A-module, i.e., the functor M ↦ S^{−1}M is exact.

Example 54.4. If A is a Noetherian local ring then the completion Â of A at its maximal ideal is flat.

Example 54.5. The first example of flatness was in Serre’s GAGA. He called a flat module an “exact couple”. The example is

A = C[x_{1}, …, x_{n}] ↪ C{x_{1}, …, x_{n}} ⊂ Â

in which C{x_{1}, …, x_{n}}, the ring of convergent power series, is flat over A.

A module B → C is faithfully flat if C ⊗_{B} M = 0 implies M = 0. Suppose A → B → C with C flat over A and C faithfully flat over B. Then B is flat over A. This is how Serre proved that his module was flat.

Theorem 54.6. Suppose A, m is a local Noetherian ring and M is a finitely generated flat module. Then M is free.

Proof. Take a minimal set of generators for M. Nakayama’s lemma tells us how this can be done. The quotient M/mM is a vector space over A/m so it has a basis, say \overline{m}_{1}, …, \overline{m}_{r}. By Nakayama’s lemma these generate M. (The module M/(m_{1}, …, m_{r}) is sent to itself by m since every x ∈ M is equivalent to an element of (m_{1}, …, m_{r}) plus something in mM. Thus M/(m_{1}, …, m_{r}) = 0.) This gives a surjection A^{r} → M. Let Q be the kernel so we have an exact sequence

0 → Q → A^{r} → M → 0.

It is a general fact that if A is Noetherian, M is flat, and

0 → R → S → M → 0

is exact then

0 → R ⊗ N → S ⊗ N → M ⊗ N → 0

is exact for any A-module N. This is proved by using the the fact that flatness implies the vanishing of Tor_{1}^{A}(M, ·).

Using the general fact tensor with k = A/m to obtain an exact sequence

0 → Q ⊗ k → k^{r} → M/mM → 0.

Since k^{r} and M/mM are k-vector spaces of the same dimension and the map k^{r} → M/mM is surjective it must also be injective. Thus Q ⊗ k = 0. But Q is finitely generated since A is Noetherian and Q is a submodule of A^{r}. Thus Nakayama’s lemma implies Q = 0 so M ≅ A^{r} is free.

Theorem 54.7. Suppose A is a d.v.r. with maximal ideal m = (t). Let M be any A-module. Then the following are equivalent.

(i) M is flat,

(ii) M is torsion free,

(iii) M ⟶^{t}M is injective.

Proof. (i)⇒(ii) Since A is a domain

0 → A ⟶^{x} A

is exact for any x ∈ A. By flatness of M

0 → M ⟶^{x} M

is exact so M is torsion free.

(ii)⇒(iii) is trivial

(iii)⇒(i) It is enough to check exactness for any ideal a = (t^{n}) ⊂ A. Why is the map M ⊗ a → M ⊗ A an isomorphism? Because under the natural identification of both modules with M the map is just multiplication by t^{n} which is injective by assumption. The diagram is

M ≅ M ⊗ a → M ⊗ A ≅ M
m ↦ m ⊗ t^{n} ↦ m ⊗ t^{n} ↦ t^{n}m

Example 54.8. If M is a projective A module then M is flat.

54.3 Algebraic geometry definitions

Definition 54.9. Let F be a coherent sheaf on X and let f : X → Y be a morphism. Then F is flat over Y if for all x ∈ X, F_{x} is a flat O_{Y,f(x)}-module under the map O_{Y,f(x)} → O_{X,x}. The morphism f : X → Y is flat if O_{X} is flat over Y.

[[“Even if f is finite, F_{x} might not be finite over O_{Y}. It is only finite over the semilocal ring.”]]

Example 54.10. 1. An open immersion U ↪ X is flat since the local rings are the same.

2. A composition of flat morphisms is flat.

3. Base extension preserves flatness. Thus if X → Y is flat and Y′ → Y then X′ =

X ×_{Y} Y′ is flat over Y′.

X′ = X ×_{Y} Y′ ⟶ X
↓  ↓
Y′  ⟶ Y

This is an exercise in local rings.

4. The product of flat morphisms is flat. [[What does this mean?]]

54.4 Families

Flatness is used for expressing the notion of a family of subschemes. For example let T be any scheme and let X ⊂ P_{T}^{n} be a closed subscheme. Then {X_{t} : t ∈ T } is the family. Here X_{t} is defined by the diagram

X_{t} = X ×_{k} Spec κ(t) ⟶ X
↓  ↓
Spec κ(t)  ⟶ T

where κ(t) is the residue field of t ∈ T.

This notion of family is bad since it allows for things which we do not want.

Example 54.11. Let T = A^{1} and X ⊂ P_{T}^{1} = A^{1} × P^{1} be the union of the coordinate axis. Then for any t ≠ 0, X_{t} is a point. But X_{0} = P^{1}.

Hard experience has lead the old pros to agree that the notion of family should require that X → T is flat. This rules out the above example since X → T is not flat. Indeed, X is defined on an open affine by xt = 0 so the coordinate ring is M = k[x, t]/(xt). The localization of M at (x, t) has torsion as a k[t]_{(t)}-module. The element x is killed by t. Since k[t]_{(t)} is a d.v.r. it follows that M is not a flat.

A vague generalization is the following. Suppose T is a nonsingular curve and X ⊂ P_{T}^{n}. Then X is flat over T iff X has no associated point (i.e., generic points of embedded and irreducible components) whose image is a closed point of T. Equivalently one could say “every component of X dominates T.” In our example the t-axis dominates T but the x-axis does not.

Example 54.12 (Bad). Let X ⊂ P_{T}^{1} = A^{1} × P^{1} be defined by (x^{2}, xt). When t ≠ 0 the fiber is P_{0} = Spec k[t] since x localizes away. If t = 0 the fiber is given by the ring

k[x, t]/(x^{2}, xt) ⊗_{k[t]} k[t]/(t) ≅ k[x]/(x^{2})

which is P_{0} with multiplicity 2 structure.

Example 54.13 (Good). Let X ⊂ P_{T}^{1} be defined by x(x − t). Then for t = a ≠ 0 the fiber is P_{0} + P_{a}. When t = 0 the fiber is 2P_{0}. This is good.

Theorem 54.14. Suppose T is a connected scheme. [[Must there be a finiteness assumption on T?]] If X is flat over T then the Hilbert polynomials P_{X_{t}}(z) ∈ Q[z] are independent of t. If T is integral then the converse is also true.

Proof. Recall that for any scheme Y,

P_{Y} (m) = h^{0}(O_{Y} (m)) for all m ≫ 0.

We just need to show h^{0}(O_{X_{t}}(m)) is independent of t for all m sufficiently large. Since T is connected we reduce to the case T is affine. Furthermore we may assume T = Spec A, with A, m a local Noetherian ring. [[“Can get from any local ring to any other by successive specializations and generalizations. Also use the fact that flatness is preserved by base extension.”]] Claims.

1. For all m ≫ 0, H^{i}(O_{X}(m)) = 0 for i > 0 and H^{0}(O_{X}(m)) is a free finitely generated A-module.

2. For all points t ∈ T and all m ≫ 0,

H^{0}(O_{X_{t}}(m)) = H^{0}(O_{X}(m)) ⊗_{A} κ(t).

Together these two claims imply the theorem.

55 Theorem about flat families

Theorem 55.1. Let X ⊂ P_{T}^{n} be a closed subscheme where T is a connected scheme of finite type over k. If X is flat over T, then the Hilbert polynomial P_{X_{t}} is independent of t ∈ T (all scheme points – not just closed points!). Conversely, if T is integral and P_{X_{t}} is independent of t then X is flat over T.

Proof. See Chapter III, section 9 of Hartshorne.

56 Examples of Flat Families

Example 56.1 (Two lines). Consider the parameterized family of curves defined as follows. For each t ∈ T = Spec k[t] associate the union of the x-axis and a line parallel to the y-axis which intersects the z-axis at z = t. Thus for t ≠ 0 the fiber X_{t} consists of the disjoint union of two lines. For t = 0 the fiber consists of two lines meeting at the origin.

A cohomological computation shows that the Hilbert polynomials are

H_{X_{t}}(z) = \begin{cases}2z^{2} + 1 − (−1), & for t ≠ 0 \\ 2z^{2} + 1, & for t = 0 \end{cases}

What is wrong? Let C′ be the subscheme of P_{T −0}^{3} defined by the ideal

I_{C′} = (y, z) ∩ (x, z − t) = (xy, xz, yz − tyw, z^{2} − tzw).

(The intersection of the two ideals is the product of the given generators since they form a regular sequence.) Since the Hilbert polynomial of C′ is constant over the fibers we know that C′ is flat over T − 0. We have the following proposition.

Proposition 56.2. Let T = Spec k[t] and let X ⊂ P_{T −0}^{n} be a closed subscheme which is flat over T − 0. Then there is a unique closed subscheme \overline{X}, flat over T, whose restriction to P_{T −0}^{n} is X. Furthermore \overline{X} is the scheme-theoretic closure of X.

Proof. See Chapter III, Section 9 of Hartshorne.

In our situation a natural guess for \overline{X} is the closed subscheme C defined by the ideal

(xy, xz, yz − tyw, z^{2} − tzw) ⊂ k[t][x, y, z, w].

As a bonus, if we can show C is flat over T then we will know that it is the scheme-theoretic closure of C′.

To test if C is flat is suffices to show that the Hilbert polynomials are constant over the fibers. We know this when t ≠ 0 so we need only check that the Hilbert polynomial at t = 0 is 2z^{2} + 2. At t = 0 the defining ideal becomes

(xy, xz, yz, z^{2}) = (x, z) ∩ (y, z) ∩ (x, y, z)^{2}.

This looks like the union of the x and y with an embedded point at the origin. The arithmetic genus of the degree d = 2 plane curve defined by (z, xy) is

p_{a} = \frac{1}{2}(d − 1)(d − 2) = 0

so it has Hilbert polynomial 2z + 1. The ith graded piece of k[x, y, z, w]/(xy, xz, yz, z^{2}) has dimension one more than the dimension of the ith graded piece of k[x, y, z, w]/(z, xy). (In degree i the first ring has w^{i−1}z whereas the latter does not.) The Hilbert polynomial is then

P_{C_{0}}(z) = (2z + 1) + 1 = 2z + 2.

Thus P_{C_{t}}(z) = 2z + 2 for all t so by the big theorem from the last lecture C ⊂ P_{T}^{3} is a flat family.

Note that although the Hilbert polynomial is constant along the fibers the cohomology of C_{0} is different than that of the other fibers. This family can sort of be thought of as 2 planes meeting at a point in 4-dimensional space.

“This is an example of the genius of Grothendieck. The Italians knew that when two planes came together the genus changed but they were not able to deal with it like Grothendieck did.”

Example 56.3 (The Twisted Cubic). This is the last example of the course. Some say this is the only example in algebraic geometry. Let T = Spec k[t]. The fiber over 1 in our family will be C_{1} ⊂ P_{k}^{3} defined by

I_{C_{1}} = (xy − zw, x^{2} − yw, y^{2} − xz).

How can C_{1} give rise to a family? One way is to assign weights to the variables then homogenize with respect to a new variable t. Assign weights as in the following table.

variable x y z w
weight 8 4 2 1

Introducing a new variable t of weight 1 and homogenizing we obtain the ideal

I = (xy − t^{9}zw, x^{2} − t^{11}yw, xz − t^{2}y^{2}) ⊂ k[t][x, y, z, w].

This ideal defines a closed subscheme C of P_{T}^{3} since I is homogeneous as an ideal in the polynomial ring k[t][x, y, z, w].

Is C a flat family? When t ≠ 0, C_{t} is a curve. But C_{0} is defined by the ideal

(xy, x^{2}, xz) = (x) ∩ (x, y, z)^{2}

so it is the entire y–z plane plus an embedded point. This is way too large so C is not a flat family. There must be some torsion. Indeed,

xyz − t^{9}z^{2}w, xyz − t^{2}y^{3} ∈ I

so

t^{2}(y^{3} − t^{7}z^{2}w) = t^{2}y^{3} − t^{9}z^{2}w ∈ I.

Thus y^{3} − t^{7}z^{2}w is a torsion element of k[t][x, y, z, w]/I over k[t]. So now add y^{3} − t^{7}z^{2}w to I. Let C′ be the closed subscheme defined by the ideal

I′ = (y^{3} − t^{7}z^{2}w, xz − t^{2}y^{2}, x^{2} − t′′yw, xy − t^{9}zw).

When t = 0 the fiber C′_{0} is defined by

(xy, x^{2}, xz, y^{3}) = (x, y^{3}) ∩ (x^{2}, xy, xz, z^{2}, y^{3}).

One can show C′_{0} is of degree 3 and has arithmetic genus 0. Thus C′ is a flat family. The basis we have given for I′ is called a Gröbner basis.

[[At this point the class came to an end. Everyone clapped, then clapped some more. It is clear that we really appreciated Hartshorne. He did a great job.]]

57 Homework problems

I did not type up solutions to the first homework set. I did type of solutions to the other homework sets.

57.1 Exercise on basic cohomology and abstract nonsense

Exercise 57.1. For each of the following categories, decide whether the category has enough projective objects (i.e. every object is a quotient of a projective object).

a) Ab(X), where X is a topological space,

b) Mod(X), where (X, O_{X}) is a ringed space,

c) Qco(X), where X is a Noetherian scheme,

d) Qco(X), where X is a Noetherian affine scheme.

Exercise 57.2. Let X = P_{k}^{1}, over an algebraically closed field k. Using only material from ChIII, §1,2 (esp. Exercise 2.2) show

a) For any coherent sheaf F on X,

H^{i}(X, F) = 0 for all i ≥ 2.

[Hint: Treat the case F torsion and F torsion-free separately; in the torsion-free case, tensor the exact sequence of Exercise 2.2 by F.]

b) Show that for all ℓ > 0, H^{1}(X, O_{X}(ℓ)) = 0.

Exercise 57.3. Let X be an integral Noetherian scheme.

a) Show that the sheaf K^{∗} (cf. II §6) is flasque. Conclude that Pic X ≅ H^{1}(X, O_{X}^{∗}).

b) Give an example of a Noetherian affine scheme with H^{1}(X, O_{X}^{∗}) ≠ 0.

Exercise 57.4. Let X be a Noetherian scheme.

a) Show that the sheaf G constructed in the proof of (3.6) is an injective object in the category Qco(X) of quasi-coherent sheaves on X. Thus Qco(X) has enough injectives.

b) Show that any injective object of Qco(X) is flasque. [Hints: The method of proof of (2.4) will not work, because O_{U} is not quasi-coherent on X in general. Instead, use (II, Ex. 5.15) to show that if I ∈ Qco(X) is injective, and if U ⊂ X is an open subset, then I|_{U} is an injective object of Qco(U). Then cover X with open affine...]

c) Conclude that one can compute cohomology as the derived functors of Γ(X, ·), considered as a functor from Qco(X) to Ab.

57.2 Chapter III, 4.8, 4.9, 5.6

57.2.1 Exercise III.4.8: Cohomological Dimension

Let X be a Noetherian separated scheme. We define the cohomological dimension of X, denoted cd(X), to be the least integer n such that H^{i}(X, F) = 0 for all quasi-coherent sheaves F and all i > n. Thus for example, Serre’s theorem (3.7) says that cd(X) = 0 if and only if X is affine. Grothendieck’s theorem (2.7) implies that cd(X) ≤ dim X.

(a) In the definition of cd(X), show that it is sufficient to consider only coherent sheaves on X.

(b) If X is quasi-projective over a field k, then it is even sufficient to consider only locally free coherent sheaves on X.

(c) Suppose X has a covering by r + 1 open affine subsets. Use Čech cohomology to show that cd(X) ≤ r.

(d) If X is a quasi-projective variety of dimension r over a field k, then X can be covered by r + 1 open affine subsets. Conclude that cd(X) ≤ dim X.

(e) Let Y be a set-theoretic complete intersection of codimension r in X = P_{k}^{n}. Show that cd(X − Y) ≤ r − 1.

Proof. (a) It suffices to show that if, for some i, H^{i}(X, F) = 0 for all coherent sheaves F, then H^{i}(X, F) = 0 for all quasi-coherent sheaves F. Thus suppose the ith cohomology of all coherent sheaves on X vanishes and let F be quasi-coherent. Let (F_{α}) be the collection of coherent subsheaves of F, ordered by inclusion. Then by (II, Ex. 5.15e) \underrightarrow{lim} F_{α} = F, so by (2.9)

H^{i}(X, F) = H^{i}(X, \underrightarrow{lim} F_{α}) = \underrightarrow{lim} H^{i}(X, F_{α}) = 0.

(b) Suppose n is an integer and H^{i}(X, F) = 0 for all coherent locally free sheaves F and integers i > n. We must show H^{i}(X, F) = 0 for all coherent F and all i > n, then applying (a) gives the desired result. Since X is quasiprojective there is an open immersion

i : X ↪ Y ⊂ P_{k}^{n}

with Y a closed subscheme of P_{k}^{n} and i(X) open in Y. By (II, Ex. 5.5c) the sheaf F on X pushes forward to a coherent sheaf on F′ = i_{∗}F on Y. By (II, 5.18) we may write F′ as a quotient of a locally free coherent sheaf E′ on Y. Letting R′ be the kernel gives an exact sequence

0 → R′ → E′ → F′ → 0

with R′ coherent (it’s the quotient of coherent sheaves). Pulling back via i to X gives an exact sequence

0 → R → E → F → 0

of coherent sheaves on X with E locally free. The long exact sequence of cohomology shows that for i > n, there is an exact sequence

0 = H^{i}(X, E) → H^{i}(X, F) → H^{i+1}(X, R) → H^{i+1}(X, E) = 0.

H^{i}(X, E) = H^{i+1}(X, E) = 0 because we have assumed that, for i > n, cohomology vanishes on locally free coherent sheaves. Thus H^{i}(X, F) ≅ H^{i+1}(X, R). But if k = dim X, then Grothendieck vanishing (2.7) implies that H^{k+1}(X, R) = 0 whence H^{k}(X, F) = 0. But then applying the above argument with F replaced by R shows that H^{k}(X, R) = 0 which implies H^{k−1}(X, F) = 0 (so long as k − 1 > n). Again, apply the entire argument with F replaced by R to see that H^{k−1}(X, R) = 0. We can continue this descent and hence show that H^{i}(X, F) = 0 for all i > n.

(c) By (4.5) we can compute cohomology by using the Čech complex resulting from the cover 𝔘 of X by r + 1 open affines. By definition C^{p} = 0 for all p > r since there are no intersections of p + 1 ≥ r + 2 distinct open sets in our collection of r + 1 open sets. The Čech complex is

C^{0} → C^{1} → ⋯ → C^{r} → C^{r+1} = 0 → 0 → 0 → · · · .

Thus if F is quasicoherent then Ȟ^{p}(𝔘, F) = 0 for any p > r which implies that cd(X) ≤ r.

(d) I will first present my solution in the special case that X is projective. The more general case when X is quasi-projective is similar, but more complicated, and will be presented next. Suppose X ⊂ P^{n} is a projective variety of dimension r. We must cover X with r + 1 open affines. Let U be nonempty open affine subset of X. Since X is irreducible, the irreducible components of X − U all have codimension at least one in X. Now pick a hyperplane H which doesn’t completely contain any irreducible component of X − U. We can do this by choosing one point P_{i} in each of the finitely many irreducible components of X − U and choosing a hyperplane which avoids all the P_{i}. This can be done because the field is infinite (varieties are only defined over algebraically closed fields) so we can always choose a vector not orthogonal to any of a finite set of vectors. Since X is closed in P^{n} and P^{n} − H is affine, (P^{n} − H) ∩ X is an open affine subset of X. Because of our choice of H, U ∪ ((P^{n} − H) ∩ X) is only missing codimension two closed subsets of X. Let H_{1} = H and choose another hyperplane H_{2} so it doesn’t completely contain any of the (codimension two) irreducible components of X − U − (P^{n} − H_{1}). Then (P^{n} − H_{2}) ∩ X is open affine and U ∪ ((P^{n} − H_{1}) ∩ X) ∪ ((P^{n} − H2) ∩ X) is only missing codimension three closed subsets of X. Repeating this process a few more times yields hyperplanes H_{1}, ⋯, H_{r} so that

U, (P^{n} − H_{1}) ∩ X, …, (P^{n} − H_{r}) ∩ X

form an open affine cover of X, as desired.

Now for the quasi-projective case. Suppose X ⊂ P^{n} is quasi-projective. From (I, Ex. 3.5) we know that P^{n} minus a hypersurface H is affine. Note that the same proof works even if H is a union of hypersurfaces. We now proceed with the same sort of construction as in the projective case, but we must choose H more cleverly to insure that (P^{n} − H) ∩ X is affine. Let U be a nonempty affine open subset of X. As before pick a hyperplane which doesn’t completely contain any irreducible component of X − U. Since X is only quasi-projective we can’t conclude that (P^{n} − H) ∩ X is affine. But we do know that (P^{n} − H) ∩ \overline{X} is affine. Our strategy is to add some hypersurfaces to H to get a union of hypersurfaces S so that

(P^{n} − S) ∩ \overline{X} = (P^{n} − S) ∩ X.

But, we must be careful to add these hypersurfaces in such a way that ((P^{n} − S) ∩ X) ∪ U is missing only codimension two or greater subsets of X. We do this as follows. For each irreducible component Y of \overline{X} − X choose a hypersurface H′ which completely contains Y but which does not completely contain any irreducible component of X − U. That this can be done is the content of a lemma which will be proved later (just pick a point in each irreducible component and avoid it). Let S by the union of all of the H′ along with H. Then P^{n} − S is affine and so

(P^{n} − S) ∩ X = (P^{n} − S) ∩ \overline{X}

is affine. Furthermore, S properly intersects all irreducible components of X − U, so ((P^{n} − S) ∩ X) ∪ U is missing only codimension two or greater subsets of X. Repeating this process as above several times yields the desired result because after each repetition the codimension of the resulting pieces is reduced by 1.

Lemma 57.5. If Y is a projective variety and p_{1}, …, p_{n} is a finite collection of points not on Y, then there exists a (possibly reducible) hypersurface H containing Y but not containing any of the p_{i}.

By a possibly reducible hypersurface I mean a union of irreducible hypersurfaces, not a hypersurface union higher codimension varieties.

Proof. This is obviously true and I have a proof, but I think there is probably a more algebraic proof. Note that k is infinite since we only talk about varieties over algebraically closed fields. Let f_{1}, ⋯, f_{m} be defining equations for Y. Thus Y is the common zero locus of the f_{i} and not all f_{i} vanish on any p_{i}. I claim that we can find a linear combination ∑ a_{i}f_{i} of the f_{i} which doesn’t vanish on any p_{i}. Since k is infinite and not all f_{i} vanish on p_{1}, we can easily find a_{i} so that ∑ a_{i}f_{i}(p_{1}) ≠ 0 and all the a_{i} ≠ 0. If ∑ a_{i}f_{i}(p_{2}) = 0 then, once again since k is infinite, we can easily “jiggle” the a_{i} so that ∑ a_{i}f_{i}(p_{2}) ≠ 0 and ∑ a_{i}f_{i}(p_{1}) is still nonzero. Repeating this same argument for each of the finitely many points p_{i} gives a polynomial f = ∑ a_{i}f_{i} which doesn’t vanish on any p_{i}. Of course I want to use f to define our hypersurface, but I can’t because f might not be homogeneous. Fortunately, this is easily dealt with by suitably multiplying the various f_{i} by the defining equation of a hyperplane not passing through any p_{i}, then repeating the above argument. Now let H be the hypersurface defined by f = ∑ a_{i}f_{i}. Then by construction H contains Y and H doesn’t contain any p_{i}.

(e) Suppose Y is a set-theoretic complete intersection of codimension r in X = P_{k}^{n}. Then Y is the intersection of r hypersurfaces, so we can write Y = H_{1} ∩ ⋯ ∩ H_{r} where each H_{i} is a hypersurface. By (I, Ex. 3.5) X − H_{i} is affine for each i, thus

X − Y = (X − H_{1}) ∪ ⋯ ∪ (X − H_{r})

can be covered by r open affine subsets. By (c) this implies cd(X − Y) ≤ r − 1 which completes the proof.

57.2.2 Exercise III.4.9

Let X = Spec k[x_{1}, x_{2}, x_{3}, x_{4}] be affine four-space over a field k. Let Y_{1} be the plane x_{1} = x_{2} = 0 and let Y_{2} be the plane x_{3} = x_{4} = 0. Show that Y = Y_{1} ∪ Y_{2} is not a set-theoretic complete intersection in X. Therefore the projective closure \overline{Y} in P_{k}^{4} is not a set-theoretic complete intersection.

Proof. By (Ex. 4.8e) it suffices to show that H^{2}(X − Y, O_{X−Y}) ≠ 0. Suppose Z is a closed subset of X, then by (Ex. 2.3d), for any i ≥ 1, there is an exact sequence

H^{i}(X, O_{X}) → H^{i}(X − Z, O_{X−Z}) → H_{Z}^{i+1}(X, O_{X}) → H^{i+1}(X, O_{X}).

By (3.8), H^{i}(X, O_{X}) = H^{i+1}(X, O_{X}) = 0 so H^{i}(X − Z, O_{X−Z}) = H_{Z}^{i+1}(X, O_{X}). Applying this with Z = Y and i = 2 shows that

H^{2}(X − Y, O_{X−Y}) = H_{Y}^{3} (X, O_{X}).

Thus we just need to show that H_{Y}^{3} (X, O_{X}) ≠ 0.

Mayer-Vietoris (Ex. 2.4) yields an exact sequence

H_{Y_{1}}^{3}(X, O_{X}) ⊕ H_{Y_{2}}^{3}(X, O_{X}) → H_{Y}^{3} (X, O_{X}) →
H_{Y_{1}∩Y_{2}}^{4}(X, O_{X}) → H_{Y_{1}}^{4}(X, O_{X}) ⊕ H_{Y_{2}}^{4}(X, O_{X})

As above, H_{Y_{1}}^{3}(X, O_{X}) = H^{2}(X − Y_{1}, O_{X−Y_{1}}). But X − Y_{1} is a set-theoretic complete intersection of codimension 2 so cd(X − Y_{1}) ≤ 1, whence H^{2}(X − Y_{1}, O_{X−Y_{1}}) = 0. Similarly

H^{2}(X − Y_{2}, O_{X−Y_{2}}) = H^{3}(X − Y_{1}, O_{X−Y_{1}}) = H^{3}(X − Y_{2}, O_{X−Y_{2}}) = 0.

Thus from the above exact sequence we see that H_{Y}^{3} (X, O_{X}) = H_{Y_{1}∩Y_{2}}^{4}(X, O_{X}).

Let P = Y_{1} ∩Y_{2} = {(0, 0, 0, 0)}. We have reduced to showing that H_{P}^{4} (X, O_{X}) is nonzero. Since H_{P}^{4} (X, O_{X}) = H^{3}(X − P, O_{X−P}) we can do this by a direct computation of H^{3}(X − P, O_{X−P}) using Čech cohomology. Cover X − P by the affine open sets U_{i} = {x_{i} ≠ 0}. Then the Čech complex is

k[x_{1}, x_{2}, x_{3}, x_{4}, x_{1}^{−1}] ⊕ ⋯ ⊕ k[x_{1}, x_{2}, x_{3}, x_{4}, x_{4}^{−1}] ⟶^{d_{0}}

k[x_{1}, x_{2}, x_{3}, x_{4}, x_{1}^{−1}, x_{2}^{−1}] ⊕ ⋯ ⊕ k[x_{1}, x_{2}, x_{3}, x_{4}, x_{3}^{−1}, x_{4}^{−1}] ⟶^{d_{1}}

k[x_{1}, x_{2}, x_{3}, x_{4}, x_{1}^{−1}, x_{2}^{−1}, x_{3}^{−1}] ⊕ ⋯ ⊕ k[x_{1}, x_{2}, x_{3}, x_{4}, x_{2}^{−1}, x_{3}^{−1}, x_{4}^{−1}] ⟶^{d_{2}}

k[x_{1}, x_{2}, x_{3}, x_{4}, x_{1}^{−1}, x_{2}^{−1}, x_{3}^{−1}, x_{4}^{−1}]

Thus

H^{3}(X − P, O_{X−P}) = {x_{1}^{i}x_{2}^{j}x_{3}^{k}x_{4}^{ℓ} : i, j, k, ℓ < 0} ≠ 0.

57.2.3 Exercise III.5.6: Curves on a nonsingular quadric surface

Let Q be the nonsingular quadric surface xy = zw in X = P_{k}^{3} over a field k. We will consider locally principal closed subschemes Y of Q. These correspond to Cartier divisors on Q by (II, 6.17.1). On the other hand, we know that Pic Q ≅ Z ⊕ Z, so we can talk about the type (a,b) of Y (II, 6.16) and (II, 6.6.1). Let us denote the invertible sheaf L(Y) by O_{Q}(a, b). Thus for any n ∈ Z, O_{Q}(n) = O_{Q}(n, n).

[Comment! In my solution, a subscheme Y of type (a, b) corresponds to the invertible sheaf O_{Q}(−a, −b). I think this is reasonable since then O_{Q}(−a, −b) = L(−Y) = I_{Y}. The correspondence is not clearly stated in the problem, but this choice works.]

(a) Use the special case (q, 0) and (0, q), with q > 0, when Y is a disjoint union of q lines P^{1} in Q, to show:

1. if |a − b| ≤ 1, then H^{1}(Q, O_{Q}(a, b)) = 0;

2. if a, b < 0, then H^{1}(Q, O_{Q}(a, b)) = 0;

3. if a ≤ −2, then H^{1}(Q, O_{Q}(a, 0)) ≠ 0).

Solution. First I will prove a big lemma in which I explicitly calculate H^{1}(Q, O_{Q}(0, −q)) and some other things which will come in useful later. Next I give an independent computation of the other cohomology groups (1), (2).

Lemma 57.6. Let q > 0, then

dim_{k} H^{1}(Q, O_{Q}(−q, 0)) = H^{1}(Q, O_{Q}(0, −q)) = q − 1.

Furthermore, we know all terms in the long exact sequence of cohomology associated with the short exact sequence

0 → O_{Q}(−q, 0) → O_{Q} → O_{Y} → 0.

Proof. We prove the lemma only for O_{Q}(−q, 0), since the argument for O_{Q}(0, −q) is exactly the same. Suppose Y is the disjoint union of q lines P^{1} in Q so I_{Y} = O_{Q}(−q, 0). The sequence

0 → O_{Q}(−q, 0) → O_{Q} → O_{Y} → 0

is exact. The associated long exact sequence of cohomology is

0 →Γ(Q, O_{Q}(−q, 0)) → Γ(Q, O_{Q}) → Γ(Q, O_{Y})
→H^{1}(Q, O_{Q}(−q, 0)) → H^{1}(Q, O_{Q}) → H^{1}(Q, O_{Y})
→H^{2}(Q, O_{Q}(−q, 0)) → H^{2}(Q, O_{Q}) → H^{2}(Q, O_{Y}) → 0

We can compute all of the terms in this long exact sequence. For the purposes at hand it suffices to view the summands as k-vector spaces so we systematically do this throughout. Since O_{Q}(−q, 0) = I_{Y} is the ideal sheaf of Y, its global sections must vanish on Y. But I_{Y} is a subsheaf of O_{Q} whose global sections are the constants. Since the only constant which vanish on Y is 0, Γ(Q, O_{Q}(−q, 0)) = 0. By (I, 3.4), Γ(Q, O_{Q}) = k. Since Y is the disjoint union of q copies of P^{1} and each copy has global sections k, Γ(Q, O_{Y}) = k^{⊕q}. Since Q is a complete intersection of dimension 2, (Ex. 5.5 b) implies H^{1}(Q, O_{Q}) = 0. Because Y is isomorphic to several copies of P^{1}, the general result (proved in class, but not in the book) that H_{∗}^{n}(O_{P^{n}}) = {∑ a_{I}X_{I} : entries in I negative} implies H^{1}(Q, O_{Y}) = H^{1}(Y, O_{Y}) = 0. Since Q is a hypersurface of degree 2 in P^{3}, (I, Ex. 7.2(c)) implies p_{a}(Q) = 0. Thus by (Ex. 5.5c) we see that H^{2}(Q, O_{Q}) = 0. Putting together the above facts and some basic properties of exact sequences show that H^{1}(Q, O_{Q}(−q, 0)) = k^{⊕(q−1)}, H^{2}(Q, O_{Q}(−q, 0)) = 0 and H^{2}(Q, O_{Y}) = 0. Our long exact sequence is now

0 →Γ(Q, O_{Q}(−q, 0)) = 0 → Γ(Q, O_{Q}) = k → Γ(Q, O_{Y}) = k^{⊕q}
→H^{1}(Q, O_{Q}(−q, 0)) = k^{⊕(q−1)} → H^{1}(Q, O_{Q}) = 0 → H^{1}(Q, O_{Y}) = 0
→H^{2}(Q, O_{Q}(−q, 0)) = 0 → H^{2}(Q, O_{Q}) = 0 → H^{2}(Q, O_{Y}) = 0 → 0

Number (3) now follows immediately from the lemma because

H^{1}(Q, O_{Q}(a, 0)) = k^{⊕(−a−1)} ≠ 0

for a ≤ −2.

Now we compute (1) and (2). Let a be an arbitrary integer. First we show that O_{Q}(a, a) = 0. We have an exact sequence

0 → O_{P^{3}}(−2) → O_{P^{3}} → O_{Q} → 0

where the first map is multiplication by xy − zw. Twisting by a gives an exact sequence

0 → O_{P^{3}}(−2 + a) → O_{P^{3}}(a) → O_{Q}(a) → 0.

The long exact sequence of cohomology yields an exact sequence

⋯ → H^{1}(O_{P^{3}}(a)) → H^{1}(O_{Q}(a)) → H^{2}(O_{P^{3}}(−2 + a)) → ⋯

But from the explicit computations of projective space (5.1) it follows that H^{1}(O_{P^{3}}(a)) = 0 and H^{2}(O_{P^{3}}(−2 + a)) = 0 from which we conclude that H^{1}(O_{Q}(a)) = 0.

Next we show that O_{Q}(a − 1, a) = 0. Let Y be a single copy of P^{1} sitting in Q so that Y has type (1, 0). Then we have an exact sequence

0 → I_{Y} → O_{Q} → O_{Y} → 0.

But I_{Y} = O_{Q}(−1, 0) so this becomes

0 → O_{Q}(−1, 0) → O_{Q} → O_{Y} → 0.

Now twisting by a yields the exact sequence

0 → O_{Q}(a − 1, a) → O_{Q}(a) → O_{Y} (a) → 0.

The long exact sequence of cohomology gives an exact sequence

⋯ → Γ(O_{Q}(a)) → Γ(O_{Y} (a)) → H^{1}(O_{Q}(a − 1, a)) → H^{1}(O_{Q}(a)) → ⋯

We just showed that H^{1}(O_{Q}(a)) = 0, so to see that H^{1}(O_{Q}(a − 1, a)) = 0 it suffices to note that the map Γ(O_{Q}(a)) → Γ(O_{Y} (a)) is surjective. This can be seen by writing Q = Proj(k[x, y, z, w]/(xy − zw)) and (w.l.o.g.) Y = Proj(k[x, y, z, w]/(xy − zw, x, z)) and noting that the degree a part of k[x, y, z, w]/(xy − zw) surjects onto the degree a part of k[x, y, z, w]/(xy − zw, x, z). Thus H^{1}(O_{Q}(a − 1, a)) = 0 and exactly the same argument shows H^{1}(O_{Q}(a, a − 1)) = 0. This gives (1).

For (2) it suffices to show that for a > 0,

H^{1}(O_{Q}(−a, −a − n)) = H^{1}(O_{Q}(−a − n, −a)) = 0

for all n > 0. Thus let n > 0 and suppose Y is a disjoint union of n copies of P^{1} in such a way that I_{Y} = O_{Q}(0, −n). Then we have an exact sequence

0 → O_{Q}(0, −n) → O_{Q} → O_{Y} → 0.

Twisting by −a yields the exact sequence

0 → O_{Q}(−a, −a − n) → O_{Q}(−a) → O_{Y} (−a) → 0.

The long exact sequence of cohomology then gives an exact sequence

⋯ → Γ(O_{Y} (−a)) → H^{1}(O_{Q}(−a, −a − n)) → H^{1}(O_{Q}(−a)) → ⋯

As everyone knows, since Y is just several copies of P^{1} and −a < 0, Γ(O_{Y} (−a)) = 0. Because of our computations above, H^{1}(O_{Q}(−a)) = 0. Thus H^{1}(O_{Q}(−a, −a − n)) = 0, as desired. Showing that H^{1}(O_{Q}(−a − n, −a)) = 0 is exactly the same.

(b) Now use these results to show:

1. If Y is a locally principal closed subscheme of type (a, b) with a, b > 0, then Y is connected.

Proof. Computing the long exact sequence associated to the short exact sequence

0 → I_{Y} → O_{Q} → O_{Y} → 0

gives the exact sequence

0 → Γ(Q, I_{Y}) → Γ(Q, O_{Q}) → Γ(Q, O_{Y}) → H^{1}(Q, I_{Y}) → ⋯

But, Γ(I_{Y}) = 0, Γ(Q, O_{Q}) = k, and by (a)2 above H^{1}(Q, I_{Y}) = H^{1}(Q, O_{Q}(−a, −b)) = 0. Thus we have an exact sequence

0 → 0 → k → Γ(O_{Y}) → 0 → ⋯

from which we conclude that Γ(O_{Y}) = k which implies Y is connected.

2. now assume k is algebraically closed. Then for any a, b > 0, there exists an irreducible nonsingular curve Y of type (a, b). Use (II, 7.6.2) and (II, 8.18).

Proof. Given (a, b), (II, 7.6.2) gives a closed immersion

Q = P^{1} × P^{1} → P^{a} × P^{b} → P^{n}

which corresponds to the invertible sheaf O_{Q}(−a, −b) of type (a, b). By Bertini’s theorem (II, 8.18) there is a hyperplane H in P^{n} such that the hyperplane section of the (a, b) embedding of Q in P^{n} is nonsingular. Pull this hyperplane section back to a nonsingular curve Y of type (a, b) on Q in P^{3}. By the previous problem, Y is connected. Since Y comes from a hyperplane section this implies Y is irreducible (see the remark in the statement of Bertini’s theorem).

3. an irreducible nonsingular curve Y of type (a, b), a, b > 0 on Q is projectively normal (II, Ex. 5.14) if and only if |a − b| ≤ 1. In particular, this gives lots of examples of nonsingular, but not projectively normal curves in P^{3}. The simplest is the one of type (1, 3) which is just the rational quartic curve (I, Ex. 3.18).

Proof. Let Y be an irreducible nonsingular curve of type (a, b). The criterion we apply comes from (II, Ex 5.14d) which asserts that the maps

Γ(P^{3}, O_{P^{3}}(n)) → Γ(Y, O_{Y} (n))

are surjective for all n ≥ 0 if and only if Y is projectively normal. To determine when this occurs we have to replace Γ(P^{3}, O_{P^{3}}(n)) with Γ(Q, O_{Q}(n)). It is easy to see that the above criterion implies we can make this replacement if Q is projectively normal. Since Q ≅ P^{1} × P^{1} is locally isomorphic to A^{1} × A^{1} ≅ A^{2} which is normal, we see that Q is normal. Then since Q is a complete intersection which is normal, (II, 8.4b) implies Q is projectively normal. Consider the exact sequence

0 → I_{Y} → O_{Q} → O_{Y} .

Twisting by n gives an exact sequence

0 → I_{Y} (n) → O_{Q}(n) → O_{Y} (n).

Taking cohomology yields the exact sequence

⋯ → Γ(Q, O_{Q}(n)) → Γ(Q, O_{Y} (n)) → H^{1}(Q, I_{Y} (n)) → ⋯

Thus Y is projectively normal precisely if H^{1}(Q, I_{Y} (n)) = 0 for all n ≥ 0. When can this happen? We apply our computations from part (a). Since O_{Q}(n) = O_{Q}(n, n),

I_{Y} (n) = O_{Q}(−a, −b)(n) = O_{Q}(−a, −b) ⊗_{O_{Q}} O_{Q}(n, n) = O_{Q}(n − a, n − b)

If |a − b| ≤ 1 then |(n − a) − (n − b)| ≤ 1 for all n so

H^{1}(Q, O_{Q}(−a, −b)(n)) = 0

for all n which implies Y is projectively normal. On the other hand, if |a − b| > 1 let n be the minimum of a and b, without loss assume b is the minimum, so n = b. Then from (a) we see that

O_{Q}(−a, −b)(n) = O_{Q}(−a, −b)(b) = O_{Q}(−a + b, 0) ≠ 0

since −a + b ≤ −2.

(c) If Y is a locally principal subscheme of type (a, b) in Q, show that p_{a}(Y) = ab−a−b+1. [Hint: Calculate the Hilbert polynomials of suitable sheaves, and again use the special case (q,0) which is a disjoint union of q copies of P^{1}.]

Proof. The sequence

0 → O_{Q}(−a, −b) → O_{Q} → O_{Y} → 0

is exact so

χ(O_{Y}) = χ(O_{Q}) − χ(O_{Q}(−a, −b)) = 1 − χ(O_{Q}(−a, −b)).

Thus

p_{a}(Y) = 1 − χ(O_{Y}) = χ(O_{Q}(−a, −b)).

The problem is thus reduced to computing χ(O_{Q}(−a, −b)).

Assume first that a, b < 0. To compute χ(O_{Q}(−a, −b)) assume Y = Y_{1} ∪ Y_{2} where I_{Y_{1}} = O_{Q}(−a, 0) and I_{Y_{2}} = O_{Q}(0, −b). Thus we could take Y_{1} to be a copies of P^{1} in one family of lines and Y_{2} to be b copies of P^{1} in the other family. Tensoring the exact sequence

0 → I_{Y_{1}} → O_{Q} → O_{Y_{1}} → 0

by the flat module I_{Y_{2}} yields an exact sequence

0 → I_{Y_{1}} ⊗ I_{Y_{2}} → I_{Y_{2}} → O_{Y_{1}} ⊗ I_{Y_{2}}

[Note: I use the fact that I_{Y_{2}} is flat. This follows from a proposition in section 9 which we haven’t yet reached, but I’m going to use it anyways. Since Y_{2} is locally principal, I_{Y_{2}} is generated locally by a single element and since Q is a variety it is integral. Thus I_{Y_{2}} is locally free so by (9.2) I_{Y_{2}} is flat.] This exact sequence can also be written as

0 → O_{Q}(−a, −b) → O_{Q}(0, −b) → O_{Y} ⊗ O_{Q}(0, −b) → 0.

The associated long exact sequence of cohomology is

0 →Γ(Q, O_{Q}(−a, −b)) → Γ(Q, O_{Q}(0, −b)) → Γ(Q, O_{Y_{1}} ⊗ O_{Q}(0, −b))
→H^{1}(Q, O_{Q}(−a, −b)) → H^{1}(Q, O_{Q}(0, −b)) → H^{1}(Q, O_{Y_{1}} ⊗ O_{Q}(0, −b))
→H^{2}(Q, O_{Q}(−a, −b)) → H^{2}(Q, O_{Q}(0, −b)) → H^{2}(Q, O_{Y_{1}} ⊗ O_{Q}(0, −b)) → 0

The first three groups of global sections are 0. Since a, b < 0, (a) implies H^{1}(Q, O_{Q}(−a, −b)) = 0. From the lemma we know that H^{1}(Q, O_{Q}(0, −b)) = k^{⊕(b−1)}. Also by the lemma we know that H^{2}(Q, O_{Q}(0, −b)) = 0. Since O_{Y_{1}} ⊗ O_{Q}(0, −b) is isomorphic to the ideal sheaf of b − 1 points in each line of Y_{1}, a similar proof as that used in the lemma shows that

H^{1}(Q, O_{Y} ⊗ O_{Q}(0, −b)) = k^{⊕a(b−1)}.

Plugging all of this information back in yields the exact sequence

0 →Γ(Q, O_{Q}(−a, −b)) = 0 → Γ(Q, O_{Q}(0, −b)) = 0 → Γ(Q, O_{Y_{1}} ⊗ O_{Q}(0, −b)) = 0
→H^{1}(Q, O_{Q}(−a, −b)) = 0 → H^{1}(Q, O_{Q}(0, −b)) = k^{⊕(b−1)}
→ H^{1}(Q, O_{Y_{1}} ⊗ O_{Q}(0, −b)) = k^{⊕a(b−1)}
→H^{2}(Q, O_{Q}(−a, −b)) → H^{2}(Q, O_{Q}(0, −b)) = 0
→ H^{2}(Q, O_{Y_{1}} ⊗ O_{Q}(0, −b)) = 0 → 0

From this we conclude that

χ(O_{Q}(−a, −b)) = 0 + 0 + h^{2}(Q, O_{Q}(−a, −b)) = a(b − 1) − (b − 1) = ab − a − b + 1

which is the desired result.

Now we deal with the remaining case, when Y is a disjoint copies of P^{1}. We have

p_{a}(Y) = 1 − χ(O_{Y}) = 1 − χ(O_{P^{1}}^{⊕a}) = 1 − aχ(O_{P^{1}}) = 1 − a

which completes the proof.

57.3 IV, 3.6, 3.13, 5.4, Extra Problems

57.3.1 Exercise IV.3.6: Curves of Degree 4

(a) If X is a curve of degree 4 in some P^{n}, show that either

1. g = 0, in which case X is either the rational normal quartic in P^{4} (Ex. 3.4) or the rational quartic curve in P^{3} (II, 7.8.6), or

2. X ⊂ P^{2}, in which case g = 3, or

3. X ⊂ P^{3} and g = 1.

(b) In the case g = 1, show that X is a complete intersection of two irreducible quadric surfaces in P^{3} (I, Ex. 5.11).

Proof. (a) First suppose n ≥ 4. If X is not contained in any P^{n−1} then since 4 ≤ n (Ex. 3.4b) implies n = 4, g(X) = 0, and X differs from the rational normal curve of degree 4 only by an automorphism of P^{4}. (I take this to mean that X is the rational normal curve of degree 4.) Thus when n ≥ 4 we have proved that case (1) occurs.

Next suppose n = 3. Then X is a degree 4 curve in P^{3}. If X is contained in some P^{2} then g = \frac{1}{2}(d − 1)(d − 2) = 3 and so case (2) occurs. If X is not contained in any P^{2} then (Ex. 3.5 b) implies

g < \frac{1}{2}(d − 1)(d − 2) = 3.

Thus g is either 0, 1, or 2. If g = 0 then X is the rational quartic curve in P^{3} which is case (1). If g = 1 then X falls into case (3). If g = 2 then O_{X}(1) is a very ample divisor of degree 4 (3.3.2) on a genus 2 curve contrary to (Ex. 3.1) which asserts that the degree of a very ample divisor on a curve of genus 2 is at least 5.

Next suppose n = 2. Then X is a degree 4 curve in P^{2} so X has genus 3 and falls into case (2).

The case n = 1 cannot occur since P^{1} contains no curve of degree 4.

(b) A curve C of genus 1 has g_{2}^{1}’s since Riemann-Roch guarantees that the complete linear system associated to any divisor of degree 2 is a g_{2}^{1}. There are infinitely many divisors of degree 2 which is not linearly equivalent. This is because P + Q ∼ P + R iff Q ∼ R. Since C is not rational there are infinitely many points Q and R with Q ≁ R.

Given two distinct g_{2}^{1}’s take the product of the corresponding morphisms to obtain a map φ : C → Q where Q is the quadric surface in P^{3}. The diagram is

\begin{matrix}C & ⟶ & P^{1} \\ ↓ & ↘ φ & ↑ \\ P^{1} & ⟵ & C_{0} ⊂ Q \end{matrix}

Let C_{0} denote the image of C under φ. Note that C_{0} is not a point so the type (a, b) of C_{0} is defined. Let e be the degree of φ. Then by analyzing how certain divisors pull back one sees that ae = 2 and be = 2. The only possibilities are that C_{0} is of type (1, 1) or of type (2, 2). If C_{0} is of type (1, 1) then C_{0} must be nonsingular because of the relation between the arithmetic genus of C_{0} and that of its normalization. So in this case C_{0} ≅ P^{1} and the projections Q → P^{1} are injective when restricted to C_{0}. This implies that the two maps coming from the distinct g_{2}^{1}’s collapse the same points, a contradiction.

Thus C_{0} is of type (2, 2) and e = 1. Because C has genus 1 and C_{0} has arithmetic genus (2 − 1)(2 − 1) = 1 it follows that C ≅ C_{0}. We now know that C embeds as a type (2, 2) curve on Q. Since a curve of type (a, a) on Q is a complete intersection we are done.

Another way to do this problem is to somehow embed X into P^{3} as a degree 4 curve then consider the exact sequence

0 → I_{X}(2) → O_{P^{3}}(2) → O_{X}(2) → 0.

Taking cohomology yields the exact sequence of vector spaces

0 → H^{0}(I_{X}(2)) → H^{0}(O_{P^{3}}(2)) → H^{0}(O_{X}(2)) → · · · .

Since X has degree 4 it follows that dim H^{0}(O_{X}(2)) = 8. Furthermore dim H^{0}(O_{P^{3}}(2)) = 10 so we see that

dim H^{0}(I_{X}(2)) ≥ 2.

This means that there exist linearly independent homogeneous polynomials f and g of degree 2 such that X ⊂ Z(f) and X ⊂ Z(g). Since X has degree 4 and Z(f) ∩ Z(g) has degree 4 the fact that X ⊂ Z(f) ∩ Z(g) implies that X = Z(f) ∩ Z(g). This is seen by looking at the appropriate Hilbert polynomials.

57.3.2 Exercise IV.3.12

For each value of d = 2, 3, 4, 5 and r satisfying 0 ≤ r ≤ \frac{1}{2}(d − 1)(d − 2), show that there exists an irreducible plane curve of degree d with r nodes and no other singularities.

Proof. I did the first few by finding explicit equations. It might have been better to do everything by abstract general methods but it was a good exercise to search for defining equations.

d = 2, r = 0: Take f = x^{2} − yz = 0. This works in any characteristic since the partials are: f_{x} = 2x, f_{y} = −z, f_{z} = −y so if f_{y} = f_{z} = 0 then z = y = 0 so x = 0. But (0, 0, 0) is not a point.

d = 3, r = 0: When char k ≠ 3 take x^{3} + y^{3} + z^{3} = 0 which is clearly nonsingular. For char k = 3 take x^{2}y + z^{2}y + z^{3} + y^{3} = 0. This is nonsingular as was proved in my solution to (I, Ex. 5.5).

d = 3, r = 1: Let f = xyz + x^{3} + y^{3}, then f_{x} = yz + 3x^{2}, f_{y} = xz + 3y^{2} and f_{z} = xy. Thus a singular point must have x = 0 or y = 0. If x = 0 then from f = 0 we see that y = 0. Thus (0 : 0 : 1) is the only singularity and it is clearly nodal. Note that this curve works in characteristic 3 as well.

d = 4, r = 0: When char k ≠ 2 take x^{4}+y^{4}+z^{4} = 0. If char k = 2 take x^{3}y+z^{3}y+z^{4}+y^{4} = 0 as in (I, Ex. 5.5).

d = 4, r = 1: If char k ≠ 2 take f = xyz^{2} + x^{4} + y^{4} = 0. Then f_{x} = yz^{2} + 4x^{3}, f_{y} = xz^{2} + 4y^{3}, and f_{z} = 2xyz. The only singular point is (0 : 0 : 1) which is a node. When char k = 2 take f = xyz^{2} + x^{3}z + y^{4} = 0. The partials are f_{x} = yz^{2} + x^{2}z, f_{y} = xz^{2}, and f_{z} = x^{3}. Thus a singular point must satisfy x = 0. Then f = 0 implies y = 0. Thus the only singular point is (0 : 0 : 1) which is a node.

d = 4, r = 2: Let C be your favorite genus 1 curve. Let D be a divisor of degree 4, then by (3.3.3) D is very ample. By Riemann-Roch,

dim |D| − dim |K − D| = 4 + 1 − 1

so |D| gives rise to embedding of C as a degree 4 curve in P^{3}. Using (3.11) project C onto the plane to obtain a plane curve of degree 4 with only nodal singularities. Since the genus of the normalization is 1 it follows that there are exactly 2 nodes.

d = 4, r = 3: Embed P^{1} as the degree 4 rational normal curve in P^{4}. Then use (3.5) and (3.10) to project into P^{2} to get a curve X of degree 4 in P^{2} having only nodes for singularities. Since

0 = g(X̃) = \frac{1}{2}(4 − 1)(3 − 1) − number of nodes

it follows that X has 3 nodes.

d = 5, r = 0: When char k ≠ 5 take x^{5}+y^{5}+z^{5} = 0. If char k = 5 take x^{4}y+z^{4}y+z^{5}+y^{5} = 0.

d = 5, r = 1: When char k ≠ 5 take f = xyz^{3} + x^{5} + y^{5} = 0. Then f_{x} = yz^{3} + 5x^{4}, f_{y} = xz^{3} + 5y^{4}, and f_{z} = 3xyz^{2}. The only singular point is (0 : 0 : 1) which is a node. In characteristic 5 let f = xyz^{3} + x^{5} + y^{5} + x^{3}y^{2}. Then f_{x} = yz^{3} + 3x^{2}y^{2}, f_{y} = xz^{2} + 2x^{3}y, and f_{z} = 3xyz^{2} so the only singular point is (0 : 0 : 1) which is clearly a nodal singularity.

d = 5, r = 2: By (Ex 5.4 a) a genus 4 curve which has two distinct g_{3}^{1}’s gives rise to a plane quintic with two nodes. The curve of type (3, 3) on the quadric surface is such a curve.

d = 5, r = 3. I have not found one yet. If I could find a degree 5 space curve of genus 3 I would win. But by Theorem 6.4 no such space curve exists.

d = 5, r = 4: Let C be a curve of genus 2. By Halphen’s theorem there exists a nonspecial very ample divisor D of degree 5. Then

dim |D| = 5 + 1 − 1 − 1 = 3

so C embeds into P^{3} as a curve of degree 5. Project to P^{2} to obtain a curve X of degree 5 whose singularities are all nodal and whose normalization has genus 2. It follows that X has 4 nodes.

d = 5, r = 5: Pick your favorite curve of genus 1. By (3.3.3) there exists a very ample nonspecial divisor of degree 5. As usual, project to obtain a degree 5 plane curve with singularities only nodes and normalization of genus 1. It follows that there must be exactly 5 nodes.

d = 5, r = 6: Embed P^{1} in P^{5} as a curve of degree 5 (Ex. 3.4), and then project it into P^{2} to get a curve X of degree 5 in P^{2} having only nodes as singularities Since

0 = g(X̃) = \frac{1}{2}(5 − 1)(4 − 1) − number of nodes

the number of nodes must be 6.

We can also obtain the following general result.

Proposition 57.7. If r and d are such that

\frac{1}{2}(d − 1)(d − 2) + 3 − d ≤ r ≤ \frac{1}{2}(d − 1)(d − 2)

then there exists a plane curve of degree d which has exactly r singularities all of which are nodes.

Proof. By (6.2) if d ≥ g + 3 then there exists a curve in P^{3} of genus g and degree d. Using (3.11) project this curve onto the plane to obtain a curve of degree d with

r = \frac{1}{2}(d − 1)(d − 2) − g

singularities all of which are nodes. We can carry out this process so long as r ≤ \frac{1}{2}(d−1)(d−2) and d ≥ g + 3, that is, as long as

r = \frac{1}{2}(d − 1)(d − 2) − g ≥ \frac{1}{2}(d − 1)(d − 2) + 3 − d.

57.3.3 Exercise IV.5.4

Another way of distinguishing curves of genus g is to ask, what is the least degree of a birational plane model with only nodes as singularities (3.11)? Let X be nonhyperelliptic of genus 4. Then:

(a) if X has two g_{3}^{1}’s, it can be represented as a plane quintic with two nodes, and conversely;

(b) if X has one g_{3}^{1}, then it can be represented as a plane quintic with a tacnode (I, Ex. 5.14d), but the least degree of a plane representation with only nodes is 6.

Proof. (a) Summary: If X is nonhyperelliptic with two g_{3}^{1}’s then X is type (3, 3) in the quadric so projecting through a point on X gives a plane quintic model with exactly two nodal singularities.

Suppose X is nonhyperelliptic and X has two g_{3}^{1}’s. Let p and p′ be the degree 3 maps X → P^{1} determined by the two g_{3}^{1}’s. Let

φ : X → P^{1} × P^{1} = Q ⊂ P^{3}

be their product. Let X_{0} = φ(X) be the image of X, and let (a, b) be the type of X_{0}. Letting e be the degree of φ we see that ea = 3 and eb = 3. This implies that either a = b = 1 and e = 3 or a = b = 3 and e = 1.

First suppose a = b = 1 and e = 3. Then X_{0} is nonsingular of genus 0 and the two projection maps X_{0} → P^{1} are injective. This implies p and p′ collapse the same points, a contradiction.

Thus a = b = 3 and e = 1. Because the arithmetic genus of X_{0} is (a − 1)(b − 1) = 4 and X has genus 4 we see that X_{0} must be nonsingular. Thus X embeds as a type (3, 3) curve on the quadric surface. Henceforth view X as embedded in Q.

Pick a point P_{0} on X ⊂ Q and fix a copy of P^{2}. Map X to P^{2} by projection through P_{0}. If two points P, Q project to the same point then P_{0}, P, Q are collinear. I claim that this implies the line L through P_{0}, P, Q is contained in Q. To see this let H by a hyperplane containing L. Then either H.Q is two copies of a line or a degree 2 curve. If H.Q is a degree 2 curve then L.(H.Q) consists of at most 2 points, a contradiction since the points P_{0}, P, Q are all contained in L.(H.Q). Thus H.Q is two copies of a line. Since P_{0}, P, Q are contained in H.Q we see that H.Q = 2L and hence that L is contained in Q. Since there are exactly 2 lines on Q through any given point of Q the image of X under projection can have at most 2 singularities.

To show that the singularities are nodes we show that three or more points can not collapse under projection. [[This is not quite enough. I do not know how to show that there is not some other unusual singularity.]] Suppose some three points are collapsed under projection. Then there exists points P, Q, R on X such that P_{0}, P, Q, R are all collinear. Now X is of type (3, 3) so X is a complete intersection Q.F_{3} where F_{3} is some degree 3 surface. Since F_{3} has degree 3 and contains P_{0}, P, Q, R it must contain the line through them. (Take a plane containing the line, and apply an argument as above.) Similarly Q contains the line through P_{0}, P, Q, R so X contains this line, a contradiction.

Since the genus of the normalization of the image X_{0} of X in P^{2} is 4 and X_{0} has exactly two nodes as singularities it follows that X_{0} has degree 5.

The converse. Suppose we are given a plane quintic curve C with two nodes and no other singularities. Then the normalization has genus

g(C̃) = \frac{1}{2}(5 − 1)(4 − 1) − 2 = 4.

Thus C represents a curve X of genus 4. Let P_{0} be one of the two nodes, then since C has degree 5 a line through P_{0} intersect C in 3 other points. This gives a degree 3 map from C to P^{1}. Since this map is defined on a nonempty open subset of X it extends to a degree 3 morphism of X into P^{3}. This gives a g_{3}^{1} on X. The other node gives a different g_{3}^{1} so X has two distinct g_{3}^{1}’s.

We must also show that X is not hyperelliptic. Suppose X is hyperelliptic so X has a g_{2}^{1}. Let p be the map to P^{1} corresponding to this g_{2}^{1} and let p′ be the map to P^{1} corresponding to some g_{3}^{1}. Let φ = p × p′ : X → Q ⊂ P^{3} be their product. Let X_{0} = φ(X) be the image of X and suppose X_{0} has type (a, b). Let e be the degree of φ. Then ea = 2 and eb = 3. Thus e = 1 so X is birational to the normalization of X_{0}. But X_{0} has arithmetic genus (2 − 1)(3 − 1) = 2 < 4 which is a contradiction.

(b) Suppose C is a nonhyperelliptic curve with exactly one g_{3}^{1}. By looking at twists of certain exact sequences we showed in class that C lies on the (singular) quadric cone. We showed furthermore that C is a complete intersection Q_{cone}.F_{3} of Q_{cone} with some cubic hypersurface F_{3}. Pick a point P_{0} on C and a copy of P^{2} ⊂ P^{3}. Projection through P_{0} defines a map C → P^{2}. If two points P, Q collapse then the three points P_{0}, P, Q are collinear. Let L be the line determined by P_{0}, P, Q. Since Q_{cone} has degree 2 and L intersects Q_{cone} in at least three points it follows that L lies on Q_{cone}. Since there is only one line through P_{0} which lies on Q_{cone} it follows that projection defines a birational map of C to plane curve C_{0} which has exactly one singularity.

If 3 points P, Q, R are collapsed by projection then the four points P_{0}, P, Q, R are collinear. Let L be the line through them. Then L must lie in F_{3} and L must lie in Q which is a contradiction. This shows that the singular point on C_{0} is a double point.

I do not know how to show it must be a tacnode.

The Converse:

If there is a quintic plane representation of degree less than 6 with only nodes then there must be at least two nodes because of the formula for the genus of the normalization. But each node gives rise to a distinct g_{3}^{1}. Since there is a unique g_{3}^{1} on X this is a contradiction.

57.3.4 Extra Problem 3, by William Stein

Suppose C is hyperelliptic and g ≥ 3. Then there does not exist a g_{3}^{1} on C.

Proof. Suppose that C has a g_{3}^{1} and let p : C → P^{1} be the corresponding morphism. Let p′ : C → P^{1} be the morphism corresponding to a g_{2}^{1} on the hyperelliptic curve C. Let

φ = p × p′ : C → P^{1} × P^{1} = Q ⊂ P^{3}.

Let (a, b) be the type of C_{0} = φ(C). If e denotes the degree of φ then ea = 2 and eb = 3. We see this by looking at how divisors of types (1, 0) and (0, 1) pull back. Thus e = 1, a = 2 and b = 3. Now C is isomorphic to the normalization of C_{0} which has arithmetic genus (2 − 1)(3 − 1) = 2 < 3 so C has genus less than 3, a contradiction.

57.3.5 Extra Problem 4, by Nghi Nguyen

If C is a non-hyperelliptic curve of genus g ≥ 4, show that C has at most a finite number of g_{3}^{1}’s.

Proof. This proof is the work of Nghi although the write up is my own. (Therefore any mistakes are my responsibility.)

Summary. For a fixed point P_{0} there are only finitely many g_{3}^{1}’s arising from a divisor D of the form D = P_{0} + Q + R. If some g_{3}^{1} is defined by a divisor D, then D − P_{0} is linearly equivalent to an effective divisor Q + R so D ∼ P_{0} + Q + R. Combining this with the first assertion implies that there are only finitely many g_{3}^{1}’s.

Step 1. Fix a point P_{0}, then we show that there are only finitely many g_{3}^{1}’s arising from a divisor D of the form P_{0} + Q + R. Suppose D = P_{0} + Q + R is a divisor such that |D| is a g_{3}^{1}.

Since C is non-hyperelliptic the canonical divisor K is very ample. Thus K − P_{0} is base point free. Let φ : X → P^{g−2} be the morphism determined by K − P_{0}. By Riemann-Roch

dim |P_{0} + Q + R| − dim |K − P_{0} − Q − R| = 3 + 1 − g

so since dim |P + Q + R| = 1,

dim |K − P_{0} − Q − R| = g − 3.

But K is very ample so dim |K − P_{0} − Q| = dim |K| − 2 = g − 3. Thus

dim |K − P_{0} − Q − R| = dim |K − P_{0} − Q|

so R is a basepoint of K − P_{0} − Q. This means that φ(R) = φ(Q).

Let X_{0} = φ(X) ⊂ P^{g−2}. Let μ = deg φ and let d = deg X_{0}. Then

μd = deg(K − P_{0}) = 2g − 3.

If μ > 1 then d ≤ \frac{2g−3}{3} since μ is not 2 because 2g − 3 is odd. But \frac{2g−3}{3} < g − 2 since g ≥ 4. Thus d < g − 2.

Let X̃_{0} be the normalization of the degree d curve X_{0} ⊂ P^{g−2}. Let h be the linear system of degree d corresponding to the morphism

X̃_{0} → X_{0} ⊂ P^{g−2}.

Then h is a linear system of degree d and dimension g − 2. Thus by (Ex. 3.4) we conclude that g−2 ≤ d. This contradicts the conclusion d < g−2 which followed from our assumption that μ > 1. Thus μ = 1.

Since μ = 1 the map φ is birational so it can collapse only finitely many points. This means that there are only finitely many choices for D = P_{0} + Q + R so that |D| is a g_{3}^{1}.

Step 2. Suppose |D| is a g_{3}^{1}. Then dim |D| = 1 and since |D| is basepoint free

dim |D − P_{0}| = dim |D| − 1 = 0.

Thus there exists an effective divisor Q + R such that

D − P_{0} ∼ Q + R,

so D ∼ P_{0} +Q+R. This shows that every g_{3}^{1} is defined by an effective divisor which contains P_{0}. By step 1 we know there are only finitely many such g_{3}^{1} so we conclude that C has only finitely many g_{3}^{1}’s.