Math 99r - Curves on algebraic surfaces

Taught by Ziquan YangNotes by Dongryul Kim

Spring 2018

This tutorial was taught by Ziquan Yang, a graduate student. The classmet on Tuesdays and Thursdays from 4:30 to 6pm. The course closed followedMumford’s Lectures on Curves on an Algebraic Surface, and there was only afinal paper on a topic related to the course.

Contents

1 January 30, 2018 41.1 Complex manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Holomorphic vector bundles . . . . . . . . . . . . . . . . . . . . . 5

2 February 1, 2018 72.1 Divisors and the Picard group . . . . . . . . . . . . . . . . . . . . 72.2 Riemann–Roch and Riemann–Hurwitz . . . . . . . . . . . . . . . 72.3 Formalism with sheaves . . . . . . . . . . . . . . . . . . . . . . . 9

3 February 6, 2018 103.1 Line bundles and Pn . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 February 8, 2018 134.1 Fiber product of schemes . . . . . . . . . . . . . . . . . . . . . . 13

5 February 13, 2018 155.1 Pullback of sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . 155.2 Quasi-coherent sheaves . . . . . . . . . . . . . . . . . . . . . . . . 155.3 Sheaf of relative differentials . . . . . . . . . . . . . . . . . . . . . 16

6 February 15, 2018 196.1 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.2 Homological algbera . . . . . . . . . . . . . . . . . . . . . . . . . 206.3 Sheaf cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1 Last Update: August 27, 2018

7 February 20, 2018 227.1 Chern classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.2 Intersection number . . . . . . . . . . . . . . . . . . . . . . . . . 237.3 Proj construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

8 February 22, 2018 268.1 Twisted sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.2 Properties of morphisms . . . . . . . . . . . . . . . . . . . . . . . 278.3 Introduction to spectral sequences . . . . . . . . . . . . . . . . . 28

9 February 27, 2018 309.1 Flat morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.2 Serre’s vanishing theorem . . . . . . . . . . . . . . . . . . . . . . 31

10 March 1, 2018 3310.1 Associated points . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.2 Criteria for flatness . . . . . . . . . . . . . . . . . . . . . . . . . . 34

11 March 20, 2018 3611.1 Cartier divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3611.2 Curves on surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 3611.3 Riemann–Roch for surfaces . . . . . . . . . . . . . . . . . . . . . 37

12 March 22, 2018 3912.1 Hodge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3912.2 Simple cases of the Picard scheme . . . . . . . . . . . . . . . . . 39

13 March 27, 2018 4213.1 m-regular sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

14 March 29, 2018 4414.1 Grassmannians and other prerequisites . . . . . . . . . . . . . . . 4414.2 Embedding in the Grassmannian . . . . . . . . . . . . . . . . . . 45

15 April 3, 2018 4615.1 Representing the curves functor . . . . . . . . . . . . . . . . . . . 46

16 April 5, 2018 48

17 April 10, 2018 4917.1 Ampleness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

18 April 12, 2018 5118.1 Representing the Picard functor . . . . . . . . . . . . . . . . . . . 51

19 April 17, 2018 5319.1 Representing the Picard functor II . . . . . . . . . . . . . . . . . 53

2

20 April 19, 2018 5520.1 Infinitesimal structure of Curve and Pic . . . . . . . . . . . . . . 55

21 April 24, 2018 5721.1 Regularity of the curve functor . . . . . . . . . . . . . . . . . . . 57

3

Math 99r Notes 4

1 January 30, 2018

The main reference will be Mumford’s Lectures on curves on an algebraic sur-face. A more general reference would be Hartshorne, or Griffiths–Harris, orComplex geometry by D. Huybrechts, or Foundations of algebraic geometry byVakil. We are going to use complex manifolds as a motivating example.

1.1 Complex manifolds

Definition 1.1. A complex manifold is a smooth manifold with holomorphiccharts (Ui, ϕi) given by the transition functions being biholomorphisms.

The chart (Ui, ϕi) is supposed to tell you what it means for a local C-valuedfunction to be holomorphic, i.e., gives the sheaf of holomorphic functions. Amap f : M → N of complex manifolds is something that pulls a holomorphicfunction back to a holomorphic function.

Example 1.2. For instance, consider the map C/〈1, i〉 → C/〈2, i〉 given byx+ yi 7→ 2x+ yi. This is a smooth nice map, but is not a morphism of complexmanifolds. The function x + yi is holomorphic is holomorphic with respect tothe complex structure of E, but its pullback 2x+ yi is not.

Definition 1.3. A holomorphic vector bundle is a complex vector bundlewith a holomorphic structure. It is going to be a complex vector bundle π :E → X where X is a complex manifold and E has a structure of a complexmanifold. Locally, it is going to be given by local trivializations (Ui, ψi)where ψi : π−1(Ui)→ Ui × Cn are isomorphisms such that the transition mapsUi ∩ Uj → GL(Cn) are holomorphic.

Because of this rigidity, exact sequences of holomorphic vector bundles mightnot split. A real manifold can carry many different complex structures. Sothings like singular cohomology H∗(−,Z) are not good enough for our pur-poses.This was actually studied classically by Riemann. If you integrate aroundpaths, we get an embedding

H1(X,Z)→ H0(X,Ω1)∨.

These numbers are called periods. This is functorial, because if you have amorphism X → Y , we get

H1(X,Z) H0(X,Ω1X)∨

H1(Y,Z) H0(Y,Ω1Y )∨.

This is a lattice in a complex vector space, and it is quite fine. But from amodern point of view, this is studying the Hodge structures.

Math 99r Notes 5

The quotient H0(X,Ω1X)∨/H1(X,Z) is a complex torus, and is a variety.

This is called the Albenese variety

Alb(X) ∼= Pic0(X) = moduli of deg 0 line bundles on X

For instance, dim Pic0(X) = dimH0(X,Ω1X) = g is the geometric genus.

In this course, we want to study the moduli of line bundles with c1 = 0 onan algebraic surface. (You can think of c1 = 0 as an analogue of deg = 0.) Onan algebraic surface X, Severi was studying the quotient group

Galg = 〈C − C ′ : C ∼alg C′〉,

Glin = 〈C − C ′ : C ∼lin C′〉.

Here, this algebraic equivalence is C ∼ C ′ there is a flat morphism X → S suchthat C and C ′ are fibers. This is a linear equivalence if S = P1.

Severi considered this group Pic0 = Ga/Gl and showed that its dimensionis dimGa/Gl = dimH1(X,OX) = b1

2 where b1 = dimQH1(X,Q). We will

construct Galg/Glin by purely algebraic methods.

1.2 Holomorphic vector bundles

Given a holomorphic vector bundle E → X and an open set U ⊆ X, let usdenote

V (U) = holomorphic sections s : U → E.

This has a natural structure of an OX(U)-module.

Example 1.4. There is the tautological line bundle OP1(−1) on P1. This is

E = ([`], x) ∈ P1 × C2 : x ∈ `.

As a sheaf E , it will send

E (U) = holomorphic maps ϕ : U → C2 such that ϕ([`]) ∈ `.

Choosing a local trivialization π−1(U) → U × C over U is equivalent tochoosing a nowhere vanishing section over U . Let me describe this local trivi-alization. There are homogeneous coordinates [z0 : z1] on P1. The open coveris given by U0 = z0 6= 0 with coordinates s = z1

z0and U1 = z1 6= 0 with

coordinates t = z0z1

. The local trivialization is given by

U0 → C2; [z0 : z1] 7→ (1, s)

and likewise U1 → C2 given by [z0 : z1] 7→ (1, t). If you do this, it is clear whatthe transition function should be. It should be given by × z1z0 .

We can also ask what the global sections are. For holomorphic vector bun-dles, global sections tend to be very finite-dimensional. A section U0 → E

Math 99r Notes 6

corresponds to a entire function f : U0 → C, and likewise U1 → E correspondsto g : U0 → C. Then the gluing condition is

g(t) =1

tf(1

t

).

You’ll quickly realize that there is no such f, g except for 0.But you can relax the condition and look for meromorphic sections. Here,

s = 1t is a meromorphic section, and it has one pole. The number of zeroes

minus the number of poles is independent of the choice of section, and so itis a natural invariant of the line bundle. This is called the degree of the linebundle.

Given two vector bundles V and W , we can take V ⊕ W , V ⊗ W , andHom(V,W ). For the tautological line bundle E → P1, we can form its dualE∨ = Hom(`,C). The global sections are going to be 2-dimensional.

Math 99r Notes 7

2 February 1, 2018

2.1 Divisors and the Picard group

Let L → X be a holomorphic line bundle over a curve X. This, with a mero-morphic section will correspond to the divisors Div(X). When we quotient thisby forgetting about sections, we get a correspondence

L→ X ←→ Div(X)/PDiv(X).

Let see how this correspondence can be described. Let K be the sheaf of themeromorphic functions on X. Then for each line bundle L, we can look at thesheaf L corresponding to this, and then we can think about the meromorphicsections

H0(X,L ⊗K ).

(Think H0 = Γ for now.) A section σ ∈ H0(X,L ⊗ K ) corresponds to anembedding L → K . Given a section σ, we can define a divisor, which is anelement of

Div(X) =⊕p∈X

Zp.

This is given byDiv(σ) = (zeroes)− (poles).

Given a divisor D, we can also define a line bundle. This is given by

U 7→ f ∈ K (U) : f + (D|U ) ≥ 0.

Then adding a point is allowing a pole at p, and subtracting a point is requirea zero at p. This describes the correspondence

L→ X with σ ∈ H0(X,L ⊗K )/C× ←→ Div(X).

Now let us see what happens when we remove the choice of a section. Forany two σ1, σ2, they differ by a meromorphic function. That is, there exists af ∈ K (X) such that σ1 = fσ2. Then we have Div(σ1) = Div(σ2)+Div(f). Wedefine the principal divisors as

PDiv = D : D = Div(f)

for a meromorphic function f . In general, we have OX(D1) ∼= OX(D2) if D1 =D2 + Div(f) because multiplication by f gives the isomorphism.

2.2 Riemann–Roch and Riemann–Hurwitz

We can define the cotangent bundle in the following way. For each coordinatechart Z, we consider the differentials dz and then glue these bundles together.

Math 99r Notes 8

This is well-defined because on triple intersections, we have the chain rule. Thiscotangent bundle is often denoted Ω1

X . If dimX = n, we denote

ωX =∧nΩ1

X

the canonical bundle. If X is a curve, ωX = Ω1X and Div(ωX) (up to PDiv)

is called the canonical divisor K.What is the canonical bundle of P1? We can pick a generic differential and

then compute its divisor. Consider the coordinate charts

s =z1

z0, t =

z0

z1.

Then s = 1t on U0 ∩U1. If we take ds, then it is going to be ds = − 1

t2 dt. So weget

ωP1 = OP1(−2).

By the way, on P1 we have D ∈ PDiv if and only if degD = 0. This meansthat the line bundle is determined uniquely by its degree. That is why weimmediately said ωP1 = OP1(−2).

Theorem 2.1 (Riemann–Roch). For any line bundle L , we have

dimH0(X,L )− dimH0(X,ωX ⊗L ∨) = degL− g + 1.

Here, g is the genus of the curve X.

We’re not going to prove it because it involves Serre duality. But let’s plugin L = OX . Then

1− h0(X,ω) = 0− g + 1

and so h0(X,ω) = g. For general varieties, this is called the geometric genus.If we put L = ω, we get

g − h0(X,OX) = degω − g + 1

and then degω = 2g−2. This is to be expected, from a topological perspective.In general, χ(M) = e(TM) = c1(ω∨). But χ(M) = 1 − 2g + 1 = 2 − 2g. Soc1(ω) = 2g − 2. A good way to remember this is that if the genus of X is high,ωX will be positive.

Given a morphism between two curves, there is a way of relating the genusof the two curves.

Theorem 2.2 (Riemann–Hurwitz). Let f : X → Y be a morphism of twocurves. Then

2gX − 2 = (deg f)(2gY − 2) +∑i

(ri − 1)

where ri is the ramification degree.

Math 99r Notes 9

I should first define the degree deg f . The easiest way is to use topology.The map f : X → Y induces a map H2(Y ;Z) → H2(X;Z). This is a mapZ → Z, and it is defined to be multiplication by deg f . Ramification degree isdefined in the following way. Locally, if f(p) = q, and z is a local coordinate at qand w is a local coordinate at p, then f∗(z) vanishes at p. If it is of order n = 1,then we say that it is unramified, and if n > 1, we say that it is ramified ofdegree n.

Proof. This should be thought of as comparing the canonical divisors of X andY . You can show that

f∗ωY ∼= ωX ⊗ OX

(−∑i

(ri − 1)Pi

).

If z is a local coordinate on Y , and w is a local coordinate on X, the bundleωY locally generated by dz and so f∗dz will be generated f∗ωY . Then this willlook like wri−1dw time some invertible thing.

On the other hand, deg(f∗L) = deg f · degL. (Or you can see this viac1f∗ = f∗c1.)

You can use this to compute the genus of a plane curve. Given an explicitcurve in P2, you project down to P1. Then you can compute all the ramificationdata, and from this determine the genus.

2.3 Formalism with sheaves

Consider C the category of open subsets of a topological space X. Given twoopen sets U ⊆ V , there is one morphism U → Y . A presheaf is a functor

F : Cop → Set/Ab/CRing/ . . . .

This means that you can restrict things on a larger open subset to a smalleropen subset. A morphism of presheaves is a natural transformation of functors.This is a global notion.

A sheaf is a presheaf satisfying the two axioms:

• If U is covered by Ui then the section on U is determined by the sectionsrestricted to Ui.

• If there are sections over Ui that agree on the intersections, then theycome from a section on U .

A compact way to say these two together is that F (U) is the equalizer

F (U)∏i F (Ui)

∏i,j F (Ui ∩ Uj).

Math 99r Notes 10

3 February 6, 2018

We are going to take a functorial point of view on schemes.

3.1 Line bundles and Pn

If you take a complex manifold M and a line bundle, and you take some sectionss0, . . . , sn such that for every point x ∈ M , there exist some si(x) 6= 0. Thenyou can define M 7→ Pn given by

M → Pn; x 7→ [s0(x), . . . , sn(x)].

This is well-defined, because whatever trivialization you choose, [s0(x), . . . , sn(x)]is going to be the same point even though the individual si(x) doesn’t makesense.

How does the converse work? Given a map ϕ : M → Pn, will you be able tofind a line bundle L with sections s0, . . . , xn? We can just pullback

L ∼= ϕ∗OPn(1), si = ϕ∗xi.

Let me now present a formalism. We define

C1 = category of “morphisms to Pn”.

The objects are morphisms M → Pn, and morphism to N → Pn are morphismsM → N that make the diagram commute. We also define

C2 = category of “pencils”

with objects L → M with `0, . . . , `n ∈ H0(M,L) such that for all x ∈ M ,`i(x) 6= 0 for some i, i.e., On+1

X L. A morphism from L → M to E → Nwould be a morphism M → N and an isomorphism L ∼= f∗E such that `i 7→ ei.

Proposition 3.1. A global section s ∈ H0(M,L ) corresponds to a morphismOX → L .

What we just discussed give an equivalence of categories between C1 andC2. Explicitly, we can map

(Mϕ−→ Pn) 7→ (ϕ∗OPn(1), ϕ∗x0, . . . , ϕ

∗xn).

In the other direction, we have

(L→M, `i) 7→ (f : M → Pn; x 7→ [`i(x)]).

So given an object X in a category C, you want to consider it as a functorHom(−, X). In general, if you have a category C, you can consider the categoryof presheaves

C = Fun(Cop,Set).

Math 99r Notes 11

There is a natural functor

C → C; C 7→ hC = HomC(−, C) : Cop → Set.

This is the Yoneda embedding. This says that the object C is completelydetermined by the functor Hom(−, C).

Given a random contravariant functor h : Cop → Set, a natural questionis whether there exists an C such that h = hC . If h = hC , we say that h isrepresented by C.

3.2 Schemes

These are just generalizations of commutative rings. For a commutative ringA, we give SpecA the set of prime ideal of A a topology with closed subsetsV (I) for ideals I ⊆ A. You can verify that V (I) is precisely the image ofSpecA/I → SpecA. Given a map ϕ : A→ B, you get a continuous map

f : SpecB → SpecA; q 7→ ϕ−1(q).

Open subsets will look like SpecA−V (I), and there is a base of distinguishedopen subsets consisting of

Uf = p ⊆ A : f /∈ p.

The subsets Uf can be identified with SpecAf . Then we can define the structuresheaf as

OX(Uf ) = Af .

Then for a general open subset, we can define OX(U) by gluing them.Now we can define schemes. A ringed space is a topological space X with

a sheaf of rings F . Then a morphism (X,F )→ (Y,G ) is given by a continuousmap X → Y along with a morphism G → f∗F of sheaves.

A locally ringed space is a ringed space (X,F ) such that Fp = lim−→p∈U F (U)

is a local ring. A morphism between two locally ringed spaces (X,F )→ (Y,G )is a map X → Y with a morphism G → f∗F such that Gp → Fp is a localhomomorphism.

Definition 3.2. An affine scheme is a locally ringed space that is isomorphicto (SpecA,OSpecA).

There is an equivalence of categories

AffSch→ CRingop; X 7→ H0(X,OX).

The real content of this is that if A → B is a ring map, and get the mapSpecB → SpecA, these consist of all morphisms as locally ringed spaces. Inparticular, I can take a pathological map like

SpecB → p, OY → f∗OX .

Math 99r Notes 12

But the thing is that if it is a local homomorphism, it can’t be anything otherthan coming from a ring homomorphism. If p is mapped to q, then we shouldhave a commutative diagram

A B

Ap Bq

such that Ap → Bq is a local homomorphism.

Definition 3.3. A scheme is a locally ringed space that is locally isomorphicto an affine scheme.

This is just like a manifold. In general, if you want to give a morphismX → Y , you need to give the data of f : X → Y and also f# : OY → f∗OX .But we can also describe this using the equivalence between affine schemes andcommutative rings. Given any f : X → Y , we can locally find a neighborhoodf(p) ∈ U ∼= SpecB ⊆ Y , and then find a neighborhood p ∈ V ∼= SpecA ⊆f−1(U) and then just describe the ring map A→ B.

In practice, we work with schemes over some base scheme S. This meansthat the objects are morphisms X → S. A scheme X can be considered as thatover SpecZ. Varieties are schemes over an algebraically closed field k = k canbe thought of as a scheme over k, with some special property.

Math 99r Notes 13

4 February 8, 2018

Let us take k = k. Consider the scheme Spec k[t]/(t2). For a A = k[x1, . . . , xn]/Iand a closed point m ⊆ A, we define the Zariski cotangent space as m/m2,and the tangent space as Homk(m/m2, k).

Proposition 4.1. There is a natural identification

HomSch/k(Spec k[t]/(t2),SpecA) ∼= Homk(m/m2, k).

Proof. The map of schemes is defined by a ring map f : A → k[t]/(t2). Giventhis, we can construct a map

m/m2 → k; x 7→ f(x).

Conversely, suppose that we have ψ : m/m2 → k. Then we get a map

f : A→ k[t]/(t2); x 7→ x+ ψ(x− x)t.

The hard thing is to verify that this is a ring homomorphism. You can verifythis.

4.1 Fiber product of schemes

Fiber products can be discussed in general categories. This is an object X×S Ysuch that for every T the following diagram can be filled in uniquely:

T

X ×S Y X

Y S

What this really means is that

hX×SY = hX ×hShY .

This should be interpreted as when tested against any test scheme T , we getthis in Set. If S is a final object, the fiber product is the same as the product.There is a natural map A×S B → A×B, and this is always a monomorphism.In the case of sets, it is

A×S B = (a, b) ∈ A×B : f(a) = f(b) ∈ S.

Proposition 4.2. Fiber products preserve monomorphisms. Composition offiber products is a fiber product.

Math 99r Notes 14

We would like to construct a fiber product of schemes. This is actuallyequivalent to proving that the functor

hX ×hShY

is representable.

Proposition 4.3. If you can cover X by Xi and each Xi ×S Y exists, thenX ×S Y exists.

Proof. The isomorphisms Uij ∼= Uji induce isomorphisms between X×SUij andX ×S Uji. Then you can glue these together.

Now cover S by Si. If Xi and Yi are the preimages of Si, then

Xi ×SiYi = Xi ×Si

(Si ×S Y ) = Xi ×S Y.

So we reduce everything to the affine case. Here, you can check that SpecA×SpecR

SpecB is just Spec(A⊗R B).

Example 4.4. This works as how you would expect. We have a pullback

Spec k[x, y] Spec k[x]

Spec k[y] Spec k.

That is, A1k × A1

k = A2k.

You can also define fibers using fiber products. If you map Spec k to aclosed point in X and take the fiber product with Y → X, you will get thefiber. Similarly, you can take intersections.

Definition 4.5. A closed immersion Y → X is a morphism that is, on topo-logical spaces, a homeomorphism onto its image, and OX → f∗OY is surjective.

Example 4.6. Consider the projection map k[x]→ k[x, y]/(y2−x). If you takethe fiber over a, you get

Ya = Spec k[y]/(y2 − a) Spec k[x, y]/(y2 − x)

Spec k[x]/(x− a) Spec k[x].

For a = 0, you get Spec k[y]/(y2).

If you think about dimkH0(Ya,OYa

), this is always 2. This can be thoughtof as the intersection number.

Math 99r Notes 15

5 February 13, 2018

5.1 Pullback of sheaves

Definition 5.1. For F : A → B and G : B → A, we say that F and G areadjoint functors if for all a ∈ A and b ∈ B,

HomA(a,G(b)) = HomB(F (a), b).

There should be compatibility conditions

Hom(a′, G(b)) Hom(F (a′), b)

Hom(a,G(b′)) Hom(F (a), b′).

Let f : X → Y be a morphism of topological spaces. If F is a sheaf on X,you can push forward to

f∗F (V ) = F (f−1(V )).

Denote the left adjoint of this by f−1, so that

Hom(f−1G ,F ) = Hom(G , f∗F ).

Concretely, we can write this as

f−1G (U) = limV⊇f(U)

G (V ).

You should verify that this is really an adjoint.If X and Y are ringed spaces, we can talk about sheaves of modules. Let

f : X → Y with OY → f∗OX . Then we can push forward an OX -module

f∗F (V ) = F (f−1(V ))

and consider as an OY -module. In this case, we have an adjoint f∗. IN partic-ular,

f∗ : Mod(Y )→ Mod(X)

is a functor. This can be actually written as

f∗G = f−1G ⊗f−1OYOX .

5.2 Quasi-coherent sheaves

In most cases, we care about quasi-coherent sheaves. Given a A-module M , wecan build out a sheaf M on SpecA, given by

M(D(f)) = M ⊗A Af .

Math 99r Notes 16

Definition 5.2. A sheaf F (over a scheme X) is quasi-coherent if you cancover X by affine opens Xi with Xi = SpecAi such that F |Xi

∼= Mi for someMi an Ai-module.

In this case, you can prove that for any SpecA ⊆ X, we will have F |SpecA∼=

M for some A-module M .Another important fact is that if X = SpecA, then there is an equivalence

of categoriesMod(X) ←→ ModA.

Moreover, it is an equivalence of categories, so that 0 → M ′ → M → M ′′ → 0is exact if and only if 0→ M ′ → M → M ′′ → 0 is exact.

If s ∈ F (D(f)), then there exists an n such that fns can be lifted to aglobal section. This can be used to prove that if

0→ F ′ → F → F ′′ → 0

with F ′ quasi-coherent over X = SpecA, then

0→ H0(X,F ′)→ H0(X,F )→ H0(X,F ′′)→ 0.

Let f : X → Y be a morphism with X = SpecA and Y = SpecB. Then wehave a map B → A. Given a N on Y , what is f∗A? We have

HomA(N ⊗B A,M) = HomB(N,MB),

and this shows that f∗N = (N ⊗B A)˜.You can prove the following:

• pullbacks of quasi-coherent sheaves are quasi-coherent

• pushforwards of quasi-coherent sheaves along quasi-compact separatedmorphisms are quasi-coherent

• kernels, cokernels, direct sums of quasi-coherent sheaves are quasi-coherent

A sheaf of ideals is a sheaf that locally looks like I an ideal of A. Thensheaves of ideals corresponds to closed subschemes. You can think of it as aclosed embedding Y → X corresponding to

0→ I → OX → f∗OY → 0.

5.3 Sheaf of relative differentials

There are also sheaves of relative differentials. If A → B is a ring map and Mis a B-module, a map d : B →M is called an A-derivation if it satisfies

d(b+ b′) = db+ db′, d(bb′) = bdb′ + b′db, da = 0.

The module Ω1B/A is characterized by

DerA(B,M) = HomB(Ω1B/A,M)

Math 99r Notes 17

for all B-module M .

B M

ΩB/A

dM

Here is another way to construct this. Consider the kernel

0→ I → B ⊗A B → B → 0

where the map B⊗AB → B is given by b⊗b′ 7→ bb′. Here, I/I2 has a B-modulestructure because B⊗AB/I = B. This is going to be generated by b⊗b′−b′⊗bwith

d : B → I/I2; b 7→ 1⊗ b− b⊗ 1.

It can be shown that (I/I2, d) realizes Ω1B/A.

For X → Y with I the sheaf of of ideals of

∆ : X → X ×Y X,

we can pull back I /I 2 and get a sheaf of relative differentials Ω1X/Y =

∆∗(I /I 2). This is something like the TM ∼= N∆(M)/M ×M and T ∗M beingequal to the conormal bundle.

If f : X ′ → X is a pullback of Y ′ → Y , then

Ω1X′/Y ′ = f∗Ω1

X/Y .

This is because formation of sheaves of relative differentials commutes withpullback. This is because you can locally show that

Ω1B′/A′

∼= Ω1B/A ⊗B B

′.

Let f : X → Y be a morphism such that f∗OY → OX . For X → Y → Z,we then get

f∗Ω1Y/Z → Ω1

X/Z → Ω1X/Y → 0.

You shouldn’t always think of Z as a point.

Example 5.3. Consider X = A2 projecting to Y = A1 projecting to Z = ∗.Then this sequence looks like

k[x, y]〈dx〉 → k[x, y]〈dx, dy〉 → k[x, y]〈dy〉 → 0.

Example 5.4. Consider the curve A = k[x, y]/(y2 − x3). Then

Ω1A/∗ = A〈dx, dy〉/(2ydy − 3x2dx).

At the singularity (x, y) = (0, 0), you see that the fiber is M/mM = k〈dx, dy〉.

Math 99r Notes 18

Example 5.5. Consider k[x]→ k[x, y]/(y2 − x). Then

Ω1A/B = (k[x, y]/(y2−x))〈dx, dy〉/(2ydy−dx, dx) = (k[x, y]/(y2−x))〈dx, dy〉/(2ydy).

This is the skyscraper sheaf at y = 0, i∗Oy=0.

We have ωY = Ω1Y/∗, but if X → Y is a map of curves, we have

0→ f∗Ω1Y/∗ → Ω1

X/∗ → Ω1X/Y → 0.

Here O(ram. divisor) is going to be Ω1X/Y . This is why we were able to prove

Riemann–Hurwitz.

Math 99r Notes 19

6 February 15, 2018

6.1 Divisors

Let me talk more about divisors. For simplicity, assume that X is a smoothvariety over k = k. Then every local ring is a UFD, and so every local ring is aDVR.

Definition 6.1. A prime divisor is a closed integral subscheme of codimension1. A Weil divisor is a linear combination

∑i niYi where Yi are prime divisors.

For f ∈ K(X)×, we define

div(f) =∑

y codim. 1

valy(f)Y

where y is the generic point of Y . (Here the local ring at y is going to be aDVR.)

Definition 6.2. A divisor is principal if it is div(f) for some f ∈ K(X)×.Define the class group as

Cl(X) = DivX/PDivX.

It is well-known that a ring A is a UFD if and only if all prime ideals ofheight 1 is principal.

Proposition 6.3. If you have Y ⊆ X, take U = X − Y . Then there exists anexact sequence

Z→ Cl(X)→ Cl(U)→ 0.

For instance, we have

Z→ Cl(Pn)→ Cl(An) = 0→ 0

and so Cl(Pn) ∼= Z.

Definition 6.4. A Cartier divisor is an element of Γ(X,K ×/O×X). It iscalled principal if it lies in the image of Γ(X,K ×)→ Γ(X,K ×/O×X).

There is a section in Hartshorne, which is to go between Weil divisors, Cartierdivisors, and invertible sheaves. From Weil divisors to Cartier divisors, you needto use a small algebraic statement that every codimension 1 integral subschemeis locally cut out by a single function.

Example 6.5. We have Cl(P1 × P1) = Z2 because P1 minus two lines is A2.

Math 99r Notes 20

6.2 Homological algbera

If you have a short exact sequence

0→ A• → B• → C• → 0

of complexes, then you get a long exact sequence

· · · → Hi(A•)→ Hi(B•)→ Hi(C•)→ Hi+1(A•)→ Hi+1(B•)→ · · · .

There is also the notion of homotopy of chain maps. If f, g : A• → B• theare called homotopic if they differ by some

f − g = dh+ hd.

In this case, f∗, g∗ : Hi(A) → Hi(B) are identical because f(a) − g(a) =dh(a) + hd(a) = dh(a) is a boundary.

Definition 6.6. In an abelian category, an injective object is an object Isuch that

0 A B

I

always exists.

Lemma 6.7. Let 0 → A → I• be an injective resolution, and 0 → B → J• beany resolution. Then we can lift

0 B J0 J1 J2 · · ·

0 A I0 I1 I2 · · ·

and moreover such lift f and f ′ are always chain homotopic.

Let F : A → B be a left exact functor, and assume that A has enoughinjectives, i.e., every object is a subobject of an injective object. Now let us takean injective resolution 0→ A→ I•, apply the functor, and take cohomology

RiF (A) = Hi(F (I•)).

This is well-defined because if you take any different resolutions, you can alwaysextend the identity map and so the chain map is homotopic to the identity map.

6.3 Sheaf cohomology

Now let X be a scheme and consider the functor

Γ : Mod(X)→ Ab; F 7→ Γ(X,F ).

Math 99r Notes 21

This is left exact, and then we define as

Hi(X,F ) = RiΓ(F ).

If X → SpecA is relative, this lands on ModA.If we don’t assume the base scheme is not affine, consider f : X → Y and

we can consider the higher direct image as Rif∗ : Mod(X)→ Mod(Y ).This is not very computable, and you can use Cech cohomology. Let U =

Ui be a cover of X. For a sheaf F , you define

Cp(U ,F ) =∏

i0<···<ip

F (Ui0···ip)

and maps d : Cp → Cp+1 as

(dα)i0···ip+1 =

p+1∑k=0

(−1)kαi0···ik···ip+1|Ui0···ip+1

.

Then Cech cohomology associated to this cover is just

Hi(U ,F ) = Hi(C∗, d).

There is always a natural map

Hi(U ,F )→ Hi(X,F ),

and if U is good enough, this is an isomorphism, and it works for quasi-coherentsheaves F .

It is a fact that a scheme X is affine if and only if Hi(X,F ) = 0 for allF ∈ Qch(X). This is called Serre’s affineness criterion.

Math 99r Notes 22

7 February 20, 2018

For a functor F : A → B, we defined RiF (A) as taking an injective resolution0→ A→ I• and taking

RiF (A) = Hi(F (I•)).

We apply this construction to a ringed space (X,OX) with an OX -module F .Then we have a functor

Γ : ModX → Ab; F 7→ Γ(X,F ).

Then if X is a scheme over SpecA then we can consider it as Γ : ModX → ModA.Similarly, you can push-forward f∗ : ModX → ModY and then we have thehigher direct images Rif∗.

Normally injective resolutions are not computable, so we introduce Cechcohomology. For U a covering, we define

CP (U,F ) =∏

i0<···<ip

F (Ui0,...,ip),

(dα)i0,...,ip+1=

p+1∑k=0

(−1)kαi0,...,ik,...,ip+1|Ui0···ip+1

.

Then we define H(U,F ) as the cohomology of this complex, and

H(X,F ) = lim−→U

H(U,F ).

Here, the limit is over refinements of covers. It is a fact that if X is affine,Hi(X,F ) = 0 for all i > 0 and F ∈ Qch(X).

Example 7.1. Let us use X = P1k and let us compute H∗(X,Ω1

X). We use thestandard covering U, V ∼= A1. Then

0→ Γ(U,Ω1)⊕ Γ(V,Ω1)→ Γ(U ∩ V,Ω1)→ 0.

The nonzero map is

d : (f(x)dx, g(y)dy) 7→(f(x) +

1

x2g( 1

x

))dx.

Here, ker d = 0 and coker d is generated by x−1dx. So

H0(X,Ω1X) = 0, H1(X,Ω1

X) ∼= k

for X = P1.

Let X be a scheme. You can easily convince yourself that

H1(X,O×X) = PicX.

Math 99r Notes 23

7.1 Chern classes

Example 7.2. Let R be the constant sheaf, and let M be a manifold. Then

H∗(M,R) = H∗dR(M).

The reason is that

0→ R→ Ω0M → Ω1

M → Ω2M → · · ·

is an acyclic resolution, because all these sheaves ΩiM are flasque. So we cancompute cohomology using this complex, and this is precisely de Rham coho-mology.

We can also consider the holomorphic case. Let us look at the complexanalytic topology here. We can consider the exponential sequence

0→ Z 2πi−−→ OXexp−−→ O×X → 1

This is exact, because this is true on a simply connected neighborhood. Thisinduces a map

c1 : Pic ∼= H1(X,O×X)→ H2(X,Z) ∼= H2sing(X,Z).

So this is a group homomorphism c1 : PicX → H2(X,Z), and is called the firstChern class. That is, c1(L1 ⊗ L2) = c1(L1) + c1(L2).

If X is a smooth projective surface over C, the group H2(X,Z) is equippedwith a Poincare pairing. That is, we have a pairing on PicX:

〈L1, L2〉 = 〈c1(L1), c2(L2)〉Poincare.

7.2 Intersection number

If s ∈ H0(X,L ⊗ K) such that s0 = M and s∞ = N are smooth complexmanifolds, then

c1(L) = [M ]− [N ].

In particular, if C and D are curves, you can convince yourself that

〈OX(C),OX(D)〉 = #(C ∩D)

is sort of the intersection number. Because (co)homology class doesn’t dependon homotopy, we see that the intersection number only depends on the linebundle.

Definition 7.3. Let C,D be two curves onX whose intersection is 0-dimensional.We define

C.D = dimkH0(C ∩D,OC∩D)

where C ∩D is the scheme-theoretic intersection.

Math 99r Notes 24

If x is a closed point in supp(C)∩ supp(D), we can define the multiplicityof C ∩D at x as

multx(C ∩D) = dimk Ox,X/(f, g).

Here, if x is the only intersection of (f) and (g), then

Am/(f, g) = (A/(f, g))m = A/(f, g).

We need that mN ⊆ (f, g), but this follows from the standard fact on Artinianrings.

We know that OX(−D) = ID is the ideal sheaf. Then

0→ ID|C → OC → i∗OC∩D → 0,

where i : C ∩D → C is inclusion. It is a fact that if X is projective and F iscoherent, then Hi(X,F ) is finite-dimensional. Then we can define the Eulercharacteristic as

χ(X,F ) =

∞∑i=0

(−1)i dimkHi(X,F ).

Then if 0→ F ′ → F → F ′′ → 0 is exact then χ(F ) = χ(F ′) + χ(F ′′).Anyways, we get

χ(C,OC) = χ(C,OX(−D)|C) + χ(C ∩D,OC∩D).

It is clear that χ(C ∩D,OC∩D) = C.D.Now let me give a different formula. We have

0→ OX(−C −D)→ O(−C)⊕ O(−D)→ OX → OC∩D → 0.

This is because locally we have

0→ (fg)→ (f)⊕ (g)→ A→ A/(f, g)→ 0.

The first part requires (f) ∩ (g) = (fg), and this follows from the fact that thelocal rings are UFD. Then we can write

χ(OC∩D) = χ(OX)− χ(OX(−D))− χ(OX(−C)) + χ(OX(−C −D)).

7.3 Proj construction

Let S =⊕

d≥0 Sd be a graded ring, so that Sd ·Sd ⊆ Sd+e. We say that an ideala ⊆ S is homogeneous if

a =⊕d≥0

a ∩ Sd.

This means that a is generated by homogeneous elements. For instance, (x +y2) ⊆ k[x, y] is not homogeneous.

Math 99r Notes 25

On projective space Pn, it doesn’t make sense to talk about values of apolynomial f at a point, but for a homogeneous polynomial f , it makes sense totalk about the vanishing locus of f . Note that products, intersections, radicals,of homogeneous ideals are homogeneous. I previously introduced a operationSpec that takes a ring to a scheme. Now I want to introduce an operation Projthat takes a graded ring to a scheme.

For a graded ring S, we define

Proj(S) = (X,OX),

where

• X as a set is the homogeneous prime ideals p ⊆ S such that p ⊇ S>0 =⊕d>0 Sd,

• the topology is generated by distinguished open subsets

Xf = p 63 f,

• the functions on the distinguished open Xf is OX(Xf ) = (Sf )0, where(Sf )0 means the degree 0 elements of Sf .

For instance, if we have S = k[x0, . . . , xn] then

OX(Xx0) = (k[xi][x

−10 ])0 = k[x1

x0, . . . , xn

x0].

We can also say (Xf ,OXf) = Spec(Sf )0. If we assume that S is finitely gener-

ated by S1 over S0, then you can show that Proj(S) is a closed subscheme ofsome PnS0

.

Math 99r Notes 26

8 February 22, 2018

Last time we defined the Proj construction. For S a graded ring, we defined ascheme

Proj(S) = (X,OX).

For f ∈ S homogeneous, we had Xf = f /∈ p and

OX(Xf ) = (Sf )0.

You can further show that

(Xf ,OX |Xf) ∼= Spec(Sf )0,

and so Proj(S) is a scheme. For example, if we take S = k[x0, . . . , xn] with thenormal grading, then

X = Proj(S) = Pnk .

8.1 Twisted sheaves

Definition 8.1. A morphism f : X → Y is said to be projective if there existsa closed immersion

X → PnZ × Y

such that its composite with the projection map is equal to f .

If I is a homogeneous ideal in S, then there is a natural map

Proj(S/I)→ Proj(S).

In general, if S T then we always get a closed immersion Proj(T )→ Proj(S).Locally, this is going to look like Spec(Sf/I)0 → Spec(Sf )0. Different ideals cangive can give you the same subscheme. For instance, if we take I ′ =

⊕d>m Id,

then(Sf/I)0

∼= (Sf/I′)0

because anything that looks like x/fn can be thought of as xfN/fm+N for largeenough N .

To classify the sheaf of ideals, we can just classify the quasi-coherent shaves.For an arbitrary graded module M , we can define a quasi-coherent sheaf

M |Xf∼= (Mf )0.

Then this can be shown to be a quasi-coherent sheaf.Proj(S) is equipped with a natural Z-family of line bundles. Here is how you

do this. For each graded module M , we can define a shifted module M(`) suchthat M(`)d = M`+d. This is a new graded module with different grading. So

we take S as a graded S-module, shift it, and take S(n). This is a line bundleon Proj(S).

Math 99r Notes 27

Example 8.2. Take S = k[x0, . . . , xn] so that Proj(S) = Pnk . We know thatPicPnk = Z. In this case, S(m)˜∼= OPn(m) for m, by comparing degree.

Unlike affine schemes, projective schemes tend to have not so many globalfunctions. Similarly, a coherent sheaf over a projective scheme tend to have“not-so-many” global sections. But this can be remedied by twisting.

Definition 8.3. For F ∈ Qch(ProjS), we define F (n) = F ⊗ OS(n).

Theorem 8.4. If F is coherent, there exists a n 0 such that F (n) is globallygenerated.

Proof. For Pn, if F |Ani

∼= Mi, then I can take sij ∈ Mi such that sij generateMi. For each sij , there exists a large n such that sijx

ni can be promoted to a

global section tij . Then pick N sufficiently large so that tij are global sectionsof F (N).

This means that we can realize F as the quotient of a direct sum of OX(n).

Definition 8.5. For F a coherent sheaf on Proj(S), we define

Γ∗(F ) =⊕n∈Z

Γ(Proj(S),F (n)),

which is a graded module over S.

Theorem 8.6. For F coherent, we have Γ∗(F )˜∼= F .

So if we apply this to I a sheaf of ideals, we see that all closed subschemes aregiven by homogeneous ideals of S. Moreover, finite type Y/ SpecA is projectiveif and only if Y ∼= Proj(S) for some S satisfying our assumption that S0 = Aand S is finitely generated by S1 over A.

8.2 Properties of morphisms

Definition 8.7. We say that a map of schemes X → Y is (locally) finite typeif for every V = SpecB ⊆ Y and every SpecA = U ⊆ f−1(V ), the induced mapB → A is a ring homomorphism of finite type, that is, A is finitely generatedover A.

Definition 8.8. We say that a morphism f : X → Y is quasi-compact if forevery V ⊆ Y , f−1(V ) is covered by finitely many affine opens.

If f : X → Y is quasi-compact and F ∈ Qch(X), then Rif∗F ∈ Qch(Y ) forall i. Higher push-forward commutes with restriction, so if V ⊆ Y then

Rif∗F |V ∼= Rif∗(F |f−1(V )).

This gives a method to compute higher push-forwards. On each open SpecB =V → Y , we have

Rif∗F |V = Hi(f−1(V ),F |f−1(Y ))˜.

Math 99r Notes 28

Definition 8.9. A morphism f : X → Y of schemes is affine if there existsan affine cover Vi such that each f−1(Vi) is affine, or equivalently, for everyaffine V ⊆ Y the preimage f−1(V ) is affine.

Let’s prove that they are actually equivalent. This is a nice application ofSerre’s affineness criterion and spectral sequences. If f : X → Y is affine, thenf∗f∗F → F is surjective for F ∈ Qch(X) and Rif∗F = 0 for i > 0.

8.3 Introduction to spectral sequences

Consider a double complex

......

. . .

E1,0 E1,1 · · ·

E0,0 E0,1 · · ·

such that the square anti-commutes. Then we can define a total complex

Tot(E∗,∗)p =⊕i+j=p

Ei,j .

A spectral sequence gives some filtered information on the cohomology of thefiltered complex.

The E0-page →E∗,∗0 is this double complex, with the differentials to hori-

zontal lines. The E1-page E∗,∗1 is the cohomology of this complex, with arrowspointing vertically. The E2-page is the cohomology of the E1-page with thearrows pointing left+up+up. At each page, we take the cohomology, so we aretaking a subquotient. Then the things get smaller, and will converge to some

→E∗,∗∞ .

The statement is that there exists a Hp(Tot) such that

→E0,p∞ • · · · Hp(Tot).

→E1,p−1∞ →E

p,0∞

Likewise, you can do the some thing with the vertical maps, and consider ↑E∗,∗.

The snake lemma is a special case of the spectral sequence. If we take adiagram

0 A′ A A′′ 0

0 B′ B B′′ 0.

α β γ

Math 99r Notes 29

If we look at the horizontal direction, →E∗,∗1 = 0 and so H∗(Tot) = 0. If we

look at the vertical direction, the E1-page looks like

kerα kerβ ker γ

cokerα cokerβ coker γ.

If we look at the E2-page, we should get

? ? ??

?? ? ?

and these can never cancel out. This shows that ? should all be 0, and ??→??should be an isomorphism. So we can put them together and get an exactsequence

0→ kerα→ kerβ → ker γ → cokerα→ cokerβ → coker γ → 0.

Math 99r Notes 30

9 February 27, 2018

Last time I wanted to show that affineness can be checked locally. Namely, iff : X → Y with Yi an affine cover such that f−1(Yi) is affine, then f if affine.We can reduce this to the case when Y is affine. I want to show that if F is aquasi-coherent sheaf on X, then Hi(X,F ) = 0 for i > 0. We are going to usethe Leray spectral sequence, and this is an instance of a Grothendieck spectralsequence.

Theorem 9.1 (Grothendieck spectral sequence). Let A F−→ B G−→ C be mor-phisms of abelian categories, and assume that A and B have enough injectives,with F sending F -acyclics to G-acyclics. Then there exists a spectral sequence

Ep,q2 = RpG RqF (A) =⇒ Rp+q(G F )(A).

Let us apply this to a special case. Let f : X → Y be a morphism of schemesover A. Then we can consider

Mod(X)f∗−→ Mod(Y )

Γ−→ ModA.

You can check that it satisfies the condition F mapping F -acyclics to G-acyclics,because it is derived pushforward is computed locally. Sow we get the Lerayspectral sequence

Hp(Y,Rqf∗F ) =⇒ Hp+q(X,F ).

If f−1(Yi) are affine opens in X, then we get Hp(Y,Rqf∗F ) = 0 for p > 0because Y is affine, and also Hp(Y,Rqf∗F ) = 0 for q > 0 because the map islocally affine. Therefore Hi(X,F ) = 0 for all i > 0.

9.1 Flat morphisms

Let me first give a motivation. We want to make sense of a “continuous familyof schemes”. For instance, we want a family of curves Ct ⊆ P2 for t ∈ T . Wecan take just Z ⊆ P2 × T , but we want this to be something like changingcontinuously. But we don’t want it to be smooth, because we want to allowramification.

Definition 9.2. Let F ∈ Mod(X) and f : X → Y . We say that F is flat overY if for all x ∈ X, Fx is flat over Of(x),Y . We say that f : X → Y is flat if OXis flat over Y .

Flatness can be checked locally. That is, M/A is flat if and only if Mp is flatover Ap for all p. This is because computing Tor commutes with localization.

Let f : X → Y be separated finite type, ν : Y ′ → Y flat, and F ∈ Qch(X).Take the fiber product

X ′ X

Y ′ Y.

g

u

f

ν

Math 99r Notes 31

Then the claim is that there exists an isomorphism

ν∗Rif∗F∼=−→ Rig∗u

∗F .

First of all, there exists ν∗f∗F → g∗u∗F because this is equivalent to

f∗F → ν∗g∗u∗F = f∗u∗u

∗F .

Hartshorne constructs this for i > 0 locally, but let’s take this for granted.Assume that Y = SpecA and Y ′ = SpecA′. Then our statement just says that

Hi(X,F )⊗A A′ ∼= Hi(X,F ′),

where F ′ = u∗F . Take an affine cover U of X, so that

Hi(X,F ) = hi(C•(U,F )).

Then U∗ = u−1(U) : U ∈ U is an affine cover of X ′, so that we can take

hi(C•(U′,F ′)) = hi(C•(U,F )⊗A A′) = Hi(X,F )⊗A A′.

9.2 Serre’s vanishing theorem

Let me just define this now, and explain its geometric meaning next time.

Definition 9.3. For X/k a projective scheme, we say that a line bundle L isvery ample if there exists an embedding X → Pn such that L ∼= i∗OPn(1).We say that L is ample if for all F ∈ Coh(X), there exists an n0 such thatfor all n > n0, F ⊗L ⊗n is globally generated.

It can be shown that L is ample if and only if L ⊗n is very ample.

Theorem 9.4 (Serre). Let X/A be projective, and let OX(1) be very ample.Then for every F ∈ Coh(X), for n 0, we have Hi(X,F (n)) = 0 for alli > 0.

We can first reduce to the case X = PrA. This is because if F is coherent onX, then i∗F is coherent on PrA and moreover

Hi(X,F ) ∼= Hi(Pr, i∗F )

by because i : X → PrA is affine.Now for n 0, we saw that F (n) is generated by global sections. Then we

have a surjectiveN⊕i=1

OX → F (n)→ 0,

and then we get some

0→ R → E =

N⊕i=1

OX(−n)→ F → 0.

Now we begin to induct. First, the base case is for i > r, we haveHi(Pr,F ) =0 for all F . The claim we want to prove is that

Math 99r Notes 32

• Hi(X,F ) is finitely generated,

• Hi(X,F (n)) = 0 for n 0.

But we have a long exact sequence

· · · → Hi(X,E )→ Hi(X,F )→ Hi+1(X,F )→ Hi+1(X,E )→ · · · .

Let X be a projective scheme over k. Let OX(1) be very ample, and letF be a coherent sheaf. Then χ(F (n)) is a polynomial, called the Hilbertpolynomial.

Theorem 9.5. Let us assume that T is integral and Noetherian. Take X ⊆ PnTa closed subscheme. Then X is flat over T if and only if χ(Xt,OXt(m)) isindependent of t.

We can more generally show that if F is a coherent sheaf over PnT , F is flatif and only if χ(Xt,Ft(m)) is independent of t. Next, we reduce to the casethat T = SpecA where A is Noetherian local. We can do this because we cancompare it with the generic point, and we can take localization.

The upshot is that checking flatness over a local ring is easy. If (A,m) is alocal ring and M is finitely generated over A, then M is flat if and only if Mis free if and only if dimkM ⊗ k = dimKM ⊗ K. We want to show that thefollowing are equivalent:

(i) F is flat over A.

(ii) H0(X,F (m)) is finitely generated ant free over A, for m 0.

(iii) Pt = χ(Xt,Ft(m)) is independent of t for m 0.

Math 99r Notes 33

10 March 1, 2018

I used the cohomology of line bundles on PnA, but let us compute this now. Wefirst have

H0(PnA,O(m)) = A〈monomials in x0, . . . , xn of degree m〉.

Also, we are going to have

Hn(PnA,O(m)) = A〈xi00 · · ·xinn : i0, . . . , in < 0, i0 + · · ·+ in = m〉.

So there is some duality, called Serre duality.Let X be a projective scheme over k, and let OX(1) be a very ample line

bundle. We have a coherent sheaf F ∈ Coh(X), and we have defined the Hilbertpolynomial as

pF (m) = χ(X,F (m)).

10.1 Associated points

Let M be a finitely generated module over a Noetherian ring A.

Definition 10.1. An associated point p ⊆ A is a point satisfying the followingequivalent conditions:

(1) p looks like p = Ann(m) for some m ∈M .

(2) ×f : Mp →Mp is never injective for all f ∈ pAp.

Once we have this, we can talk about global notions.

Definition 10.2. Let X be Noetherian scheme and F be a coherent sheaf onX. Then Ass(F ) is the set of points x ∈ X such that ×f : Fx → Fx is neverinjective for f ∈ mx,X ⊆ Ox,X .

You can show that Ass F is a finite set.

Lemma 10.3. Let L be a line bundle, and s ∈ Γ(X,L ), then ×s : F → F⊗Lis injective if and only if s does not vanish on any Ass(F ).

We also call Ass(OX) the associated points of X.

Example 10.4. Let X = Spec k[x, y]/(x2, xy). This looks like (x) ∩ (x, y)2.There are going to be two associated points: the generic point and the point(x, y). This is because xf = 0 for all f , and so multiplication by f is neverinjective. An associated point that is not a generic point is called an embeddedpoint.

Let us go back to our Hilbert polynomial. Assume that k is infinite, by basechange. Choose a hyperplane x = 0 that does not pass through any x ∈ Ass(F ).Then we have a map

F (−1) → F ; s 7→ s⊗ x.

Math 99r Notes 34

We can then take0→ F (−1)→ F → G → 0.

The point here is that dim supp(G ) = dim supp(F )− 1 unless F = 0, and then

χ(G (m)) = χ(F (m))− χ(F (m− 1)).

Example 10.5. For X = Pnk an F = OX , we have

χ(X,OX(m)) =∑i

(−1)idimHi(X,OX(m)) =

(n+m

n

).

We still can distill some invariants out of this. For instance,

pOX(0) = χ(X,OX), deg pOX

= dimX.

Example 10.6. Consider a hypersurface X ⊆ Pnk of degree d. Then there is adefining exact sequence

χ(X,OX(m)) =

(n+m

n

)−(n+m− d

n

).

In particular, if n = 2 then we have

χ(X,OX(m)) =−d2 + 3d

2+md.

For m = 1, we get

χ(X,OX) =−d2 + 3d

2= 1− dimH1(X,OX) = 1− dimH0(X,ΩX).

So if X is smooth, we get

g =1

2(d− 1)(d− 2).

Also, note that the leading coefficient of the polynomial times (n − 1)! is thedegree of the embedding.

10.2 Criteria for flatness

Let F be a coherent sheaf over X = PnT , where T = SpecA with A Noetherianlocal. We want to show that the following are equivalent:

(i) F is flat over T ,

(ii) H0(X,F (m)) is a free A-module of finite rank, for m 0,

(iii) χ(Xt,FXt(m)) is independent of t.

Math 99r Notes 35

Let us first show (i) ⇒ (ii). Let U be an affine cover. We know thatHi(X,F (m)) = 0 for m 0 and i > 0, so

0→ H0(X,F (m))→ C0(U,F (m))→ C1(U,F (m))→ · · ·

is exact for m 0. But all of the Ci are flat. It is also true that 0 → M ′ →M →M ′′ → 0 with M and M ′′ flat implies that M ′ is flat. Then we inductivelyget H0(X,F ) is flat. But for a finitely generated module over a Noetherian localring, flat is equivalent to free.

Now let us show (ii) ⇒ (i). Take m0 such that (ii) holds for m ≥ m0. Thenfor

M =⊕m≥m0

H0(X,F (m)),

we have F = M .For (ii) ⇒ (iii), it suffices to show that pt(m) = rankAH

0(X,F (m)). Nowconsider a set of generators for m ⊆ A and take

Aq → A→ kt → 0.

Then we getF q → F → Ft → 0.

For m 0, we will get an exact

H0(X,F (m))q → H0(X,F (m))→ H0(X,Ft)→ 0,

and by freeness, we have

H0(X,F (m))q → H0(X,F (m))→ H0(X,F (m))⊗A kt → 0.

So we get the isomorphism we want.

Math 99r Notes 36

11 March 20, 2018

Today we are finally going to talk about curves on surfaces.

11.1 Cartier divisors

Definition 11.1. A Cartier divisor is an element of Γ(X,K ×/O×X). It iscalled effective if its local equations are holomorphic.

Cartier divisors are in one-to-one correspondence with Weil divisors. Thisis because the short exact sequence 1→ O×X → K ×

X → K ×X /O×X → 0 induce a

connecting homomorphism

H0(K ×X /O×X)

δ−→ H1(O×X) = Pic(X).

You can pull back Cartier divisors. Assume that g : X → Y , and assumethat if x ∈ Ass(X) then g(x) /∈ supp(D). Then for local equations fi, you cancheck that g∗(fi) are locally not a zero divisor, because locally, the zero divisorsof A are

⋂p∈Ass(A) p. In particular, if g is flat, this condition is automatically

satisfied.Let g : X → X be a flat map, and D ⊆ X be a Weil divisor, and D be the

corresponding Cartier divisor. The following conditions are equivalent:

(i) D → Y is flat,

(ii) for all x ∈ X, a local equation F at x is not a zero divisor in the ringOx,X ⊗Oy

κ(y) for y = g(x),

(iii) Ass(g−1(y)) ∩ supp(D) = ∅,(iv) F is not in any associated prime ideal of B/pB, for any p ⊆ A prime.

For instance, let us show that (iv) implies (i). Locally let us write g : SpecB →SpecA, and we want to show that B/(F ) is flat over A. We have

0→ B×F−−→ B → B/(F )→ 0,

and want to show that Tor1A(B/(F ), A/p) = 0 for all p ⊆ A. But we have

0 = Tor1A(B,A/p)→ Tor1

A(B/(F ), A/p)→ B/pB → B/pB.

But the map B/pB → B/pB given as multiplication by F is injective.

11.2 Curves on surfaces

Let X/k be a scheme. There are functors

CurveX : Sch/k → Set; S 7→

relative effectiveCartier divisors on X ×k S

PicX : Sch/k → Set; S 7→ line bundles on X × S/Pic(S).

Math 99r Notes 37

There is an obvious natural transformation CurveX → PicX given by sendingD to OX×S(D).

For surfaces, we don’t have degree, but there is still going to be a similarnotion. Define

Picτ (X) = L ∈ Pic(X) : L.L′ = 0 for all L′ ∈ PicX.

We then define the numerical class as Num(X) = PicX/Picτ X. This is alsocalled the Neron–Severi group. This is finitely generated over Z, and embedsinto H2(X;Z).

Proposition 11.2. Both functors can be written as

CurveX =∐

ξ∈Num(X)

CurveξX , PicX =∐

ξ∈Num(X)

PicξX .

11.3 Riemann–Roch for surfaces

Before proving this, let us talk more about curves on surfaces.

Theorem 11.3. Let C ⊆ X be a nonsingular curve. Then

2gC − 2 = C.(C +K)

where K is the canonical divisor.

Proof. We have 0→ IC/I 2C → Ω1

X/k|C → Ω1C/k → 0 and then

ωC ∼= ωX |C ⊗ OX(C)|C .

The degree of the left hand side is 2gC − 2, the degree of the right hand side isK.C + C.C.

With this, I can give a Riemann–Roch for surfaces. Recall that for curves,we had

χ(L ) = deg L − g + 1.

The Euler characterstic is some invariant of the line bundle plus some invariantof the curve.

Theorem 11.4 (Riemann–Roch for surfaces). For L a line bundle over X,

χ(L ) =1

2L .(L −K) + χ(OX).

Proof. We first reduce to the case when L = OX(C). Then we have 0 →OX(−C)→ OX → i∗OC → 0, so

0→ OX → OX(C)→ OC ⊗ OX(C)→ 0.

Now it suffices to compute χ(OC ⊗ OX(C)). But note that i∗(OX(C)|C) =i∗OC⊗OX(C). (This is because for finite morphisms, Hi(X,F ) = Hi(Y, g∗F ).)

Math 99r Notes 38

So we can compute Euler characteristic of OX(C)|C instead. But we haveRiemann–Roch for curves, and we have

χ(OX(C)|C) = C2 − C.(C +K) + 2

2+ 1 =

1

2C.(C −K).

Lemma 11.5. Let S be connected. For line bundle L1,L2 over X × S, theintersection number L1,s.L2,s stays constant in s.

This implies that the numerical class stays the same, because we can takeL2 to be the pullback of a line bundle on X.

Proof. Recall that the intersection number can be computed as

L1.L2 = χ(OX)− χ(L ∨1 )− χ(L ∨2 ) + χ(L ∨1 ⊗L ∨2 ).

But the coefficient of the Hilbert polynomials say constant.

In general, Pic(X × Y ) 6= Pic(X) × Pic(Y ). Here is an example. Let E bean elliptic curve, and look at the maps

Pic(E)2 → Pic(E × E)→ Pic(E)2

that compose to the identity. But there are other stuff in Pic(E × E). Forinstance, take n : E → E and look at the graph Γ([n]) ⊆ E × E. Then theimage of this to Pic(E)2 should be (0, n-torsions). If we map to Pic(E×E), Iclaim that we get a different line bundle. To see this, intersect with the diagonal.The new bundle we have is going to have n2 + 1, and the intersection numberof the original graph is (n− 1)2.

Math 99r Notes 39

12 March 22, 2018

For X/C projective, we have Hodge theory.

12.1 Hodge theory

For all k, we have

Hk(X;C) =⊕i+j=k

Hi,j

where Hi,j is canonically identified with Hj(X,ΩiX). Then

Hksing(X,Z)⊗Z C→ Hk(X,R)⊗C → Hk(X,C)

are isomorphisms.Recall that there is an exponential sequence

0→ Z→ OXexp−−→ O×X → 0.

Then we get a long exact sequence

· · · → H1(X,Z)→ H1(X,OX)→ H1(X,O∗X)δ−→ H2(X,Z)→ H2(X,OX)→ · · · .

This map δ i the map taking a line bundle to its numerical class. But if youtake the conjugate, we see that

im δ ⊆ ker(H2(X,C)→ H0,2 ⊕H2,0) = H1,1.

The Lefschetz (1, 1)-theorem states that in fact we have im δ = H2(X,Z)∩H1,1.On the other hand, the kernel is going to be

Picτ (X) = ker δ = H1(X,OX)/H1(X,Z).

But H1(X,Z) injects into H1(X,OX) and

rankH1(X,Z) = dimCH1(X,C) = dimRH

0,1.

This shows that the quotient is actually going to be a torus with a complexstructure.

12.2 Simple cases of the Picard scheme

Let us go back to the natural transformation of functors

CurveX → PicX .

Consider a point in PicX(k) = Pic(X), which is a line bundle L . The set ofdivisors D that map to L can be described as

H0(X,L )/k× = PH0(X,L ).

Math 99r Notes 40

We can check that this is the correct fiber on all S-points. Consider L as afunctor sending S to a point, with a natural transformation L → PicX givenby picking out OS L over S ×X. Then we can take the fiber

LinSysL

= L ×PicXCurveX .

Proposition 12.1. LinSysL∼= hPH0(X,L ).

Proof. We first haveLinSys

L(S) = D ⊆ X × S

such that D are relative effective Cartier divisors over S, such that OX×S(D) ∼=p∗1L ⊗ p∗2K for some K ∈ Pic(S). Likewise, by definition, we have

hPN−1(S) = K /S with s1, . . . , sn ∈ H0(K) such that OnS K/ ∼= .

Note that, this is the same as a section

s ∈ H0(X × S,L K ) = H0(X,L )⊗H0(S,K )

that is nonvanishing, up to scalar multiplication. So then we can take the zerolocus and get a divisor in D×S. One nontrivial thing to check is that K1

∼= K2

on S if and only if L K1∼= L K2. This can be checked by pulling back to

S.

Now we can compute these. For instance, suppose we want to computePicP2 . We know that its k-points is Z, but we don’t know if there are otherthings happening. We would like to say that PicP2 =

∐Z Spec k.

Theorem 12.2. If H1(X,OX) = 0, then Pic(X × S) = Pic(X)× Pic(S).

A corollary is that

PicX =∐

ξ∈Pic(X)

Spec k.

(More precisely, it is represented by that thing on the right.) To prove thetheorem, we need the following fact from Lecture 7 (in Mumford’s book).

Theorem 12.3. Let F be flat over Pn×S. If for some s0 ∈ S, Hj+1(Pns0 ,FPns0

) =0, then there exists an open U on which

g∗Rjp∗F ∼= Rjq∗h∗F

for any

Pn × T Pn × S

T S

q

h

p

g

with T → S factoring through U and any F on Pn × S.

Math 99r Notes 41

Proof. We only need to show that anything in Pic(X×S) comes from Pic(X)×Pic(S). Let L be a line bundle on X×S, and let s ∈ S. Consider the inclusionσ : X → X × S with the point s. Then we can define

M = L ⊗ (p∗1σ∗L )∨.

Then M |Xs∼= OXs and H1(Xs,MXs) = 0.

By the theorem we stated, we can take T = s, and because H1(Xs,MXs),we obtain

p1∗M ⊗ κ(s) ∼= H0(S,MXs).

Then because Γ(U, p1∗M )→ p1∗M ⊗ κ(s) is surjective for some affine open U ,any section in H0(S,MXs

) lifts to a section in H0(U ×X,M ). Suppose we lift1 to α. This gives a homomorphism

Φ : p∗1σ∗M →M |U×X .

This is means that it is an isomorphism on a neighborhood of XS . Because Xis quasi-compact, we can use the tube lemma to find an open neighborhood Wof s such that Φ is an isomorphism on W × S.

Now we can find a covering Wi of S, such that for each Wi × S we havean isomorphism p∗1σ

∗M ∼= M on each Wi ×X. Then we can glue them to getthat M ∼= OX K for some line bundle K on S.

Math 99r Notes 42

13 March 27, 2018

Last time there was this theorem about Pic(X × Y ) ∼= Pic(X) × Pic(Y ) ifH1(X,OX) = 0. Given L on X×S, we want to find L and K such that X andS such that L = LK.

So we defined L = L |Xsfor some s, and then defined

Ms = L ⊗ (p∗1Ls)∨.

Using that H1(X,OX) = 0, we showed that we can trivialize this on X ×U forsome neighborhood U of s. Then because H0(X × (U ∩ U ′),O×) = H0(U ∩U ′,O×), every transition function can be descended to a transition function onS. Moreover, because S is connected, this means that the isomorphism class ofL |Xs is locally constant in S, and thus Ms is really just M . Therefore we canpatch these local sheaves together to get what we want.

So when H1(X,OX) = 0, we have that Pic(X) is a discrete scheme. Infact, we will later identify H1(X,OX) with the tangent space of Pic(X) at theidentity.

13.1 m-regular sheaves

Definition 13.1. A coherent sheaf F on Pn is calledm-regular ifHi(Pn,F (m−i)) = 0 for all i > 0.

Serre vanishing tells us that Hi(Pn,F (m)) vanishes for m 0, but this istelling us when this vanishes.

Proposition 13.2. If F is m-regular, then

(a) H0(Pn,F (k − 1))⊗H0(Pn,OPn(1)) H0(Pn,F (k)) for all k > m.

(b) Hi(Pn,F (k)) = 0 for all i > 0 and k + i ≥ m.

Proof. Take a hyperplane H that avoids the associated points of F . Then wehave an exact sequence

0→ OPn(−H)→ OPn → i∗OH → 0,

and tensoring with F (k) gives

0→ F (k − 1)→ F (k)→ F (k)⊗ i∗OH → 0.

For k = m− i, we have

→ Hi(Pn,F (m− i))→ Hi(Pn,FH(m− i))→ Hi+1(Pn,F (m− i− 1))→ · · · .

Here, we are using induction on n. If F is m-regular, the first and third termsvanish, so the second term also vanishes. This shows that FH is also m-regular,because we can compute cohomology wherever. So from the exact sequence

Hi+1(F (m− i− 1))→ Hi+1(F (m− i))→ Hi+1(FH(m− i)),

Math 99r Notes 43

the first and third vanishes by induction, and so the second vanishes. Thisproves (b).

For (a), we consider the diagram

H0(F (k − 1))⊗H0(O(1)) H0(FH(k − 1))⊗H0(OH(1))

H0(F (k)) H0(FH(k)).

The top horizontal map is surjective, because H0(F (k− 1))→ H0(FH(k− 1))is already surjective, which is because H1(F (k − 2)) = 0. We want to showthat the left vertical map is surjective. But the kernel of the bottom map isH0(F (k− 1)), and by construction, the image of the left vertical map containsit.

Theorem 13.3. For every n, there exists a polynomial Pn(x0, . . . , xn) such thatany ideal sheaf I on Pn is Pn(a0, . . . , an)-regular, where ai are defined by

χ(I (m)) =

n∑i=0

ai

(m

i

).

Proof. Take Z be the subscheme corresponding to I . Take a hyperplane Hthat avoids the associated points of Z. Then we again have

0→ I (m)→ I (m+ 1)→ (I ⊗OH)(m+ 1)→ 0.

Here, IH is an ideal sheaf of H. Apply the induction hypothesis to IH . Thenthen first we have, for m ≥ m1 − 2,

0→ H0(I (m))→ H0(I (m+1))→ H0(IH(m+1))→ H1(I (m))→ H1(I (m+1))→ 0.

Also, for i ≥ 2, we get

0→ Hi(I (m))→ Hi(I (m+ 1))→ 0

for m ≥ m1 − i.Now, we want to show that H1(F (m)) stabilizes after some bounded time.

From the short exact sequence, we see that either

H0(I (m+ 1))→ H0(IH(m+ 1))

is surjective or dimH1(I (m+1)) < dimH1(I (m)). So after some time, it willstabilize at 0, and from on this point, all H0 are going to be surjective. Thisshould happen in approximately H1(I (m))-time, but we know that

dimH1(I (m)) = dimH0(I (m))− χ(I (m)) ≤ dimH0(O(m))− χ(I (m))

because Hi(I (m)) vanish for i ≥ 2 for sufficiently large m, and I is an idealsheaf. Then dimH0(O(m)) is a polynomial in m, so the stabilization occurs inpolynomial time.

Math 99r Notes 44

14 March 29, 2018

Today we are finally going to construct some schemes. We introduced twofunctors

CurvesX =∐

ξ∈NS(X)

CurvesξX , PicX =∐ξ

PicξX .

But we can use a coarser decomposition. Let X ⊆ Pnk so that we have a veryample line bundle OX(1). Then each line bundle have a Hilbert polynomial,and then we can write

CurveX =∐P

CurveX .

This is indeed a coarser decomposition because Riemann–Roch tells us thatthe Hilbert polynomial hF (m) = χ(F (m)) is determined by the intersectionpairing.

14.1 Grassmannians and other prerequisites

Classically, the Grassmannian G(n, r) is the space of r-dimensional spaces inAn+1. The moduli interpretation is that it sends S to the set

E rank r locally free with sections s0, . . . , sn ∈ Γ(E ) satisfying On+1 E .

You can construct G(n, r) using the Plucker embedding to P(··· ). This is sup-posed to send a r-dimensional space to v1 ∧ · · · ∧ vr.

Proposition 14.1. Let G be a scheme over k, and let A ⊆ B : (Sch/k)op → Set,with the inclusion transformation A → B factoring through hG. Assume thatfor all α ∈ B(S) there exists a subscheme Y ⊆ S such that for all g : T → S,

g∗(α) ∈ A(T ) ⊆ B(T ) ⇔ g factors through Y.

Then there exists a G0 ⊆ G such that A ∼= hG0.

Definition 14.2. Let S be a scheme. A stratification of S is a set S1, . . . , Smsuch that each Si ⊆ S is locally closed and S =

∐i Si at the level of points.

Proposition 14.3. Let F/Pn × S be a coherent sheaf. Then there exists astratification of S such that for every T → S, the pullback g∗TF is flat over Tif and only if g factors through

∐i Si.

Pn × T Pn × S

T S

gT

p

g

Here is a trivial example. Consider the skyscraper sheaf on A1k at some point.

You can make the sheaf flat on A1k \ 0 and 0.

Math 99r Notes 45

14.2 Embedding in the Grassmannian

Recall what we proved last time.

Theorem 14.4. For all n, there exists a polynomial Fn(x0, . . . , xn) such thatfor every I on Pn, it is Fn(a0, . . . , an)-regular where χ(I (m)) =

∑ai(mi

).

So given a Hilbert polynomial P , there exists a m0 such that all I withHilbert polynomial P is m0-regular. For curves, you can do this if you use Serreduality and

dimH1(L ) = dimH0(ω ⊗L ∨) = 0

if deg L ≥ degω.Fix a Hilbert polynomial P and choose a m0 sufficiently large such that

(i) If D ⊆ X is any curve with Hilbert polynomial P , then OX(−D) is m0-regular. So H1(X,OX(−D +m0)) = H2(X,OX(−D +m0)) = 0.

(ii) H1(O(m0)) = 0.

(iii) OX(−D +m0) is spanned by global sections.

Then using the long exact sequence for 0 → OX(−D + m0) → OX(m0) →OD(m0)→ 0, we get that H1(X,OD(m0)) = 0.

Suppose that D ⊆ X ×S is a family of curves, giving Hilbert polynomial P .Then p∗OD(m0) is locally free of rank χ(OX(m0))−P (m0). By the theorem ofGrothendieck, H1(Xs,ODs

(m0)) = 0 implies that

p∗OD(m0)⊗ κ(s)→ H0(Xs,ODs(m0))

is an isomorphism. Also, if we apply the theorem, to H2(X,OX(−D+m0)) = 0,then we get that we can compute R1p∗OX×S(−D +m0) locally. This gives

R1p∗OX×S(−D +m0) = 0.

Also, F globally generated means that p∗p∗F → F is surjective. So

p∗p∗OX×S(−D +m0) OX×S(−D +m0)

is surjective.Fix a basis e0, . . . , eN of H0(OX(m0)). Then

0→ p∗OX×S(−D +m0)→ p∗OX×S(m0)σ−→ p∗OD(m0)→ 0.

So we have produced a locally free p∗OD(m0) of rank r, and (N + 1) sectionssi = ∆(1⊗ ei) of p∗OD(m0).

Math 99r Notes 46

15 April 3, 2018

Last time, we had to pick m sufficiently large so that O(−D +m0) is spannedby global sections for all D with given Hilbert polynomial.

Proposition 15.1. If F/Pn is m-regular, then F (k) is generated by its globalsections for k ≥ m.

Proof. It suffices to show for k = m. We know that we have a surjection

H0(Pn,F (k − 1))⊗H0(Pn,OPn(1)) H0(Pn,F (k))

for k > m. By Serre vanishing, we know that

H0(Pn,F (k))⊗ OPn F (k)

for k 0. Then we twisted this to

H0(Pn,F (m))⊗H0(Pn,OPn(1))⊗k−m ⊗ O(m−k)Pn F (m)⊗ OPn(m− k).

But this factors through H0(Pn,F (m))⊗ OPn .

15.1 Representing the curves functor

Last time, we construct a natural transformation

CurvePX(S)→ hG(N,r),

for a given Hilbert polynomial P . This curve functor is given by

CurvePX(S) = D ⊆ X × S relative Cartier divisor

so that D/S is flat. On the other hand, the functor represented by the Grass-mannian is

Hom(S,G(N, r)) = locally free E of rank r with ON+1S E / ∼= .

To go from D to E , we choose m0, which only depends on P , such thatp∗OD(m0) is locally free and R1p∗OX×S(−D+m0) = 0. We have a short exactsequence

0→ OX×S(−D +m0)→ OX×S(m0)→ OD(m0)→ 0.

If we push forward along p, we have vanishing of R1p∗, so we have

0→ p∗OX×S(−D +m0)→ OS ⊗H0(X,OX(m0))→ p∗OD(m0)→ 0.

Now we can take a basis e0, . . . , eN of H0(X,OX(m0)) and then consider thelast map ON+1

S OD(m0).

Math 99r Notes 47

We are going to apply the formal lemma from last time here. Consider thediagram

CurvePX hG(n,r)

AllSubschX .

Φ

Ψ

We still need to define the natural transformation Ψ : hG(N,r) → AllSubsch. IfI have a surjection,

0→ K → OS ⊗H0(X,OX(m0)) E → 0

I can take the kernel. Then pulling back to X × S gives

p∗K → p∗p∗OX×S(m0)→ p∗E → 0.

Then if I twist by −m0, I get

p∗K (−m0)→ p∗p∗OX×S → OX×S .

Then I can take the image I = im(p∗K (−m0)) ⊆ OX×S .Now we need to check the formal condition. Given g : T → S and some

subscheme Z ⊆ X × S, we want to find a Y such that the pullback of Z toX × T is a relative Cartier divisor with Hilbert polynomial P if and only ifg factors through Y . Let Z correspond to an ideal sheaf I ⊆ OX×S . By flatstratification, there exists a Y ⊆ S such that I |Y is flat with Hilbert polynomialfor Z ⊆ X × S is defined by

χ(OZ×ST (n)) = χ(OX(n))− P (n).

The next step is to restrict Y to make sure that Z is a Cartier divisor. Notethat being a Cartier divisor is open, so that if Zt ⊆ Xt is Cartier then thereexists a neighborhood U such that Z is a Cartier in t. In fact, it suffices tofind a neighborhood of Xt such that for each point x over t, we can find aneighborhood V such that Z ∩ V is cut out by one equation in V .

Consider a point t ∈ T that is cut out by mt, so that Xt is also cut out bymt. For x ∈ Zt, consider Ox, and let Ix ⊆ Ox to be the ideal associated to Z.The n Ix is cut out by one equation in Ox/mtOx. So we can write

Ix + mtOx = (f) + mtOx

for some f ∈ Ox. Here, we may as well assume f ∈ Ix.Now we have an exact sequence

0→ Ix/(f)→ Ox/(f)→ Ox/Ix → 0.

Tensoring with k(t) gives

Tor1Ot

(Ox/Ix, k(t))→ Ix/(f)⊗k(t)→ Ox/(f) +mtOx → Ox/Ix +mtOx → 0.

The first term vanishes by flatness. The last map is an isomorphism by thecondition. So we have Ix/(f) ⊗ k(t) = 0. By Nakayama, we get Ix/(f) = 0,and so Z is cut out by a single polynomial.

Math 99r Notes 48

16 April 5, 2018

So we constructed the scheme representing the curves functor last time. Ourgoal was to represent CurvePX . These are divisors D ⊆ X such that P (n) =χ(OX(−D+n)). Then we found out that there is a m0 such that OX(−D+m0)are spanned by global sections. Using this we constructed a functor

CurvePX → hG(n,r).

Then we used flatness stratification and that flatness argument to show that itis representable. You can even show that this functor is going to be closed inG(n, r).

Math 99r Notes 49

17 April 10, 2018

Our next goal is to construct the Picard functor. Again, we reduce it to con-structing it for one numerical class. But here, for any two numerical class ξ andξ′, we have a non-canonical isomorphism

PicξX∼= Picξ

′

X .

So we will construct PicξX for some ξ such that any L with ξ is very ample and0-regular.

17.1 Ampleness

Recall that L /X is ample if for all F ∈ Coh(X), there exists an n only de-pending on F such that L n′ ⊗F is globally generated for all n′ ≥ n.

Theorem 17.1. If L is ample, then L n is very ample for n 0.

Proof. Let U be an affine neighborhood of P . Let Y = X − U and considerIY ⊗L n globally generated for n 0. Then we get s ∈ Γ(IY ⊗L n) suchthat s 6= 0 at P . But we can view IY ⊗L ⊗n ⊆ L ⊗n and then s vanishes onY as a section of L ⊗n.

Now let Xs be the locus of where s 6= 0. This is in U , but is cut out be oneequation, so it is affine. If we choose n large enough, we can cover X with Xsi .Now we look at bij generators of Γ(Xsi ,OXsi

). Then cij = sni bij ∈ Γ(X,L n)

defines the closed embedding X → PN .

There is a cohomological criterion for ampleness.

Theorem 17.2. L /X is ample if and only if, for all F ∈ Coh(X) there existan n such that for n′ ≥ n we have Hi(X,L n ⊗F ) = 0 for i > 0.

Theorem 17.3 (Nakai–Moishezon criterion). Let X/k be a surface. Then L /Xis ample if and only if L 2 > 0 and L .C > 0 for every irreducible curve C.

Positivity comes from intersecting two effective curves.

Proof. Suppose that L is ample. We might as well assume that L = O(D) forsome D effective. Then we can consider D as a hyperplane section. This showsthat D2 > 0 and D.C > 0 as well. The other direction is more interesting. Weuse Riemann–Roch

h0(L )− h1(L ) + h2(L ) =L 2 −L .K

2+ χ(OX).

We first show that L ⊗OD is ample on D. First we replace D with its reducedDred, because this does not change ampleness. Then we see that OX(D) ⊗OD|Ci = OCi(D ∩ Ci) which is positive because D ∩ Ci > 0.

Math 99r Notes 50

Then we show that L n is generated by global sections so that we haveϕ : X → PN . We have then

0→ H0(X,L n−1)→ H0(X,L n)→ H0(X,L n ⊗ OD)

→ H1(X,L n−1)→ H1(X,L n)→ H1(x,L n ⊗ OD)→ · · · .

For n 0, the last term is zero, so the map H1 → H1 is surjective. Then forlarger enough n, the dimension cannot decrease, so we have

0→ H0(X,L n−1)→ H0(X,L n)→ H0(X,L n ⊗ OD)→ 0.

Since L n ⊗ OD is globally generated over D, we can take si ∈ H0(X,L ⊗OD) that do not vanish simultaneously on D. Then we can lift them to si ∈H0(X,L n). Then we can add the one section that only vanishes on D.

The claim is that this has finite fibers. If not, there exists a curve C suchthat ϕ(C) is a point. But then nD.C = 0 because a point C and a generalhyperplane does not intersect. So the morphism ϕ : X → PN is quasi-finite andprojective, so it is finite. Now for ϕ : X → Y a finite morphism, L/Y is ampleif and only if ϕ∗L is ample. (This follows from the Leray spectral sequence andthe cohomological criterion for ampleness.)

Lemma 17.4. Let H be ample and D be any divisor such that D.H > 0 andD2 > 0. Then for n 0, we have nD is effective.

Proof. By Serre duality, we have

h2(X,OX(nD)) = h0(X,OX(K − nD)).

But the right hand side vanishes if nD.H > K.H. Also in Riemann–Roch, wecan make the χ(nD) large enough, and then h0 − h1 > 0 implies h0 > 0.

Math 99r Notes 51

18 April 12, 2018

For L : V → V an endomorphism, its top wedge power is 1-dimensional, so weget a determinant. We want to construct the Picard functor.

18.1 Representing the Picard functor

The strategy is that if the natural map

CurveξXΦ−→ PicξX

admits a section s, then we can realize P (ξ) as a pullback functor

P (ξ) C(ξ)

C(ξ) C(ξ)× C(ξ).

(1,f)

∆

Then because C(ξ) is representable, this is a fiber product of just schemes.So we need to find a section s of Φ. Choose ξ such that L with ξ is very

ample and 0-regular. Now what is a section? For each line bundle L/X ×S, weneed a Cartier divisor D ⊆ X × S such that

OX×S(D) = L⊗ p∗2M

for some M . This association should be moreover functorial and well-defined inthe sense that if L′ = L⊗ p∗2M ′ then D is the same as for L.

Let me first explain Mumford’s construction. Take E = p2∗L and Lx = i∗x(L)for the some x ∈ X and ix : S → X × S. Then we have a morphism E → Lxinduced by evaluation at x. Because X×S → S is flat, we have that E is locallyfree. Let r = rank E and pick r − 1 points x1, . . . , xr−1. Then we have a map

E →r−1⊕i=1

Lxi

and taking the (r − 1)th wedge power gives

∧r−1E →

r−1⊗i=1

Lxi.

Taking the dual gives (r−1⊗i=1

Lxi

)∨→ E ⊗ (

∧rE )∨,

and we have

OS → E ⊗ (∧rE )∨ ⊗

r−1⊗i=1

Lxi .

Math 99r Notes 52

Then we can pull this back to get

OX×S = p∗2OS → p∗2E ⊗ p∗2(

(∧rE

)∨⊗r−1⊗i=1

Lxi

)→ L⊗ p∗2(· · · ).

This gives the desired section of the line bundle of the form L⊗ p∗2M .We need to check that if L′ = L ⊗ p∗2M ′ then we get the same divisor. To

do this, we need the projection formula.

Proposition 18.1 (projection formula). On locally ringed spaces, let f : X →Y be a morphism with F a sheaf on X and E be a locally free sheaf of finiterank on Y . Then

f∗F ⊗OYE ∼= f∗(F ⊗OY

f∗E ).

If we replace L with L⊗ p∗2M ′, from

L⊗ p∗2(

(∧rp2∗L)∨ ⊗

r−1⊗i=1

Lxi

),

we get a bunch of p∗2M′, but they cancel out. Functoriality is clear.

But what is this construction really doing? Let’s just look in the case ofS = pt. Here, E = H0(X,L). If r = dimH0(X,L) and I have some pointsx1, . . . , xr−1, I can take the kernel of

H0(X,L)→⊕

Lxi∼= kr−1; s 7→ (s(x1), . . . , s(xr−1)).

This is going to be a 1-dimensional subspace generically, and the vanishing locusof this is going to be my curve. That is, this is the curve passing through allthe r − 1 points x1, . . . , xr−1.

But the problem is that generically this will give us a unique curve, but itmight be the entire surface. Can we choose r − 1 points so that for any linebundle the curve is uniquely determined this way?

Example 18.2. Let C be a genus 2 curve, and let us look at the degree 2 case.In this case, if degL = n then we can compute dimH0(L) = 2 for n = 2 anddimH0(L) = n − 1 for n > 2. Whatever P1, . . . , Pn−2 we pick, we can takeQ1, Q2 such that OC(Q1 +Q2) = K. Then for L = P1 + · · ·+ Pn−2 +Q1 +Q2,we will have

s ∈ H0(L) : s(Pi) = 0 = H0(L(−P1 − · · · − Pn−2)) = H0(OC(Q1 +Q2))

a dimension 2 space.

So things don’t work well.

Math 99r Notes 53

19 April 17, 2018

Let C be a curve of genus g and let d be some large number. There is a morphism

Cd → PicdC ; (x1, . . . , xd) 7→ OC(x1 + · · ·+ xd)

and then we can look at the graph Γ ⊆ PicdC × Cd. The fiber of the projectionmap Γ→ PicdC has fiber

ΓL = PH0(C,L).

It turns out that dim PicdC = g, so that we have

h0(L)− 1 ≥ d− g.

We can use this to give a new perspective on the counterexample given lasttime. Given a curve C of genus ≥ 2, can you choose points x1, . . . , xd−2 suchthat for all L/C of degree d, vanishing on xis imposes exactly d − 2 linearconditions? We can consider

Z = (L, (x1, . . . , xd−2)) : h0(L(−∑xi)) ≥ 2 ⊆ PicdC × Cd−2.

This can be thought of as a bad locus. Here, the image of the projectionZ → PicdC is going to be a closed subscheme, because it is the line bundles withcohomology jumps. The fibers are going to be dimZL = d− 4, and so we havedimZ = d − 2. This is equal to the dimension of Cd−2. So we can’t concludethat such points d1, . . . , d− 2 exists by just dimension counting.

For general g ≥ 2, we need Brill–Noether theory. For C of genus g, so thath0(L) = d+1−g, we want to pick d−g points. So we look at Z ⊆ PicdC×Cd−g.Again, if we look at ZL, we can use Bril–Noether to get

codimZ ZL = codim PicdC = 2.

Then we have dimZ = d − 2, and so the map to Cd−g is more likely to besurjective.

19.1 Representing the Picard functor II

Anyways, we were trying to represent PicξX for a numerical class ξ. Our strategyis the same, but now we need to do better than just taking d−2 points. So takeG to be some parameter space, and we want a rule for getting a line bundle.Mumford adds more degree of freedom to G so that dimZ < dimG.

Choose Nr − 1 points and group them into N − 1 sets of r points and 1 setof r − 1 point. Let us write this

Y = (x1,1, . . . , x1,r), . . . , (xN,1, . . . , xN,(r−1)).

For the last (r − 1) points, do what we previously did, which is to take

σN ∈ H0(X × S,L⊗ p∗2(∧rE )−1 ⊗ [

⊗r−1i=1LxN,i

]).

Math 99r Notes 54

For the set of ρ points in Y, we look at the map∧rE →

⊗ri=1 Lxk,i

and look atthe corresponding section

σk ∈ H0(X × S,L⊗ p∗2(∧rE )−1 ⊗ [

⊗ri=1Lxk,i

]).

Now we consider

σY = σ1 ⊗ · · · ⊗ σN−1 ⊗ σN ∈ H0(X × S,L⊗ p∗2(∧rE )−N ⊗ [

⊗k,iLxk,i

]).

Theorem 19.1. For suitable choices of ξ and N and Nr− 1 point and scalarsaY , the sum

∑aYσY is flat over S if and only if

∑aYσY does not vanish

identically over any s ∈ S.

I won’t prove this, but here is the basic idea. Consider a polarized surface(X,OX(1)). A 0-cycle U is called λ-independent if for all C ⊆ X, we have

deg(U ∩ C) ≤ λ(degC)2.

We also say that a 0-cycle U of Pn is strongly stable if for all hyperplane H,

deg(U ∩H) ≤ degU

n+ 1.

Then the thing we want to prove can be formulated in terms of these propertiesand you play with Riemann–Roch.

Next time we are going to study the tangent space of PicX . Because TpXis Hom(Spec k[ε]/ε2, X), the tangent space TLPicX(k) is the deformation of Lover Spec k[ε]/ε2.

Math 99r Notes 55

20 April 19, 2018

How do we differentiate? We can define the tangent space as

HomSch/k(Spec k[ε]/ε2, X) = TxX.

How does the left hand side have a structure of a k-vector space? In general,schemes don’t have pushouts, but

Spec k Spec k[ε1]/(ε21)

Spec k[ε2]/(ε22) Spec k[ε1, ε2]/(ε21, ε1ε2, ε22)

is a pushout diagram. So if we have two tangent vectors, we get a mapSpec k[ε1, ε2]/(ε21, ε1ε2, ε

22) → X, and then we can compose with the “diagonal”

Spec k[ε]/ε2 → Spec k[ε1, ε2]/(ε21, ε1ε2, ε22) induced by ε1 7→ ε, ε2 7→ ε.

20.1 Infinitesimal structure of Curve and Pic

Here, the only thing we know about CurveX and PicX are the functorial prop-erties. So we need to abstractly talk about deformations. Let X be a surfaceand D ⊆ X be a curve with Num(D) = ξ. This can be considered as a points ∈ CurveσX(k). What is the tangent space at s? By definition, we have

HomSch/k(Spec k[ε]/ε2,CurveξX) =

D ⊆ X × T effective relative Cartier

divisor with D0 = D

,

where we write T = Spec k[ε]/ε2.The claim is that we can also describe this space as

H0(D,ND/X) = H0(X, i∗ND/X).

In particular, if H0(X,ND/X) = 0 then you cannot move your curve infinitesi-mally. This occurs when you blow up a surface at a curve and get E.E = −1.Then H0(P1,OP1(−1)) = 0.

The first thing to note is that X and XT = X × T are the same as topo-logical spaces. Let’s first construct the map for relative Cartier divisors toH0(X, i∗ND/X). A relative Cartier divisor is represented by (Ui, fi) func-tions on an open cover. Then we can write

fi = gi + εhi

for gi ∈ Γ(Ui,O×X) and hi ∈ Γ(Ui,OX), because fi has to be invertible. Because

our fiber D is fixed, gi are already determined. We also need the compatibilityconditions, fi and fj should differ by a unit. So we should be able to write

gi + εhi = (aij + εbij)(gj + εhj)

Math 99r Notes 56

and then gi = aijgj and hiaijhj + bijbj for aij ∈ Γ(Ui ∩ Uj ,O×X) and bij ∈Γ(Ui ∩ Uj),OX). We can write this as

higi− hjgj

=bijaij

.

The claim now is that (Ui, hi

gi) defines global section in ND/X . The normal

bundle can be described as

0→ OX → OX(D)→ i∗ND/X → 0,

so that ND/X = OD ⊗ OX(D) and locally 0 → A → 1fA → A/(f) ⊗A 1

fA.

Now the difference between hi/gi and hj/gj comes from OX(Ui∩Uj) and so thesections glue well.

The converse construction goes this way. If s ∈ Γ(X, i∗ND/X), then for Uian affine cover of X we have

0→ OX(Ui)→ OX(D)(Ui)→ i∗ND/X(Ui)→ 0

and so we can lift s|Uito hi/gi a section of OX(D)(Ui). Then we can trace back.

If we writehigi− h′ig′i

= ci ∈ OX(Ui)

then you can show that

(gi + εhi) = (di + εcidi)(g′i + εh′i)

for di the unit with gi = dig′i. So gi + εhi is well-defined as a section in

Γ(Ui,K∗XT/O∗XT

). So we get an effective Cartier divisor.Over a local ring, Cartier divisors and flatness interact very well. Here is

what I mean. Let (A,m) be a local ring and let X be flat over A. ConsiderD ⊆ X a closed subscheme.

Proposition 20.1. (1) If D is flat and D is flat and D0 is a Cartier divisor,then D is also a Cartier divisor. (This we proved when representing thecurves functor.)

(2) If D is a Cartier divisor, then D is flat if and only if D0 6= X0.

Proof. We may assume that X = SpecB so we have A→ B. Let D be B/(f).We want to show that B/(f) is flat over A, and then it suffices to show thatTorA1 (B/(f), A/m) = 0 because A is a local ring. If we use 0 → B → B →B/(f)→ 0, then we get

0→ TorA1 (B/(f), A/m)→ B ⊗A/m ×f−−→ B ⊗A/m→ · · · .

Because f /∈ mB, multiplication by f is injective.

So for S = CurveξX and D ⊆ X corresponding to s ∈ S, we have constructedan isomorphism

ρ : TsS → H0(D,ND/X).

Once we show that CurveξX is non-singular, then we can compute its dimensionas the dimension of the tangent space.

Math 99r Notes 57

21 April 24, 2018

Suppose D corresponds to s ∈ CurveξX(k). We identified a first-order infinites-imal deformation of D inside X with the global sections of the normal bundle.This is also the same as the Zariski tangent space of the scheme TsCurveξX .

21.1 Regularity of the curve functor

Today we are going to first study regularity of CurveξX when ξ is sufficientlyample. This notion of regularity comes from dimension theory. Let (A,m) be aNoetherian local ring. We say that the local ring is regular if

dimA = dimk m/m2.

This is equivalent to there being a regular sequence (f1, . . . , fd) such that fispan m/m2.

Example 21.1. Consider Spec k[x, y]/(xy), localized at (x, y). This is notregular.

We want to talk about regularity of CurveξX , but then we need to formulatea functorial characterization of regularity. Let A be a k-algebra and A/m = k.Then A is regular if and only if for every R S with R,S Artin rings over k,

A

R S

we have Hom(A,R) Hom(A,S). Here we don’t worry too much because k isalgebraically closed.

A curve D ⊆ X is semi-regular if

H1(OX(D))→ H1(ND/X)

is zero.

Theorem 21.2 (Severi–Kodaira–Spencer). CurveξX is regular at s if char k = 0and D is semi-regular.

Let us denote test Artinian rings by (A,m) and (B, n). Write 0 → I →A → B → 0. Given a relative Cartier divisors DB ⊆ X × SpecB, we wantto extend it to DA ⊆ X × A extending DB . Denote DB by (Ui, Fi) withFi ∈ Γ(Ui,OX ⊗B) and Fi = GijFj for Gij ∈ Γ(Uij , (OX ⊗B)×).

We note that we may assume that dimk A− dimk B = 0, because we can tothis inductively. Let η be a element of A \ B so that ηA = ker(A → B). Westart with arbitrary liftings Fi and Gij of Fi and Gij . Then

Fi ∈ Γ(Ui,OX ⊗A), Gij ∈ Γ(Uij , (OX ⊗A)×).

Math 99r Notes 58

We can do this because A B implies A× B×.Let us write

Fi −GijFj = ηhij

for hij ∈ Γ(Uij ,OX ⊗A). Then we get

η(hij +Gijhjk) = ηhik + (Gik −GijGjk)Fk.

If we divide by η and then reduce modulo m, we get

hij +G0ijhjk = hik +

G0ik −G0

ijG0jk

ηF 0k (mod m).

If we divide by F 0i , then we get

hijF 0i

+hjkF 0j

=hikF 0i

+1−G0

ijG0jkG

0ik

η

in Γ(Uijk,ND/X). This implies that hij/F 0i is a Cech 1-cocycle of ND/X .

This shows that being able to choose Fi and Gij such that hij = 0 is equivalentto this cocycle being 0 in H1(ND/X).

Now we have 0→ OX → OX(D)→ ND/X → 0, and so we have

H1(O(D))0−→ H1(ND/X) → H2(OX).

So we an test something in H1 being zero by looking at its image in H2(OX).Our equation tells us that the image is 1

η (1−GijGjkGkj). This is a cobound-

ary because 1η (1− (G0

ij)−1Gij) gives a 1-cycle.

Here is one interesting thing you can see. Let ξ be an ample class. Thereis the natural map CurveξX → PicξX and the fiber is just PH0(OX(D)). So weshould have

dimPH0(OX(D)) + dimH1(OX) = dimH0(ND/X).

Here, H1(OX) is the tangent space of Pic. You can also see this from

0→ H0(OX(D))

H0(OX)→ H0(ND/X)→ H1(OX)→ 0.

Index

adjoint functor, 15affine morphism, 28affine scheme, 11Albenese, 5ample, 31associated point, 33

canonical bundle, 8canonical divisor, 8Cartier divisor, 19, 36Chern class, 23class group, 19closed immersion, 14complex manifold, 4cotangent space, 13

degree, 6divisor, 7

effective divisor, 36embedded point, 33Euler characteristic, 24

finite type, 27flat, 30

geometric genus, 8Grassmannian, 44Grothendieck spectral sequence, 30

higher direct image, 21Hilbert polynomial, 32holomorphic vector bundle, 4homogeneous ideal, 24homotopic, 20

injective object, 20inverse image sheaf, 15

λ-independent, 54Leray spectral sequence, 30locally ringed space, 11

multiplicity, 24

Neron–Severi group, 37Nakai–Moishezon criterion, 49numerical class, 37

period, 4presheaf, 9prime divisor, 19principal divisor, 7, 19projection formula, 52projective, 26pull-back sheaf, 15push-forward sheaf, 15

quasi-coherent, 16quasi-compact, 27

ramification, 9regular, 57regularity of sheaves, 42relative differentials, 17represented functor, 11Riemann–Roch, 8

for surfaces, 37ringed space, 11

scheme, 12semi-regular curve, 57sheaf, 9sheaf cohomology, 21sheaf of ideals, 16stratification, 44strongly stable, 54

tangent space, 13

very ample, 31

Weil divisor, 19

Yoneda embedding, 11

59