Introduction

There will be no specific text in mind. Sources include Griffiths and Harris aswell as Hartshorne. Others include Mumford’s ”Red Book” and ”AG I- ComplexProjective Varieties”, Shafarevich’s Basic AG, among other possibilities.

We will be taught to the test: that is, we will be being prepared to takeorals on AG. This means we will de-emphasize proofs and do lots of examplesand applications. So the focus will be techniques for the first semester, and thesecond will be topics from modern research.

1. Techniques of Algebraic Geometry

(a) Varieties and Sheaves on Varieties

i. Cohomology, the basic tool for studying these (we will start thisearly)

ii. Direct Images (and higher direct images)iii. Base Changeiv. Derived Categories

(b) Topology of Complex Algebraic Varieties

i. DeRham Theoremii. Hodge Theoremiii. Kahler Packageiv. Spectral Sequences (Specifically Leray)v. Grothendieck-Riemann-Roch

(c) Deformation Theory and Moduli Spaces

(d) Toric Geometry

(e) Curves, Jacobians, Abelian Varieties and Analytic Theory of ThetaFunctions

(f) Elliptic Curves and Elliptic fibrations

(g) Classification of Surfaces

(h) Singularities, Blow-Up, Resolution of Singularities

2. Problems in AG (mostly second semester)

(a) Torelli and Schottky Problems: The Torelli question is ”Is the mapthat takes a curve to its Jacobian injective?” and the Schottky Prob-lem is ”Which abelian varieties come from curves?”

(b) Hodge Conjecture: ”Given an algebraic variety, describe the subva-rieties via cohomology.”

(c) Class Field Theory/Geometric Langlands Program: (Curves, VectorBundles, moduli, etc) Complicated to state the conjectures.

(d) Classification in Dim ≥ 3/Mori Program (We will not be talkingmuch about this)

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(e) Classification and Study of Calabi-Yau Manifolds(f) Lots of problems on moduli spaces

i. Moduli of Curves (See Angela Gibney’s Course)ii. Moduli of Bundles (related to Geometric Langlands)

Assume knowledge of manifolds and cohomology and DeRham theory, cupproduct, some Sheaf theory, some Kahler manifolds. From algebra, rings andmodules, projective modules, etc.

1 Varieties and Sheaves over C1.1 First Definitions

Definition 1.1 (Affine Variety). Affine Space An = Cn as a topological space.An affine variety Z ⊂ An is the subset where a collection of polynomials vanish.

Example 1.1. Am ⊂ An for m < n

Example 1.2 (Quadrics). A quadric Q ⊂ An, the set of points ξ|q(ξ) = 0for q a quadratic polynomial.

If n = 1, then the 0-dimensional quadric is either two points or a doublepoint in A1, these are the cases where q(x) = x2 − 1 and q(x) = x2.

If n = 2, we get a conic over R, which are the cases of the ellipse, parabola,hyperbola, two lines, line, point, and the empty set which correspond to 4x2 +y2−1, y−x2, x2−y2−1, x2−y2, x2, x2 +y2 and x2 +y2 +1. Over C, however,the ellipse, hyperbola and the empty set all become the same thing smooth conic,S2 \ 2 points. The parabola is S2 \ pt. There still remain the singular conic(pair of lines/point) and the double line

Example 1.3 (Affine Plane Curves). Let C ⊂ A2 be the set of points (x, y) ∈A2|f(x, y) = 0 where f is polynomial in x, y of some degree. Assume that Cis nonsingular, that is, f = ∂f

∂x ,∂f∂y = 0 have no common solutions. So C is a

complex submanifold of A2.In degree 1 there is 1, in degree 2 there are 2, in degree 3 the question is

more complicated.

Definition 1.2 (Projective Space). Projective space Pn: as a set, Pn = An+1 \0/ ∼, where (x0, . . . , xn) ∼ (λx0, . . . , λxn) where λ ∈ C∗. That is, the set oflines through 0 in An+1.

Alternate description Pn =∐ni=0 Ui/ ∼ where Ui ' An and then glued

together.

We will be working over C. We should think of An = Cn not as a linearspace but as an algebraic manifold.

Definition 1.3 (Affine Algebraic Manifold). An affine manifold is a subsetX ⊂ An defined by polynomial equations fi = 0 which is nonsingular. Thatis, it is a complex analytic submanifold of An.

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Definition 1.4 (Projective Algebraic Manifold). A projective manifold is acomplex analytic submanifold realized as a subset of Pn given by Fi(x) = 0where Fi are homogeneous polynomials.

Lemma 1.1. The union of two disjoint affine submanifolds of An is an affinesubmanifold.

Proof. The union X ∪ Y with X defined by fi and Y defined by gi, is definedby figj.

Definition 1.5 (Projective Variety). A projective variety is X ⊂ Pn for somen given by Fi = 0 for some collection of homogeneous polynomials in n + 1variables.

Now that we have objects, we need to ask about maps. In particular, whatsort of equivalence do we want? We could use diffeomorphism, equivalenceof complex manifolds, or we could attempt to define equivalence of complexvarieties algebraically. We will use all of these. The algebraic notion is definedas follows:

Definition 1.6 (Morphism). A morphism of varieties f : X → Y is a contin-uous map which is given locally coordinatewise by rational functions.

An isomorphism of varieties is a morphism with a two-sided inverse.

We will refine this notion a bit later, once we have the notion of a locallyringed space.

Now consider the set of all affine cubics. This is A10, or better P9, becausef and af define the same variety for a 6= 0.

Theorem 1.2 (Existence of the j-invariant). There exists an explicit morphismj : P9 → P1 = A1∪∞ such that f = 0 (a projective plane cubic) is nonsingulariff j is finite and f1 = 0 is isomorphic to f2 = 0 iff j(f1) = j(f2).

If f is taken to be an affine plane cubic, then there are three such equationssatisfying the same conclusion.

Proof. Theorem IV.4.1 in Hartshorne.

We will talk more about the j-invariant later. It leads to parameter spacesand moduli spaces. The parameter space is all possible objects (ie, A10), and themoduli space is roughly the set of isomorphism classes which is also an algebraicvariety, in this case, A1.

The biggest difference between varieties and algebraic manifolds is that va-rieties may be singular:

Definition 1.7 (Singular). Let V be a variety define by polynomials f1, . . . , fk(in some affine or projective space). Then p ∈ V is singular iff f(p) = ∂f

∂xi(p) =

0 for all i. Otherwise, p is nonsingular.

In fact, we now come naturally to the notion of reducibility versus irre-ducibility:

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Definition 1.8 (Reducible). A variety V is reducible if V = V1 ∪ V2 for V1, V2

both proper subvarieties of V . Otherwise it is irreducible.A variety is locally reducible if on some open subset (often in the complex

topology) it is reducible.

Note that any reducible variety is locally reducible.

Example 1.4 (Singularity Types). 1. xy = 0, the union of two lines. Thisis globally reducible.

2. y2 = x2(x − 1), the nodal cubic. This is globally irreducible, but locallyreducible.

3. y2 = x3, the cuspidal cubic. Take a map C2 → P1 by (x, y) 7→ y/x. Ifwe throw away the origin, this maps the cubic to C \ 0, and the mapextends continuously but not algebraically to the origin to make C ' C astopological spaces.

As an unfortunate fact of language, there is a notion that something bereduced, which is very different from being reducible.

Definition 1.9 (Reduced). An ideal I ⊂ C[x1, . . . , xn] is said to be reduced (orradical) if C[x1, . . . , xn]/I has no nilpotent elements.

Look at the intersection of y = x2 and y = 0. This is a nonreduced intersec-tion, because V (y − x2) ∩ V (y) = V (y − x2, y) = V (x2).

Roughly speaking, an affine or projective variety is a subset of An or Pndefined by an ideal which is reduced and irreducible. (Here, irreducibility isa new requirement. We will revise the old definition, without the irreduciblerequirement, to be an algebraic set, rather than a variety.)

Example 1.5. In dim = 0, a single point is a variety, 2 points are a variety,but a double point is not.

Example 1.6. In dim = 1, a line is, a smooth cubic is, a nodal cubic is, acuspidal cubic is, unions of lines intersecting are not.

Roughly speaking once again, affine varieties correspond to Stein spaces andprojective varieties correspond to compact manifolds.

Definition 1.10. Let X be a topological space. Then we define a category withobjects the open sets of X and morphisms U → V corresponding specifically toU ⊂ V .

1.2 Sheaves

We need a good way of collecting local data. The next two definitions give usthis:

Definition 1.11 (Presheaf). A presheaf on a topological space is a contravariantfunctor on the above mentioned category.

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Definition 1.12 (Sheaf). A sheaf of abelian groups on a topological space is apresheaf F satisfying an additional two axioms:

1. If U is an open set and Vi is an open cover of U , s ∈ F (U) is anelement such that s|Vi

= 0 for all i, then s = 0.

2. If U is an open set and Vi is an open cover of U , if si ∈ F (Vi) for eachi with the property such that for each i, j, we have si|Vi∩Vj = sj |Vi∩Vj thenthere exists s ∈ F (U) such that s|Vi = si for all i.

We think of the first of these conditions as uniqueness and the second asexistence.

Example 1.7. Let F (U) = 0 if U = ∅ and = C else. This is a presheaf but nota sheaf. The sheafification (this is defined in Hartshorne, II.1.2) of this presheafis the sheaf of locally constant functions.

Definition 1.13 (Subsheaf). S ⊂ F is a subsheaf iff for all U , S (U) ⊂ F (U)is a subobject.

Definition 1.14 (Quotient). A quotient sheaf F → Q is a sheaf such that forall U , Q(U) is a quotient of F (U).

Let i : U → X be an open inclusion of topological spaces, then if F is asheaf on X, then we define i∗F = F |U , which is a sheaf on U .

In the other direction, we take a sheaf G on U and i∗G (W ) = g(U ∩W ) withW ⊂ X open. We could also do extension by zero, which is G (U ∩W ) = G (W )if W ⊂ U and otherwise is 0. Extension by zero is denoted by i!G

Example 1.8. Let X = A1 and U = A1 \ 0 with F = OX on A1. Theni∗F = F |U = OU . i∗OU (W ) =holomorphic functions on W \ 0, and allowarbitrary singularities at the origin. However, i!OU (W ) = 0 if 0 ∈W , and elsethe holomorphic functions on W .

Quotients are not necessarily sheaves, and so we need sheafification andcohomology theories.

Definition 1.15 (Pushforward and Pullback). Let f : X → Y be a morphismof varieties, F a sheaf on X and G a sheaf on Y . Then the pushforward ordirect image of a sheaf is f∗F (U) = F (f−1(U)), and the pullback is f∗G (W ) =G (f(W )).

For this, we need to make sense of G (V ) where V ⊂ Y is a non-open subset.Do it by defining G (V ) = lim−→U⊃V G (U).

In particular, Gp = lim−→U3p G (U) is the stalk of G at p, which consists ofgerms of elements of G at p.

Definition 1.16 (Ringed Space). A ringed space (X,O) is a pair with X atopological space and O a sheaf of rings.

A locally ringed space is a ringed space such that the stalks are local rings.

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Example 1.9. So a C∞ manifold is a locally ringed space locally isomorphicto (Rn, C∞).

Definition 1.17 (Algebraic Variety). An algebraic variety is a locally ringedspace that is locally isomorphic to an affine variety.

Definition 1.18 (Scheme). A scheme is a locally ringed space that is locallyisomorphic to SpecR for some ring R.

Next we look at divisors and line bundles. To do so, we need the followingdefinition:

Definition 1.19 (Sheaf of Modules). Let (X,OX) be a locally ringed space.Then a sheaf of OX-modules F is a sheaf consisting of OX(U)-modules F (U)for each open set, with the restriction maps homomorphisms of modules com-patible with the change of rings.

We say that an OX-module is locally free of rank n if there exists an opencover Ui of X such that F |Ui = O⊕nUi

for each i.

We can perform all the usual construction involving modules with OX -modules, by performing them on each open set and then sheafifying.

1.3 Invertible Sheaves and Divisors

We will focus on locally free sheaves of rank 1. These are called invertible, andthis is justified as follows:

Proposition 1.3. Let L be a locally free sheaf of rank one on a locally ringedspace (X,OX). Then there exists G a locally free sheaf of rank one such thatF ⊗ G ' OX .

Proof. Exercise.

Definition 1.20. Given D ⊂ X, we take O(D)(U) = f ∈ O(U \D)|poles areof order ≤ 1 on D. This is a sheaf

Example 1.10. D = ∅, then O(D) = O.

Example 1.11. If X = A1, and D = (0), then U = A1\0, and so O(D)(U) =f(x)/xk|f ∈ O(U), k ∈ N.

Now we will define subsets (in fact, with multiplicities) such that O(D) isparticularly well behaved.

Definition 1.21 (Prime Divisor). A prime divisor is a codimension 1 subvari-ety.

These are analogous to primes in number theory, as the prime numbers arethe codimension 1 irreducible subsets of Spec Z.

Definition 1.22 (Weil Divisor). A Weil divisor on X is an element of the freeabelian group generated by the prime divisors. We denote this group by DivX.

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Definition 1.23 (Cartier Divisor). A divisor is Cartier if it is locally definedby a single equation.

On a smooth variety, any Weil divisor is Cartier, on a singular variety, thingsare more complicated.

If D =∑niDi is a divisor, then we define O(D) to be the sheaf of functions

with poles of order at worst ni on Di.

Proposition 1.4. Let D be a Weil divisor. Then O(D) is an invertible sheafif and only if D is Cartier, and O(−D) is its inverse.

Proof. Exercise.

We have the following corollary:

Corollary 1.5. The set of isomorphism classes of invertible sheaves on a varietyX is a group, which we shall denote by Pic(X).

We will often call locally free sheaves vector bundles and, in particular,invertible sheaves will be called line bundles.

In the future, we will talk about how to take a line bundle and get a divisor,along with equivalence relations on divisors.

We should now introduce the notation Γ(U,F ) for F (U). This is an im-portant piece of notation because it makes it easier to write the global sectionfunctor Γ(X,−).

1.4 The Rational map associated to a Line Bundle

First we need to know what a rational map is.

Definition 1.24 (Rational Map). A rational map f : X1Y is a morphismU → Y for some open U ⊂ X

Now, let X be a variety and L be an invertible sheaf on X. Then Γ(X,L ) isa vector space, which, for reasons that will be clear later, we sometimes denoteby H0(X,L ). Define h0(X,L ) to be the dimension of this vector space. Thenwe get a morphism X \BL → Ph0−1, that is, a rational map X → Ph0−1.

The crude way to find this map is to choose a basis s1, . . . , sh0 for H0(X,L )(we are implicitly assuming h0 is finite, which it is, but we have not provedthis) and define x 7→ (s1(x) : . . . : sh0(x)) where defined, by also choosing anisomorphism C ' Lx, to get an (h0 − 1)-tuple of numbers.

As changing this isomorphism results in multiplication by a nonzero scalar,the point of Ph0−1 is independent of the isomorphism. We define BL = x ∈X|s(x) = 0 for all sections s, and so we get a map ϕL : X \BL → Ph0−1.

Example 1.12 (Veronese Map). Let X = Pn and D = dH for H a hyperplane.Then the morphism ϕO(D) is called the Veronese map, and has BO(D) = ∅ ford ≥ 1.

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Example 1.13 (Segre Map). Let X = Pr×Ps, and D = H1 +H2 where H1 is ahyperplane in Pr and H2 is a hyperplane in Ps, that is, O(D) = π∗OPn1 (H1)⊗π∗2OPn2 (H2). Then BO(D) = ∅. The map ϕO(D) is called the Segre Map,

A more natural definition of ϕL is to also give it dependency on V ⊂Γ(X,L ) and call it ϕL ,V : X \BL ,V → P(V ∗) by x 7→ s ∈ V |s(x) = 0.

The most important sheaves, in practice, have the following nice property:

Definition 1.25 (Coherent Sheaf). A sheaf F of OX-modules is coherent if itis ”finitely presented.” That is, there is an exact sequence L1 → L0 → F → 0where Li is a locally free sheaf of finite rank.

Under nice circumstances which will hold for this course, coherence is equiv-alent to ∀x ∈ X, there exists U 3 x with O⊕mU → O⊕nU → F |U → 0 is exact.We will generally use this version.

Example 1.14 (Coherent Sheaves). 1. OX .

2. Any locally free sheaf of finite rank

3. If F ,G are coherent, so is F ⊕ G and F ⊗ G .

4. If i : Y → X is an inclusion of a subvariety, and F is a coherent sheafon Y , then i∗F is coherent.

In contrast to coherent sheaves, a sheaf is quasicoherent if it is coherentor if it is ”infinitely generated” in some sense (that we will not define at themoment).

It is a fact that direct images only preserve quasi-coherence in general, butthat if the map is proper, then it preserves coherence. (Proper means thatinverse image of compact is compact)

2 Cohomology

2.1 Cech Cohomology

We want to define Hi(X,F ) for any sheaf of abelian groups F . Because wewill be making use of it in the future, we introduce the following:

Definition 2.1 (Constant Sheaf). Let X be a topological space and let A be anabelian group. Then A denotes the constant sheaf, which is the sheafification ofthe constant presheaf. This is the sheaf of locally constant functions X → A.

We want our definition to gives us that Hi(X,A) = Hi(X,A), as defined inalgebraic topology.

The standard way of defining a cohomology theory for some object is toassociate to the object a chain complex, and then take the cohomology of thiscomplex.

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Definition 2.2 (Cech Cohomology of a Cover). Let X be a topological spaceand F a sheaf of abelian groups on X. Let U be an open cover with orderedindex set I.

Define Cp(U ,F ) =∏i1,...,ip∈I F (Ui1,...,ip). We define d : Cp → Cp+1 by

(dα)i0,...,ip =∑pk=0(−1)dαi0,...,ik,...,ip |Ui0,...,ip

.This gives the Cech complex, and so we get notions of Cech cocycles and

coboundaries, and so we define Cech Cohomology of the sheaf and the cover tobe cocycles modulo coboundaries.

Definition 2.3 (Cech Cohomology). If V is a refinement of U , we say that V >U , and this defines a poset. If V > U then we get a map C∗(U ,F )→ C∗(V,F ),which in fact induces a map Hp(U ,F ) → Hp(V,F ). Taking the direct limit,we define H∗(X,F ) = lim−→U H

∗(U ,F ).

Proposition 2.1. Let A be an abelian group. Then H∗(X,A) = H∗(X,A).

Proof. Omitted.

So when is a cover U good enough? Direct limits are hard to compute, sowe would like to know when Hp(U ,F )→ Hp(X,F ) is an isomorphism.

Proposition 2.2. If X is a variety and F is quasi-coherent, then if Ui0,...,ip isalways affine, then Hp(U ,F )→ Hp(X,F ) is an isomorphism.

To get a better understanding of this, we need to look at acyclicity in general.Now we pause to note a common element of all cohomology theories, in the formwe will be using:

Theorem 2.3 (Mayer-Vietoris Sequence). Let 0 → A → B → C → 0 be ashort exact sequence of sheaves. Then there exists a long exact sequence 0 →H0(X,A )→ H0(X,B)→ H0(X,C )→ H1(X,A )→ . . ..

2.2 Derived Functor Cohomology

Cech Cohomology is good for computations, but for theoretical purposes, thereis a different cohomology theory that is often better. We will assume familiaritywith abelian categories, which are categories that behave like the category ofabelian groups.

Definition 2.4 (Left Exact Functor). A functor F : C → D of abelian categoriesis left exact if 0→ A→ B → C → 0 short exact implies that 0→ FA→ FB →FC is exact.

Example 2.1 (Global Sections). The functor Γ(X,−) from sheaves on X toabelian groups is a left exact functor.

So we will define right derived functors, which fix the failure of exactness,though this will take some effort.

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Definition 2.5 (Injective Objects). An object I in an abelian category is in-jective if hom(−, I) is an exact functor.

A category C has enough injectives if for each object A, there exists an in-jective object I and an exact sequence 0→ A→ I.

This implies that every object has an injective resolution, that is:

Lemma 2.4 (Injective Resolutions). Let A be an object in a category withenough injective. Then there are injective objects Ij and an exact sequence0→ A→ I0 → I1 → I2 → . . ..

Definition 2.6 (Right Derived Functor). Let F : C → D be a left exact functorand C have enough injectives. Then we define the right derived functors RiF tobe obtained as follows:

Let A be an object of C. Then take an injective resolution 0 → A → I∗ ofA. Then RiF (A) = Hi(I∗).

The right derived functors satisfy the Mayer-Vietoris Sequence, and so be-have as we would hope a cohomology theory should.

Proposition 2.5. The category of abelian groups and the category of sheavesof abelian groups both have enough injectives.

Proof. Omitted.

Definition 2.7 (Sheaf Cohomology). Let Hi(X,F ) to be the ith derived functorof the global section functor.

And so now we have a second definition of cohomology.

Definition 2.8 (Acyclic). An object is acyclic if nonzero cohomology all van-ishes.

The following lemma is very helpful in actually using derived functor coho-mology:

Lemma 2.6. Injective objects are acyclic.Flasque sheaves are acyclic for Γ(X,−). (A sheaf is flasque if the restriction

maps are all surjective.)

Example 2.2. Injective sheaves are flasque.

Theorem 2.7 (Leray). If U is an acyclic cover, then the natural map fromH∗(U ,F ) to H∗(X,F ) is an isomorphism.

Example 2.3. Hartshorne showed that algebraically, affine varieties are acyclic(when looking at quasi-coherent sheaves)

Example 2.4. Analytically, we have that Stein manifolds are acyclic.

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Note that there are two different notions of acyclic here, though they areclosely related. A space is acyclic if quasicoherent sheaves on it are acyclic assheaves.

The analytic analogue of flasque is ”fine”, that is, a sheaf which always haspartitions of unity.

Definition 2.9 (Partition of Unity). A sheaf of abelian groups has partitionsof unity if for all open covers of open subsets, any s ∈ Γ(U) can be written as∑si for si ∈ Γ(Vi) with Vi ⊂ Ui.

Example 2.5. For X any manifold, true for sheaf of C∞ functions but notanalytic or algebraic.

Next we note the following:

Theorem 2.8. Let M be a topological space, and U an acyclic cover for F .Then Hi(U ,F ) ' Hi(M,F )

And finally, we will move toward a statement of the Serre Duality Theorem.On any X a nonsingular variety dimX = n, there is a vector bundle of rank

n, the tangent bundle TX. This defines a locally free sheaf of rank n, T . ThenH om(T ,O) = Ω1 is the sheaf associated to the cotangent bundle. We defineΩi =

∧i Ω1.

Definition 2.10 (Canonical Bundle). Let X be a nonsingular variety of dimen-sion n. Then the canonical bundle KX =

∧n Ω1.

The canonical bundle is a line bundle.

Theorem 2.9 (Serre Duality Theorem). Let X be a nonsingular variety ofdimension n and L a line bundle. Then Hn(X,L ) ' H0(X,L −1 ⊗KX)∗.

2.3 Cohomology through Forms

As we are working over C, we have an additional option for defining the coho-mology of our spaces.

Definition 2.11 (deRham Cohomology). Let X be a real manifold. Then thedeRham Complex is A 0 → A 1 → . . . the complex of sheaves of C∞ p-formswith the exterior derivative. The cohomology of this complex (as a complex, notas sheaves) is the deRham Cohomology of X, denoted H∗dR(X).

The following is extremely important:

Lemma 2.10 (Poincare). If X = Rn, then the sequence A 0 → A 1 → . . . isquasi-isomorphic to the sequence R→ 0→ 0→ . . ..

All quasi-isomorphism means is that there exists a map of complexes whichinduces isomorphisms on cohomology groups. This is stronger than merelyhaving identical cohomology groups.

A consequence of Poincare’s lemma, with quite a bit of additional work is

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Theorem 2.11 (deRham). Let X be a C∞ real manifold, then Hn(X,R) 'HdR(X)

More generally, to a complex of sheaves C∗ we will assign its hypercohomol-ogy H∗(X,C∗), but not yet. In the meantime, we will state several propertiesof hypercohomology:

Proposition 2.12. 1. If C∗ is just a single nonzero sheaf, then H∗(X,C∗) =H∗(X,F ). If the Ci are acyclic, then Hi(X,C∗) = Hi(Γ(Ci)).

2. If C∗1 → C∗2 is a quasi-isomorphism, then H(C1) = H(C2).

Now we look at Dolbeault Cohomology.Let X be a complex manifold. Now define a sequence with A p the holomor-

phic p-forms.

Lemma 2.13. The sequence A 0 → A 1 → . . . is quasi-isomorphic to O → 0→0→ . . ..

More generally, we define A p,q to be the sheaf of p-holomorphic and q-anti-holomorphic forms. Then A p,0 → A p,1 → . . . with ∂ is quasi-isomorphic toΩp.

Theorem 2.14 (Dolbeault’s Theorem). Let Hp,q

∂(X) = Hq(Γ(Dolp)) where

Dolp is the sequence defined above. Then there is an isomorphism Hq(X,Ωp)→Hp,q

∂(X).

Contrast this with the Hodge Theorems. Let M be a compact, complex,Hermitian manifold, then the Laplacian is ∆ = ∂∂ ∗+∂ ∗ ∂. Define a harmonic(p, q) form to be α ∈ A p,q with ∆α = 0.

Theorem 2.15 (Hodge). Every ∂ cohomology class contains a unique elementof shortest norm (the norm is induced from Tangent Bundle to Cotangent Bun-dle to forms), and this element is harmonic.

Also, Harp,q(M)→ Hp,q

∂(M) is an isomorphism of finite dimensional vector

spaces.

There exists a similar real Hodge Theorem, but we are mostly concernedwith the complex case.

Corollary 2.16. If X is a real Riemannian manifold, then Hp(X,R) can becomputed by restricting to harmonic p-forms.

Similarly, if X is complex.

A more powerful version is

Theorem 2.17. If X is a compact Kahler manifold (a complex Riemannianmanifold with a symplectic form such that all three structures are compatible)then we have the decomposition Hn(X,C) = ⊕p+q=nHp,q(X) = Hp,q

∂(X) =

Harp,q(X).Additionally, Hp,q(X) = Hq,p(X).

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Example 2.6. If X is a curve of genus g, then H0(X,C) = C = H0,0

C2g = H1,0 ⊕H0,1, and C = H2(X,C) = H1,1 = H1(Ω1) = H1(K).So the interesting things happen in H0,1 and H1,0.H1,0 = H0(Ω1) = H0(K) and H0,1 = H1(O) and Serre Duality says that

they are dual and so have the same dimension, and so H1,0 and H0,1 havedimension g.

So this says that if you vary the complex structure on a Riemann surface,the holomorphic and antiholomorphic parts of H1 change.

On a general Hermitian manifold, 6 ∃ a relation between the ∆∂ and ∆d. Theassumption you need is that the manifold is Kahler. Then ∆∂ = ∆∂ = 1

2∆d.

Example 2.7. A projective manifold is Kahler.

Corollary 2.18. Hn(M,Ωn) = Hn,n ' C where n = dimM .

We need to note that ∆ commutes with ∗ and ∂, where ∗ is the Hodge star.

Corollary 2.19. Hq(M,Ωp)×Hn−q(M,Ωn−p)→ Hn(M,Ωn) is a perfect pair-ing.

That we have the Kunneth Formula also follows from this.

Corollary 2.20. b2k+1(M) is even.b2k(M) > 0.

Proof.∑2k+1p=0 hp,2k−p = 2

∑bp=0 h

p,2k+1−p.

To write all of this information, we use the Hodge Diamond.h0,0

h1,0 h0,1

......

...

hn,n−1 hn−1,n

hn,n

Not all Hodge diamonds occur, there are constraints.Hp,q(Pn)? Well, Hp,p(Pn) = C and else 0.What is Hp,q for a Riemann Surface of genus g?

Definition 2.12 (Calabi-Yau Manifold). A three dimensional complex manifoldis Calabi-Yau if Ωn = K = O. Sometimes we also assume that hp,0 = 0 for0 < p < n.

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If we combine this with Serre Duality, we find the hodge diamond:

1

0 0

0 a 0

1 b b 1

0 a 0

0 0

1

Mirror Symmetry says that if there is a Calabi-Yau manifold with a, b, thenthere is one with b, a.

2.4 Algorithm for the Cohomology of a smooth hypersur-face

Fix the topological type of M . Vary the complex structure by a smooth andproper map M → B, with fibers Mb. Then Hi(Mb,C) are locally independentof b.

The family of cohomologies Hi(Mb,C) forms a vector bundle over B calledRiπ∗C = V , the higher direct images of C.

This vector bundle has a natural flat connection ∇ : V → V ⊗Ω1 called theGauss-Manin Connection. Eg, V = VZ ⊗ C where VZ = Riπ∗Z.

How do the Hp,q vary? Hp,0 varies holomorphically. But Hp,q doesn’t ifq > 0.

Let F i = ⊕p+q=n,p≥iHp,q. Then we have a filtration of F 0 = Hn.We will find an algorithm to calculate hp,q of a hypersurface, H∗(Pr,O(n))

and dR, Dolbeault and Cech cohomologyCohomology of a HypersurfaceWe will need a topological result, the Lefschetz hyperplane theorem.

Theorem 2.21 (Lefshetz Hyperplane). Let Y ⊂ PN a submanifold and X =Y ∩H a hyperplane section which is nonsingular. Then Hi(Y,Z) → Hi(X,Z)is an isomorphism for 0 ≤ i < n, and is injective for i = n.

Example 2.8. Let X ⊂ P2 a curve. Then Z = H0(P2,Z) ' H0(X,Z) and0 = H1(P2,Z) → H1(X,Z) is injetive, we can do this by taking a Veroneseembedding of P2 into PN , where N depends on the degree of X.

Example 2.9. Let X be a surface in P3. Then 0 = H1(P3,Z) ' H1(X,Z),and so surfaces in P3 are simply connected.

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Example 2.10. If X is a three fold in P4, then H1(X,Z) = 0 and H2(X,Z) =Z.

Our goal is to work out the Hodge diamond of a smooth surface X ⊂ Pn+1.It has ones down the middle and zeros elsewhere in the top half and bottom half,the only thing we don’t know is the middle (longest) row, hn,0, hn−1,1, . . . , h0,n.

The algorithm produces things called the primitive Hodge numbers hp,q0 =dim(Hp,q(X)/Hp,q(Y )).

The Hodge decomposition Hn(X,C) give topological information indepen-dent of the complex structure of X.

Subgroups Hp,q vary real analytically, Hn,0 varies holomorphically, H0,n

varies antiholomorphically.F iHn = ⊕p≥iHp,q determines the Hodge Filtration of Hn.Note that Hp,q = F p ∩ Fn−p.We have π : X → B a family, and we look at the sheaf ΩpX /B . This is

a vector bundle over X . We push it forward via π to B, then taking derivedfunctors of π∗, we get Rqπ∗Ω

pX /B a sheaf of OB-modules, and in fact, locally

free over B whose fiber at b ∈ B is Hq(Xb,ΩpXb

) = Hp,q(Xb).Compare this with Rnπ∗C = V , which is a locally constant sheaf of C-vector

spaces on B.Then V can be thought of as a vector bundle on B with a flat connection.

If we forget the connection, we get a vector bundle, or a sheaf of OB-modules.Formally, V = V ⊗C OB .

The issue is that the natural map H p,q → V is no holomorphic.Define F q = ⊕p≥iH p,q, the image in V of each F i is a locally free sheaf of

OB-modules.The upshot is that V = F 0 ⊃ F 1 ⊃ . . . and F iis a holomorphic subbundle

of V , and H p,q ' F p/F p+1.So things vary holomorphically as quotients, but not as subbundles.The connection ∇ sends V → V ⊗ Ω1

B and F i → F i−1 ⊗ Ω1B , and this is

called Griffiths Transversality.This implies that H p,q →H p−1,q+1 ⊗ Ω1

B which is O-linear.Equivalently, TB ⊗H p,q →H p−1,q+1, as TB = H1(Xb, TXb

), we have thatH1(X,TX)⊗Hq(X,Ωp)→ Hq+1(X,Ωp−1). This map is just the cup product.

So let’s guess a formula for the Hp,q.Hn,0 = H0(X,Ωn) = H0(X,KX) = H0(X,KPn+1(d)|X) = H0(X,OX(d −

n− 2)) = H0(Pn+1,OPn+1(d− n− 2) = Sd−n−2 where S = C[x0, . . . , xn+1] thehomogeneous coordinate ring of Pn+1

We have 0→ O(−n−2)→ OP(d−n−2)→ OX(d−n−2)→ 0. The first termis in fact KP, and so we have 0→ H0(KP)→ H0(O(d− n− 2))→ H0(OX(d−n− 2))→ H1(KP)→ . . . from Mayer-Vietoris. H0(KP) = Hn+1,0(P), etc.

So we now know that Hn,0 = Sd−n−2, Hn,0 ⊕ H1(TX) → Hn−1,1 andHn−1,1 ⊗H1(TX)→ Hn−2,2.

What is H1(TX)? It’s related to Sd, in fact, it is a quotient of Sd by thosedeformations of the equation which do not change the complex structure.

15

As GL(n+ 2) acts on Sd, what we want to do is divide out by this action, iff = f(x0, . . . , xn+1), and we have T0GL(n+ 2) = gl(n+ 2) the n× n matrices.Then Eij is a basis elemtn of gl(n+ 2).

How does it act? By exp(tEij). Differentiating with respect to t the functionf exp(tEij) gives us xi ∂f∂xj

.

So changing f 7→ f+∑ij gijxi

∂f∂xj

ε for infinitesimal ε is the result of GL(n+2) action, and hence does not change the complex structure.

We need to check two things:

1. All deformations of the complex structure on X come from varying thecoefficients of the defining equation in Pn+1

2. All trivial deformations (on the complex structure) in Pn+1 come from theaction of GL(n).

To check these, just calculate H1(X,TX). How? On projective space, wehad the Euler sequence and now we restrict to X and get short exact sequencesand then the long exact sequence on cohomology should yield these facts.

The easiest answer would be H1(X,TX) = Sd, and VHS: Hn,0 = Sd−n−2,Hn−1,1 = S2d−n−2, etc. This is not quire right, though.

Let J be the ideal in S generated by the first partials of f . Then R = S/Jis a graded ring, the Jacobian ring. The answer is in fact Hp,q

0 = Rdq+d−n−2.Why? Because H1(X,TX) isn’t Sd, but is rather Sd/Jd = Rd, because we

can’t have things that are divisible by the partials.

3 Curves

First we must define the objects of study.

Definition 3.1 (Curve). A curve C is a nonsingular one dimensional projectivecomplex variety.

The first fact we need to know about curves is as follows:

Proposition 3.1. Let X be a curve and L be a line bundle on X. Then thereexists a divisor D such that O(D) ' L .

Note that this divisor is not unique.

Definition 3.2 (First Chern Class). Let L be a line bundle on a curve. ThenL ' O(D) for some D =

∑Ni=1 nipi, where pi ∈ C are points. Now we define

c1(L ) = deg L =∑Ni=1 ni.

Proposition 3.2. The first Chern class c1(L ) is well-defined, that is, it doesnot depend on the divisor chosen to represent L .

In fact, we can determine this value without directly using divisors:

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Proposition 3.3. Let U ⊂ C be an open set such that L |U ' OU . Then lets ∈ L (U) and identify it with its image under the isomorphism, as a rationalfunction. Then deg L =

∑p∈X degp(s).

And now we will state a powerful theorem which we will not prove.

Theorem 3.4 (Riemann-Roch). Let C be a curve and L a line bundle on C.Then h0(C,L ) − h1(C,L ) = deg L − g + 1, where g is the genus of C (as aRiemann Surface).

We take a moment now to define the Euler characteristic of a sheaf F on aspace X to be χ(F ) =

∑∞i=0(−1)ihi(X,F ).

Serre Duality applied to the Riemann-Roch formula gives us that h0(C,L )−h0(C,L −1 ⊗KC) = deg L − g + 1.

Assuming this, we now have enough to make some real statements.

Theorem 3.5 (Computation of Cohomology for Curves). Let C be a curve,OC its structure sheaf, T the tangent sheaf and K the canonical sheaf. Then wehave

deg h0 h1

O 0 1 gK 2g − 2 g 1T 2− 2g 0 + ε 3g − 3 + ε

where ε = 0 if g ≥ 2 and ε = 1 if g = 1, and ε = 3 if g = 0.

Proof. Apply Riemann-Roch to O, then h0(O) − h1(O) = 0 − g + 1. As anyglobal holomorphic (and thus regular) function on a compact Riemann surfaceis constant, h0(O) = 1, so we get h1(O) = g. Now we apply Serre Duality, andget h0(K) = g.

Apply Riemann-Roch to K, then h0(K) − h1(K) = degK − g + 1 givesg − 1 = degK − g + 1, so degK = 2g − 2.

Apply Riemann-Roch to T now, and we get h0(T )−h1(T ) = 2−2g−g+1 =3− 3g, as deg T = −deg T−1 = degK. If g ≥ 2, then deg T < 0, so h0(T ) = 0,because otherwise you’d have a function with zeroes but no poles on a compactRiemann surface.

Assume that g ≥ 2, then we get −h1(T ) = 3− 3g, so h1(T ) = 3g − 3, so weget for g ≥ 2 the following table.

We must still check the genus 1 and 0 cases.If g = 1, consider K. degK = 0 and h0(K) = 1 > 0. This implies that

K ' O, and therefore T = O.If g = 0, we get h0(T )− h1(T ) = 2− 0 + 1 = 3, h1(T ) = h0(K2) = 0, and so

h0(T ) = 3 as degK < 0, h1(T ) = 0.

3.1 Rational Curves

The ε above is dimH0(T ), and H0(T ) is the space of holomorphic vector fieldson C, which is also the dimension of the automorphism group of C. Though

17

we have not proved it, it is true that Aut(C) = PGL(2,C) for g = 0 and is ofdimension 3, if C has genus 1, then Aut(C) = C ×Z/2, which has dimension 1,and if g ≥ 2, then the group of automorphisms is finite and so has dimension 0.

The reason for that is that a vector field is an infinitesimal automorphism,that is, they arise as derivatives of one-parameter families of automorphisms.And Tid(AutC) = H0(C, TC).

Just for simpicity, we make the following definition:

Definition 3.3 (Rational Curve). A curve of genus zero is said to be a rationalcurve.

We can, in fact, prove the genus zero case fairly easily, if we assume thatAut P1 = PGL(2,C), though the use of the following

Proposition 3.6. Any rational curve is isomorphic to P1.

Proof. Let g = 0. Let L = O(p) for some p ∈ C. Then degL = 1. Riemann-Roch says that h0(L) − h1(L) = 2. By Serre Duality, h1(L) = h0(K ⊗ L−1),which has degree -3, and so vanishes, so h0(L) = 2, h1(L) = 0. So when C is ofgenus 0, and p ∈ C, then h0(O(p)) = 2 and h1(O(p)) = 0.

What is the base locus? Well, a section of O(p) is a function with a poleat p of order at most 1. There is a two dimensional space of functions thatvanishes at deg(OC(p)) = 1 points. Take a basis f, g, each vanishes at a uniquepoint. Sections are independent iff p1 6= p2, and so BL = ∅. So we get a mapφL : C → P1 defined everywhere. This has the property that ∀x ∈ P1, ∃s asection of L which vanishes precisely on φ−1

L (x). That is, φ−1L (x) = pt. So this

map is a bijective algebraic map.We must still check that the differential of this map is everywhere nonzero.

That is, φL is an analytic local isomorphism. In particular, it is a cover but P1

is simply connected, so C = P1.

With the degree of a line bundle of such importance, we make the followingdefinition:

Definition 3.4 (Stratification of Pic(C)). We define the set Picn(C) to be theset of line bundles of degree n on C.

Also we note the following:

Definition 3.5 (Abel-Jacobi Map). There is a function αC : C → Pic1 Cdefined by αC(x) = O(x). We call this the Abel-Jacobi map.

So what is Pic1(C)? Is α injective? Surjective?For rational curves, it cannot be injective, because, in fact, ∀p, q, α(p) =

α(q). We saw that ∀q ∈ C, there exists a section of O(p) such that s vanishesprecisely on q. It is surjective, because of linear equivalence, which is defined asfollows:

Definition 3.6 (Linear Equivalence). On a curve, two divisors D and D′ arelinearly equivalent D ∼ D′ if D−D′ = (f) for some rational function f , where(f) =

∑p∈C ordp(f)p.

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We can conclude that if g = 0, then C ' P1, so Pic1(C) = pt and Pic(C) =Z by the degree map.

Let L = OC(dp) for p ∈ C a rational curve. If d < 0, then BL = C and nomap. If d = 0, then L is trivial and the base locus is empty, and φL : C → pt.If d > 0, then BL = ∅ and we get φL : C → Pd. This is always an embedding,and it is called the d-uple embedding, or the Veronese embedding, as definedearlier.

3.2 Elliptic Curves

Definition 3.7 (Elliptic Curve). An elliptic curve is a curve of genus onetogether with a marked point, (C, x).

Over C, any curve of genus 1 has points and any two points are equivalent.Over Z, or C[x], or function fields this may not be the case, though we won’timmediately say what these mean.

We will now look at Pic(C) for elliptic curves. First we will do somethingthat works for any genus:

Proposition 3.7. Pic(C) ' H1(X,O∗).

And so we can do the following:

Corollary 3.8. Pic0(C) ' Cg/Z2g, and Pic(C) = Pic0(C)× Z.

Proof. By the proposition, a line bundle on C corresponds to an element inH1(C,O∗). We define a map O

exp→ O∗ by taking holomorphic functions f on Cto e2πf . The kernel of this map is Z, and as a sheaf map it is surjective, so wehave a short exact sequence of sheaves 0→ Z→ O

exp→ O∗ → 0.This gives us a long exact sequence on cohomology, 0→ H0(Z)→ H0(O)→

H0(O∗) → H1(Z) → H1(O) → H1(O∗) → H2(Z) → 0. We can break this upinto 0→ Z→ C exp→ C∗ → 0 and 0→ Z2g → Cg → Pic(C)→ Z→ 0.

This tells us that we have a map Pic(C)→ Z which is surjective. This is thedegree map or the first Chern class. The kernel is Pic0(C), which is Cg/Z2g.

So Pic0(C) = (S1)2g topologically, and Pic(C) = Pic0(C) × Z, because theextension splits (not naturally though).

Moving back to elliptic curves, we let L ∈ Pic0(C).

Lemma 3.9. For a line bundle L of degree zero on an elliptic curve, we have

h0(L ) =

1 L = O

0 else.

Proof. Assume not, then s ∈ L (C) is a section which has no poles, and so if ithas degree 0, it must have no zeros. So it has global sections C and thus mustbe trivial, which is a contradiction.

Also, Riemann-Roch tells us that h0 − h1 = deg L − g + 1 = 0, so h1 = 0also.

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Lemma 3.10. For d > 0, if L ∈ Picd(C) where C is an elliptic curve, thenh0(C) = d.

Proof. We know that h1(L ) = h0(L −1 ⊗ K) by Serre Duality. The sheafK ⊗L −1 has degree −d, and so has no global section, and thus h1(L ) = 0.

Applying Riemann-Roch, we get that h0(L )− h1(L ) = h0(L ) = deg L −g + 1 = d− 1 + 1 = d.

What about KC? By the computation of cohomology, degKC = 0, soh0(Kc) = g = 1 and so by Lemma 3.9, we have that KC = OC .

For an element L ∈ Pic1(C), BL is a point and ϕL : C \ pt → pt.This suggests that we define a map β : Pic1(C)→ C by L 7→ BL .

Corollary 3.11. The Abel-Jacobi map for a genus 1 curve C is an isomorphismC ' Pic1(C)

Proof. α and β are inverse maps, which can be seen by explicit calculation.

This is at least topologically true.

Definition 3.8 (Jacobian). The Jacobian of a curve is Pic0(C).

Now we can see that C ' Pic1(C) ' Jac(C), which is a g dimensionalcomplex manifold (in fact a variety). The corollary in fact says that there is anisomorphism of varieties C ' Jac(C) if g = 1.

Lemma 3.12. Let L ∈ Pic2(C), then BL = ∅.

Proof. Looking at L ∈ Pic2(C), we know h0(L ) = 2 and h1(L ) = 0, so anysection will vanish on two points. BL will then be either one point or theempty set. In fact, it will always be empty, because the Abel-Jacobi map is anisomorphism.

So for any L ∈ Pic2(C), we get a map ϕL : C → P1. What is ϕ−1L (pt)?

It is two points or else a point with multiplicity two. This tells us that thetopological degree is 2. We can also think about this as a branched cover of P1,cut along curves between 0 and 1 and between ∞ and a point λ.

Definition 3.9 (Branch Point). Let C → C ′ be a map of curves of topologi-cal degree d. Then a point p ∈ C ′ is a branch point if f−1(p) does not havecardinality d.

We say that a map has ramified behavior over branch points and unramifiedbehavior away from them (this is provisional, and not a rigorous definition).

So now we must ask how many branch points does ϕL have?

Theorem 3.13 (Hurwitz Formula). Let f : C → C ′ be a map of curves oftopological degree d. Then χ(C) = dχ(C ′) − |B|, where B is the set of branchpoints of f .

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Thus, we have χ(C) = 2χ(P1) − b, and so b = 4. As P1 has a three dimen-sional automorphism group, we can take three of these points to be 0, 1,∞, andso the fourth branch point λ determines C, up to an action of S3.

Theorem 3.14. If L ∈ Pic3(C), then BL = ∅ and ϕL : C → P2 is anembedding of C as a cubic plane curve.

Proof. Let L ∈ Pic3(C), then h0(L ) = 3 and BL = ∅, so we get a mapϕL : C → P2. ϕL (C) is a planar cubic curve, because L has degree three. Wemust show that if p 6= q ∈ C, then ϕL (p) 6= ϕL(q) and for all p ∈ C, dpϕL 6= 0,so it is an embedding.

Assume ϕL (p) = ϕL (q), that is, for all s ∈ Γ(L ), s(p) = 0 iff s(q) = 0.Consider L (−p) which has degree 2, and has no base locus. In particular,q /∈ BL (−p), and so there exists s′ ∈ Γ(C,L (−p)) such that s′(q) 6= 0. Notethat L (−p) → L is an inclusion, so s′ determines a section s ∈ L with svanishing at q1, q2 not equal to q and s vanishes at q1, q2, p but not at q.

Now for dp(ϕL ). Assume that dpϕL = 0 for some p. Then for all sectionss ∈ Γ(L ), p ∈ Z(s) ⇐⇒ 2p ∈ Z(s). The same argument arrives at acontradiction here as well.

Thus, every genus 1 curve is a smooth plane curve of degree 3. We will in thefuture need the Adjunction formula, which is a bit more general than Hurwitz’sFormula. Another special case of the Adjunction Formula is as follows:

Proposition 3.15 (Adjunction for Subvarieties). Let X be any projective non-singular variety and L ∈ Pic(X). Then for any s ∈ Γ(X,L ) and Y = s = 0nonsingular, we have KY = (KX ⊗L )|Y .

This allows us to prove a converse for Theorem 3.14.

Theorem 3.16. Any smooth plane curve of degree three is an elliptic curve.

Proof. We will need the fact that KP2 = O(−3). We want to apply Adjunctionfor X = P2 and L = O(3), as the sections of L will be cubics. Then KY =O(−3 + 3) = O. And so, the degree of KY = deg O = 0 = 2g − 2, and sog = 1.

To make this fully rigorous, we need to prove the Adjunction formula andto compute KP2 .

3.3 Curves of Genus Two

Let g = 2. Then degK = 2, h0(K) = 2. What is BK? We have two independentsections, s1, s2, say si vanishes on Di. Then degDi = 2.

Lemma 3.17. D1 ∩D2 = ∅

Proof. If p ∈ BK , then D1 = p + q and D2 = p + v. Consider K(−p) =K ⊗ O(−p). Di − p give two independent sections, so h0(C,K(−p)) ≥ 2. LetL = K(−p), then L had degree 1, and h0(L) = 2, so ϕL : C → P1 has degree 1,and so it is an isomorphism, contradicting the genus.

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More generally, the following is true:

Theorem 3.18. Let g ≥ 1. Then a line bundle with deg L = 1 cannot haveh0(L ) > 1.

So we now have ϕK : C → P1, which realizes any curve C of genus 2 as adouble cover of P1.

By Hurwitz’s Formula χ(C) = dχ(P1)− b. For ϕL a double cover, this givesb = 6.

It is a fact that the dimension of the moduli space of curves of genus g is3g−3+ε. For g = 2, this is 3. Another way to see this is that there are 6 branchpoints, so six degrees of freedom, less three degrees of freedom from Aut P1, andso the moduli of genus 2 curves has dimension 3.

Proposition 3.19. If C is a curve of genus 2 and L ∈ Pic0(C), then h0(L ) =1 L = OC

0 else, and if L ∈ Pic1(C), then h0(L ) =

1 L ∈ α(C)0 else

where α is

the Abel-Jacobi map.

Proof. Let p ∈ C. Then 1 ≥ h0(O(p)) ≥ h0(O) = 1. Conversely, for L ∈ Pic1,if it has a section, then that section must vanish at some p, and so L ' O(p).

In general, L ∈ Pic1(C) then h0(L ) = 1 if and only if L is in the image ofα(C).

Lemma 3.20. For g ≥ 1, the Abel-Jacobi map α is injective.

Proof. This is a consequence of the fact that if a curve has points p, q ∈ C suchthat p is linearly equivalent to q, then C is isomorphic to P1. See Hartshornefor proof.

We also have the following fact:

Lemma 3.21. If g > 0 and d ≥ 1, then h0(L ) ≤ deg L for any line bundleL .

Proof. If L has degree d and h0(L ) 6= 0, then choose a section s. Let p be apoint where s(p) = 0. Then we have 0→ L (−p)→ L → Cp → 0, where Cp isthe sheaf with value C if p ∈ U and 0 else. We also have 0 → H0(L (−p)) →H0(L ) → C and so we have h0(L ) ≤ h0(L (−p)) + 1. The lemma follows byinduction.

Example 3.1. If L = K, g = 2, then we have that h0(L ) = 2 and deg L = 2.Serre Duality then tells us that h1(L ) = h0(K ⊗L −1), and this will then havedegree 0, and so h1(L ) = 0 unless L = K, and h1(K) = 1.

Riemann-Roch says that h0(L )−h1(L ) = d−g+ 1 = 1, and so h0(L ) is 1or 2, and it is 2 iff L = K, and otherwise is 1. So h0(L ) 6= 0 for L ∈ Pic2(C)with C a genus 2 curve.

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In particular, for g = 2 and L ∈ Pic2(C), we have h0(L ) ∈ 1, 2 if L 6= K.What about if we have deg L = 3? We have entered the stable range: that

is, whenever d > 2g − 2. Calling it this is justified by the following:

Proposition 3.22. For any g, if d = deg(L ) > 2g−2, then h0(L ) = d−g+1.

Proof. Riemann-Roch says that h0(L )− h1(L ) = d− g + 1 and Serre Dualitygives h1(L ) = h0(K ⊗L −1) which has negative degree and so is zero, and soRiemann-Roch reduces to the desired quantity.

Thus, for g = 2, d ≥ 3, there is a unique stratum: all of Picd(C).We want to look at embeddings, and so we must continue on to d = 4. Then

h0(L ) = 3, and so we get ϕL : C \BL → P2. For general L , the base locus isempty.

So we get ϕL : C → P2. The degree of the image is 4, because L ∈ Pic4(C).The image cannot be smooth! This is a consequence of the following corollaryof the Adjunction formula:

Corollary 3.23. Let C be a smooth plane curve of degree d. Then g =(d−1)(d−2)

2 .

Proof. Let C be a plane curve of degree d, that is, it is the zero set of a section ofOP2(d). Then the Adjunction formula gives us that degKC = deg(KP2 ⊗O(d)).That is, 2g − 2 = deg(O(−3)⊗ O(d)) = deg(O(d− 3)|C). That gives 2g − 2 =d(d− 3), which can be solved to give g = (d−1)(d−2)

2 .

This means that if d = 4, we must have g = 3 for a smooth plane quarticcurve. In fact, for p, q ∈ C

Claim: There exist p, q ∈ C such that generically p 6= q and ϕL(p) = ϕL(q),but a special case is p = q and dϕL|p = 0.

However, if L had degree 5, it will embed C into P3 for L a generic linebundle.

3.4 Hyperelliptic Curves

We note first that for any g ≥ 2, h0(K) = g, deg(K) = 2g − 2.

Definition 3.10 (Hyperelliptic). A curve is hyperelliptic if there exists a degreetwo map π : C → P1.

Theorem 3.24. Either C is hyperelliptic in which case ϕK is two to one ontoa rational curve in Pg−1, or else ϕK : C → Pg−1 is an embedding.

To justify this at all, we first need the following lemma.

Lemma 3.25. Let C be a curve g(C) ≥ 2 and K its canonical divisor, thendegK = 2g − 2 and h0(K) = g. Then BK = ∅.

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Proof. Say p ∈ BL, then that is the same as H0(K(−p)) → H0(K) (which isinjective) is an isomorphism. This would imply that h0(K(−p)) = h0(K) = g.So p is a base point iff h0(K(−p)) = g. But h0(K(−p)) = dg − g + 1 +h1(K(−p)) = 2g − 2− 1 + g + 1 + h1 = g − 1.

So we now have a canonical map ϕK : C → Pg−1. And we claim that eitherϕK is an embedding or else it is a 2 to 1 map onto a rational normal curve,which is the Veronese embedding of P1 under ϕO(g−1).

Before we prove the theorem, we will look at the hyperelliptic case in moredetail.

In the case of a hyperelliptic curve, we should make the provisional definitionthat a point mapped to a branch point is called a ramification point, R, theramification divisor, is their sum and B is the sum of the branch points, thebranch divisor. Things are really a bit more complicated, but this much is truein the hyperelliptic case.

In the case of π : C → P1 a double cover ramified at 2g + 2 points of C, theAdjunction formula gives KC = π∗KP1(R).

More generally, even with the better definition of ramification, the followingis true:

Theorem 3.26. Let π : X → Y be finite map between smooth projective curves.Then we have KX = π∗KY (R). If we define π∗OP1(1) to be H, then degH =2 and h0(H) = 2, and so we have KC = H−2(R). In fact, KC = Hg−1.Equivalently, O(R) = Hg+1.

Now we will prove the characterization of canonical maps:

Proof. Assume that ϕK(p) = ϕK(q) for some p 6= q. This means that anydifferential form will vanish at p iff it vanishes at q. So H0(K(−p− q)) includesinto each H0(K(−p)) and H0(K(−q)) and each of those includes into H0(K),and the maps commute. So h0(K(−p− q)) = g− 1 as it is equal to 2g− 2− 2−g + 1 + h0(O(p + q)) and O(p + q) has dimension 2, and we will call this linebundle H.

By definition, degH = 2 and we know that h0(H) = 2, and so ϕH : C → P1

is a double cover. If x ∈ P1, then ϕ−1H (x) = r + s. Take ϕK(p) = ϕK(q)

to be y, then O(y) = O(x) on P1, and so ϕ∗H(O(y)) = ϕ∗H(O(x)), and soO(r+s) = O(p+q). This says that these two sheaves both have two dimensionsof global sections.

Now we note that ϕK(p) = ϕK(q) iff h0(O(p + q)) = 2, and so ϕK(r) =ϕK(s), and so the map ϕK factors through P1 via ϕH . And so the claimholds.

The arguments so far have been very repetitive, and we can encode this andstop worrying about it using the Geometric form of Riemann Roch:

Theorem 3.27 (Geometric Riemann-Roch). Given p1, . . . , pd ∈ C, with g ≥ 1,so h0(C,O(p1, . . . , pd)) = 1+the number of linear relations among the imagesϕK(pi).

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If you lift the points to their spans in Cn+1, then this number is just d −dim spanp1, . . . , pd

Example 3.2. If d = 1, then h0(O(p)) = 1 as there are no relations.

If d = 2, then h0(O(p+ q)) =

1 if ϕK(p) 6= ϕK(q)2, if ϕK(p) = ϕK(q)

.

If d = 3, then h0(O(p+ q + r)) =

1, generically2, C is trigonal3, never

Definition 3.11 (Trigonal). A curve is trigonal if it has a map of degree threeto P1.

Proposition 3.28. A curve is trigonal iff ϕK(C) sits on some ruled surfaceand intersects each ruling in 3 points.

We now prove the Geometric Riemann-Roch Theorem:

Proof. Take h0(O(p1 + . . .+ pd)) = 1 + d− 1− dim span(ϕK(pi)) = d− (g− 1−h0(K(−

∑pi))). By Serre Duality, this is d− g + 1 + h1(O(

∑pi)), and so it is

equivalent to standard Riemann-Roch with Serre Duality.

3.5 The Abel-Jacobi Map

Recall that the Abel-Jacobi map is α : C → Pic1(C) defined by α(p) = O(p).In fact, we can generalize this to get αd : Cd → Picd(C) by (p1, . . . , pd) 7→O(p1 + . . .+ pd). Even better, this map factors through Symd(C), just becauseaddition of divisors is commutative. So now we ask: what are the fibers of αdlike?

Theorem 3.29 (Abel). αd(p1+. . .+pd) = αd(p′1+. . .+p′d) iff O(p1+. . .+pd) 'O(p′1 + . . .+ p′d), and so the fibers are projective spaces.

We can interpret this as follows: Choose a basis ω1, . . . , ωg of the differentialson C, then α(p) = α(p0) ⇐⇒

∫ pp0ωi = 0 for all i for some path.

We will look at the Abel-Jacobi map α : C → Pic1(C) = Cg/Λ in moredetail. First we note that a choice of a base point p0 ∈ C determines anisomorphism Pic1(C) → Pic0(C) by L 7→ L(−p0). One interpretation of α isas a multivalued function C → Cg. It can be expressed in terms of a basisω1, . . . , ωg of H0(C,KC). So the Abel Jacobi map is p 7→ (

∫ pp0ω1, . . . ,

∫ pp0ωg).

With the algebraic definition of αd, Abel’s Theorem is obvious. Analytically,it is significantly less so.

Example 3.3. If g = 1, look at the elliptic integral∫ pp0

dx√x3+ax+b

. What Abel’s

Theorem says here, look at∑3i=1

∫ pi

p0dx√

x3+ax+b, and we can take the plane cubic

y2 = x3 + ax + b, and so the answer of the integral doesn’t depend on whichthree points we take if they are collinear.

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This connects because dx/y is a global 1-form on an Elliptic Curve (becausedx/y = 2dy/(3x3 + a)).

We will now prove Abel’s Theorem from the analytic definitions.

Proof. Let D =∑pi and D′ =

∑p′i with O(D) ∼= O(D′) = L . Then h0(L ) ≥

2. Consider the function on P(H0(C,L )) that takes D 7→∫Ddp0

ω. This is amultivalued holomorphic function on Pn.

This implies that it is a single valued function in the universal cover, whichis Pn itself, and hence is constant.

Proposition 3.30. The map dα : TpC → Tα(p) Pic1(C) ' Cg = H0(C,KC)∗

takes C → Pg−1 and is equal to ϕK .

Proof. ϕK is the map that takes p 7→ (ω1(p), . . . , ωg(p)), and α is given by theintegrals of the ωi, and so differentiating to get dα returns the values of theωi.

We can restate the analytic proof as starting with Cd → Symd αd→ Picd(C),and ω is a 1-form on C. We can think of it as a 1-form on Picd(C).

Formally, α : C → Pic1(C) induces α∗ : H1(C) → H1(Pic1(C)) an isomor-phism, and so α∗ : H1(Pic1(C))→ H1(C) is an isomorphism.

So α∗1ω is a holomorphic 1-form on Symd C. We’ve assumed that there existsP ⊂ Symd(C) with α∗dω|P ≡ 0.

4 Vector Bundles

4.1 Standard Bundles

First, we look at line bundles.

Lemma 4.1. The degree map from Pic(Pn)→ Z is an isomorphism.

Proof. This follows from the fact that if X ⊂ Pn is a degree d hypersurface,then the d-uple Veronese embedding on Pn takes X to a hyperplane section,and so X is linearly equivalent to dH where H is a hyperplane in Pn.

Example 4.1. OPn(1) is the hyperplane bundle, and it is a generator of Pic(Pn).For d ≥ 0, the sections of OPn(d) are the homogeneous polynomials of degree din n+ 1 variables. Thus, h0(O(d)) =

(n+dn

).

Example 4.2. Let T be the tangent bundle. The sections are∑aijxi

∂∂xj

. Notethat Pn = P(V ), and we can get an exact sequence 0 → O → OPn(1) ⊗C V →TPn → 0.

Definition 4.1 (Euler Vector Field). The Euler vector field is e =∑xi

∂∂xi

which is a vector field on V \ 0 which is tangent to the fibers of projectivization.

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The first map in the short exact sequence above is e. This gives the abovesequence the name ”The Euler Sequence.” It determines the tangent space ofPn by (O(1) ⊗C V )/O where the top is a sum of dimV copies of O(1) modulothe diagonal embedding of O.

Example 4.3. For n = 1, the Euler sequence is 0 → O → O(1) ⊕ O(1) →

O(2)→ 0 with the first map (x2,−x1) and the second being(x1

x2

).

Proposition 4.2. Let K be the canonical bundle on Pn. Then K = O(−n− 1)

Proof. Dualizing the Euler sequence we get 0 → Ω1Pn → O(−1) ⊗C V

∗ → O →0. Taking determinants, which is additive for short exact sequences, we getdet(Ω1) ⊗ det(O) = det(O(−1) ⊗C V

∗). The left is K and the right is thedeterminant n+ 1 copies of O(−1), and so is O(−n− 1).

Definition 4.2 (Normal Bundle). Let X be a variety and D ⊂ X a smoothdivisor. Then we have a short exact sequence 0 → TD → TX |D → N → 0 forsome vector bundle N . We call N the normal bundle.

We define N∗, the dual of N , which satisfies 0 → N∗ → Ω1X |D → Ω1

D → 0to be the conormal bundle.

Theorem 4.3 (Adjunction Formula). Let X be a variety, D ⊂ X a divisor.Then KD = KX(D).

Proof. Let f be a local defining equation for D. Then df is a local generatingsection for N∗. Change the defining equation f ′ = gf , then df ′|D = fdg|D +gdf |D = gdf |D, so conclusion is that df transforms in the same way as f .

So N∗ ∼= (OX(D))|D. Take determinants to get KX |D ∼= KD ⊗ OX(−D)and so KD = KX(D).

Example 4.4. For X = P2, and D a smooth curve of degree d, we get OX(D) =OP2(d) and so KX = OP2(−3) and KD

∼= OP2(d− 3)|D.

Definition 4.3 (Projection from a Point). Let D ⊂ Pn be a divisor given byf = 0, and let p ∈ Pn \ D. We define the projection from a point map to bePn \ p → Pn−1 by taking a line through p to its intersection with Pn−1. Incoordinates, we can take p = (1 : 0 : . . . : 0) and so π(x0, . . . , xn) = (x1, . . . , xn).

Proposition 4.4. This π is the projectivization of the linear projection An+1 →An whose kernel is the line in An+1 corresponding to p.

We call this map a linear projection. We will study π|D : D → Pn−1.This map is a branched covering of degree d. We are interested in determin-

ing B, the branch locus.

Remark 1. The branch locus of this map is called the apparent contour due toits connection to how we look at things and the use of projective geometry todraw things in perspective.

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Definition 4.4 (Polar Equation). As D is given by (f = 0), we have df =(∂f∂x0

, . . . , ∂f∂xn

)and we define dpf = 〈p, df〉 =

∑pi

∂f∂xi

. As a function of thexi, this is called the polar equation of f at p. The degree of the polar equationis d− 1.

Now, R, the ramification divisor, has codimension 2 in Pn, and it is easierto write down its equations that those of B, the branch divisor.

Lemma 4.5. R is a complete intersection, and is given by f = dpf = 0.

Note that this is simply the condition for a multiple root in the case of apolynomial in one variable, f = f ′ = 0.

Example 4.5. If we have n = 2, then D is a curve of degree d in the plane.Then the number of ramification points is d(d− 1), which is also the number ofbranch points. For general n, this argument gives degB = d(d− 1).

Proposition 4.6. Let C be a nonsingular conic. Then dpf ∩ C is the set ofpoints of C whose tangent lines contain p.

Example 4.6. For C ⊂ P2 a conic, the polar map (as a function of p ratherthan xi) takes P2 \ C to (P2)∗. For instance, let f = x2

0 + x21 + x2

2 and letp = (p0, p1, p2), then dpf = x0p0 + x1p1 + x2p2 so we get a line for each p, andfor each line there is a p which gives it.

Proposition 4.7. Given f ∈ Sym2(V ∗) you have a map dpf : V → V ∗ (as afunction of p) which is an isomorphism converting a point to a hyperplane. Thismap extends to all of P2 and takes points on the conic to their tangent lines.

4.2 Direct Images

Example 4.7. Let f : P1w → P1

z where the subscripts are the names of thecoordinates. Let f be given by z = w2. Then what is f∗O? It is a rank twovector bundle on P1

z, and in particular, it is an OP1z-module. We do an even-odd

decomposition.Then f(w) = g(w2) + wh(w2) = g(z) + wh(z), and g ∈ Γ(OP1

z) and h ∈

Γ(OP1z(−1)). g is arbitrary and h must vanish at ∞. Thus π∗OP1

w= OP1

z⊕

OP1z(−1).

We want to generalize this example, and get the following:

Theorem 4.8. Let π : P1 → P1 be the squaring map. Then

π∗O(d) =

O(d2

)⊕ O

(d−2

2

)d is even

O(d−1

2

)⊕ O

(d−1

2

)d is odd

Proof. The proof is as in the example, by using an even-odd decomposition.

Even more generally:

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Theorem 4.9. Let π by the nth power map P1 → P1. Then π∗O(d) is a directsum of n line bundles, with π∗(O(−1)) = ⊕ni=1O(−1) and an increase in dincreases the degrees of the terms of the image so that none has degree morethan one greater than any of the others.

Proof. We take f(w) 7→∑n−1i=0 w

igi(z). So now we get conditions on g comingfrom f ∈ Γ(O(d)), specifically a pole at infinity with n ord∞(g) + i ≤ d.

Now there are n cases, for π∗O(−1), we get a direct sum of n O(−1)’s. Aswe increase degree, we increase the degree of the summands alternately, so thatnone has degree more than one greater than any of the others.

deg(π∗F ) = deg F − (n − 1), π∗F is a direct sum of line bundles andthe distribution of degrees is as tight as possible (only achieves two adjacentvalues)

More generally, π : P1w → P1

z any algebraic map of degree n, then z = π(w) =P (w)/Q(w) where degP ≤ n, degQ ≤ n, relatively prime and at least one hasdegree equal to n, and Q is the unique polynomial vanishing at π−1(∞) and Pis the unique on vanishing on π−1(0). To use this to find them, we use Lagrangeinterpolation.

So in general, π∗(L ) = ⊕Li, and deg π∗L = deg L − (n − 1) and thedegrees will be bunched closely.

4.3 Split Sequences and Grothendieck’s Theorem

Definition 4.5 (Split Sequence). Let 0→ S → V → Q → 0 be a short exactsequence of vector bundles (as sheaves).

We say this splits if V = S ⊕Q, this is equivalence to there being a maps : Q → V which composes to the identity on Q.

Proposition 4.10. On the long exact sequence, H0(Q) → H1(S ) is the zeromap if the sequence splits.

If the coboundary map is zero, does the sequence split?

Example 4.8. Look on P1 where we have 0 → O(−1) → O ⊕ O → O(1) → 0given by maps (y,−x) and

(xy

). This is certainly not a split sequence. The map

δ : H0(O(1))→ H1(O(−1)) is a map from a 2 dimensional vector space to a 0dimensional vector space.

Conclusion: Split⇒ δ = 0 but not vice versa.

However, there is a variant that works.

Theorem 4.11. Let 0→ S → V → Q → 0 be a short exact sequence of vectorbundles. Then this sequence splits if and only if for all L ∈ Pic(X), the mapH0(Q ⊗L )→ H1(S ⊗L ) induced by taking the tensor product and then thelong exact sequence on cohomology is zero.

Proof. Let L be a line bundle. Then 0→ S ⊗L → V ⊗L → Q ⊗L → 0 isalso exact. This sequence is split if and only if the original one was.

Remainder of proof omitted.

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More generally, we are talking about the extension problem.

Lemma 4.12. Let Q be a vector bundle and Q∗ its dual. Then there exists anatural map O → Q ⊗Q∗.

Definition 4.6 (Ext). Let 0 → S → V → Q → 0 be a short exact sequenceof vector bundles with Q a line bundle. Then we define Ext(Q,S ) = H1(Q∗⊗S ) = H1(H om(Q,S )), as δ : H0(O) → H1(Q∗ ⊗S ) is equivalent to givingδ ∈ H1(Q∗ ⊗S ).

So then by definition, Ext(Q,S ) consists of all extensions of Q by S , andH1(H om(Q,S )) is a cohomology group that classifies extensions.

Theorem 4.13 (Grothendieck). Every vector bundle on P1 is a direct sum ofline bundles.

Proof. The idea is that, given V , we can find a line bundle contained in V , L ,which is maximal of degree d.

(Note that a subsheaf which is locally free might not be a subbundle, de-pending on whether the quotient is locally free)

A locally free subsheaf of a locally free sheaf is a subbundle iff it is maximal.Use Riemann-Roch to prove that the set of possible degrees is bounded

above. Now replace V by V ⊗L −1. Equivalently, d = 0, and than tensoringwith L −1 then there are no global sections left.

If d > 0, then the sections of V cannot vanish anywhere.So we have 0 → H0(V ) ⊗C O → V → Q → 0 gives a long exact sequence

0 → H0(V ) ' H0(V ) → H0(Q) → H0(V ) ⊗ H1(O), but the isomorphismcausesH0(Q) = 0, and so Q is a vector bundle of lower rank. Then by induction,Q = ⊕iO(−di) for di > 0.

So we have now that V can be written as an extension 0 → ⊕jO → V →⊕iO(−di)→ 0

We claim that this extension splits.All such extensions split, because

Ext(⊕iO(−di),⊕jO) = ⊕i,j Ext(O(−di),O) = ⊕ijH1(P1,O(di)) =

⊕ijH0(P1,K(−di))∗ = ⊕ijH0(O(−di − 2))∗ = 0

.

4.4 Jumping Phenomenon

Definition 4.7 (Family of Vector Bundles). A family of vector bundles param-eterized by A1 is a vector bundle on X × A1. We define Vt = i∗tV a vectorbundle on X.

Proposition 4.14. There exists a family of vector bundles such that s 6= 0, t 6= 0implies Vs ' Vt but Vt 6' V0 for t 6= 0.

Equivalently, the topology on isomorphism classes of vector bundles on Xis non Hausdorff. The point V0 is in the closure of V1.

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It looks like the quotient A1/C∗, or in general An/C∗, which is Pn−1∐0.

Example 4.9. Let E be an elliptic curve. Then π : E → P1 is two to one. LetL ∈ Pic0(E) ' E be a line bundle. Then we have 0→ E → Pic(E)→ Z→ 0 ashort exact sequence. So what is π∗L ? It’s a locally free sheaf on P1 of rank 2.

So it is O(a) ⊗ L . This has deg(π∗O) = deg L − 12 (degR) = −2, so

h0(P1, π∗L ) = h0(E,L ) =

1 L = O

0 L 6= O.

However, also h0(P1,O(a + 1)) =

a+ 1 a ≥ −10 a ≤ −1

= max0, a + 1. So

we are left with max(0, a + 1) + max(0, b + 1) ∈ 0, 1, and a + b = −2. Wewill write b = −2− a, and so we have max(0, a+ 1) + max(0,−(a+ 1)) is 1 orzero. So this is |a+ 1|. If L = O then |a+ 1| = 1, and so a = 0 or a = −2. IfL 6= O, then |a+ 1| = 0, so a = −1.

Alternatively, tensor with π∗OP1(1), π∗(L ) = O(a) ⊕ O(b), and π∗(L ⊗π∗O(−1)) = O(a+ 1)⊕ O(b+ 1).

This is an application of the following:

Theorem 4.15 (Projection Formula). Let π : X → Y and let F a sheaf on Xand G a sheaf on Y . Then π∗(F ⊗ π∗G ) = π∗F ⊗ G .

Corollary 4.16. h0(E,L ⊗ π∗O(d)) = h0(P1,O(a+ d)) + h0(P1,O(b+ d)) ford >> 0 this gives 0 = a+ b+ 2 so a+ b = −2.

And, to phrase it as a theorem rather than a remark that has come up,

Theorem 4.17 (Degree Shift). Let π : X → Y a morphism and L a linebundle on X. Then deg(π∗L ) = deg L − 1

2 degR.

Corollary 4.18. χ(π∗L) = χ(L).

5 Moduli and Deformations

We are interested in looking at families of objects. This leads to notions ofmoduli spaces, and to infinitesimal moduli, that is, deformations.

Definition 5.1 (Deformation Space). The deformation space of X is Def(X) =T[X]moduli of X ′s.

Proposition 5.1. Let C be a smooth projective curve and let L be a linebundle. Then Def(L ) = H1(X,OX)

Proof. Let X be a variety, and L ∈ Pic(X), what is TL Pic(X)? Here we havea solution to the nonlinear problem, that is, Pic(X) = H1(X,O∗X).

To specify L ∈ Pic(X), give an open cover U and gluing function gij ∈Γ(Uij ,O∗) satisfying a cocycle condition.

31

A transition function is a 1-cochain, and a cocycle condition is a 1-cocycle,and so fixing U , L corresponds to a set of consistent transition functions. Thisis more like a line bundle with a trivialization on Ui.

So how about infinitesimal deformations?Start with L 7→ gij. We want to deform to first order. Rather than

looking at X → Spec C as a complex scheme, we look at X 7→ Spec C[ε].We have natural maps C[x] → C[ε] → C, and so Spec C → Spec C[ε] →

Spec C[x].So we want to replace gij with gij + εhij .So we want to know what the consistent ways of writing hij ’s. So what does

the cocycle condition say about the hij ’s?(gij + εhij)(gjk + εhjk) = (gik + εhik)To zeroth order, this gives gijgjk = gik, which is just saying that gij is a

line bundle, and so is satisfied.To first order, it says gijhik + hijgik = hik.And we don’t care about the second order terms or higher.The set of solutions are then the solutions to a linear equation, and so we

have linearized the problem. Let us define Hij = g−1ij hij .

Multiply by gki(that is, divide by gik). Note that gij/gik = gkj , and so nowthe equation is Hjk +Hij = Hik.

This cocycle condition gives H1(X,O).

Example 5.1. So we have objects line bundles on X, moduli space Pic(X) =H1(X,O∗) and deformations H1(X,OX).

Example 5.2. If we look at rank n vector bundles on a fixed variety X, then themoduli space is Bunn(X) = H1(X,GLn(O)), and the deformations are elementsof H1(X,End(V )) = H1(X,V ⊗ V ∗) at V .

Example 5.3. If our objects are curves of genus g, then Mg, then the tangentspace at C a curve is T[C](Mg) = H1(C,TC) = H0(C,K⊗2

C )∗ by Serre Duality,for g ≥ 2.

Example 5.4. Let X be any variety, and MX the moduli space of varietieswith the same underlying complex topology, then TX(MX) = H1(X,TX).

Example 5.5 (Principal G-bundles). Then look at the moduli space BunG(X),and TV (BunG(X)) = H1(X,Aut(V )) where V ∈ BunG(X).

Example 5.6. Fix Y a variety and vary X a subvariety. Then we have themoduli space MX⊂Y and TX(MX⊂Y ) = H0(X,NX/Y ).

Example 5.7. Varying both Y and X. This is no good, but instead we can lookat varying both X and a L a line bundle on X. There should be a map to MX

forgetting L , but there shouldn’t be a map to the moduli of L .

32

6 Lecture

Last time: How to calculate hp,q for a hypersurface in Pn+1.Answer: If p + q 6= n, then we have δp,q. The interesting case is p + q = n.

Then hp,q ' Rd−n−2+pd where R = S/J , the Jacobian.

Example 6.1. Let S = C[x, y] and f =∏di=1(x− aiy).

Take as an example f = xd+yd. Then J = (xd−1, yd−1), and so dimC R =(d− 1)2. (We are taking C[x, y] as the affine plane here)

Proposition 6.1. dimR is constant as f varies as long as X remains smooth.Similarly for Rk.

So in this case, the hodge numbers are topological invariants. However, thisis not true in general, sometimes it depends on the complex structure.

Now let X be a plane curve in P2, then h1,0 = Rd−n−2 = Rd−3 and h0,1 =R2d−3. By Serre duality, dimRd−3 = dimR2d−3.

Again, consider f = xd + yd + zd. Then if d = 5, we have g = 6.R0 = C · 1, has dimension 1. R1 = Cx + Cy + Cz has dimension 3. R2 =

Cx2 + Cy2 + Cz2 + Cxy + Cxz + Cyz is dimension 6. R3 is generated by cubicmonomials, and there are 10. R4 has dimension 12. R5 has dimension 12, R6

is 10, R7 is 6, R8 is 3 and R9 is 1.Start with a smooth hypersurface of degree d and dimension n. Variation

of Hodge structure gives vector spaces Rd−n−2 thgouth Rd(n+1)−n−2. We havebilinear maps Rd × Rd−n−2 → R2d−n−2, etc, and the top piece has dimension1.

Can you recover X (ie, its equation f) from this bilinear data?Sometimes the answer is no.

Example 6.2. Let n = 2, d = 3, that is, we have a plane cubic curve, anElliptic Curve.

We’re given Rk, which are all one dimensional. There’s no informationhere, any two elliptic curves give the same map.

However, looking at a plane quintic, there are four different groups andnontrivial maps between them, and we can do it.

In most cases, the answer is yes.Next: Calculate Hi(Pr,O(n)) from the definition.Take the standard open cover of Pr by copies of Ar.Then take F = ⊕n∈ZO(n). We have S = C[x0, . . . , xr] = H0(Pr,F ) =

⊕n∈ZH0(Pr,O(n)).

C∗(U ,F ) :∏ri0=0 Sxi0

→∏

0≤i0,i1≤r Sxi0 ,xi1→ . . ..

And so for r = 1, H0(F ) = S = C[x, y], H1(F ) = x−1y−1C[x−1, y−1]. Andnow, H1(P1,O(n)) consists of the strictly negative monomials of degree −n.

The idea in general is that the Cech complex decomposes into a direct sumof subcomplexes, one for each monomial.

The part of the Cech complex for a given monomial is the Koszul Complexfor this Vector Space.

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That is, C → V →∧2

V →∧3

V → . . .. All that we must do is computethe cohomology of this complex. This complex is exact except at the beginningand end...

Wait...The answer is SUPPOSED to be...???

7 WEDNESDAY

Calculation: Hk(Pr,O(n)).S = C[x0, . . . , xr] ⊂ C[x±1

i ] = Sx0,...,xr .Let R = 0, . . . , r and I ∈ ZR, with xI monomials. Then J ⊂ R we get a

localization SJ by inverting xi for i ∈ J .Sj = ⊕|J|=jSJ . The Cech complex for the standard U and the quasicoherent

F = ⊕nO(n) is S1 → S2 → S3 → . . .→ Sr+1.We augment the complex and have S0 → S1 → . . . → Sr+1, and this aug-

mented complex C∗ has decomposition e∗ = ⊕IeI where eI∗ is SI0 → SI1 → . . .→SIr+1 where I = I−

∐I+ and I− = r ∈ R|Ii < 0 and I+ = I \ I−. a = |I+|.

The coefficient of xI in SJ is 0 or C (if J ⊃ I−) and the coefficient in Sj isC, J |J ⊃ I−, |J | = j.

So the complex is in fact 0→ 0→ . . .→ 0→ C→ Cr+1−a = V →∧2

V →. . .. So the coefficient in Sj is

∧|J|−aV .

We’ve identified the spaces. What are the maps? V has a basis of degree1 monomials not in I. V 3 e is a sum of these monomials with coefficient 1.Then 1 7→ e 7→ v ∧ e and v1 ∧ v2 7→ v1 ∧ v2 ∧ e.

This is the Koszul Complex. In a different basis, we take e = (1, 0, . . . , 0)and V = Ce⊕ V0. Then

∧kV = e

∧k−1V0 ⊕

∧1V0.

And so we get maps∧k

V0 → e∧k

V1, etc. And these maps make thecomplex exact, unless 0 = dimV = |I+|. That is, unless I < 0 strictly.

And so H∗(CI∗ ) =

C ∗ = r, I− = R

0 elseand H∗(C∗) =

0 ∗ 6= r

1Qxi

C[x−1i ] ∗ = r

.

And so hr(Pr,O(−(r + 1) − n)) = h0(Pr,O(n)). And as O(−r − 1) = K, thisgives us Serre Duality for Pn.

Now we will sketch proofs of deRham, Dolbeault and Cech theorems.dR: On an algebraic manifold X, look at 0→ C→ A0 → A1 → . . .. We want

to break dR into a sequence of short exact sequences, 0→ C→ A0 → Z1 → 0,0 → Z1 → A1 → Z2 → 0, etc. So for each we get a long exact sequence oncohomology. Check things carefully, it works out by induction.

Similarly for Dolbeault. We get an acyclic (fine) resolution of Ωp, and so webreak into SES, and then use long exact sequence of cohomology.

Similarly for Cech, II.3.5 in Hartshorne.Let M be an A module. Then M is a sheaf of OX modules forA = Γ(X,OX).We get for f ∈ A, Xf = x ∈ X|f(x) 6= 0, and Γ(Xf , M) = Mf . Then

sheafify. A sheaf F of OX -modules is quasicoherent if locally F |Ui= Mi|Ui

.

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Coherent is the same, but with Mi finitely generated. (This is in the Zariskitopology)

Proposition 7.1. Take M = Γ(X,F ) and M = F . On X = SpecA, thefunctor M 7→ M from A-modules to quasicoherent sheaves on SpecA is anequivalence of categories with inverse functor Γ(X,−).

Take an injective resolution of M , 0 → M → I∗. Sheafifying, we get 0 →M = F → I∗ is an exact sequence of q.c. OX -modules. And so F has aresolution by injectives, flasques, acyclic, and so H∗(X,F ) = H∗(Γ(I∗)).

The proof imitates the proof for dR and Dolbeault.

8 FRIDAY

Today and Wednesday we will do Derived Functors and CategoriesThen cancel class for a conference at IASAnd we will finish up on Chern Classes, K-Theory, and Grothendieck-Riemann-

RochOn Monday 11/19, we will finish talking about deformation theory and the

Torelli Problem11/26 no class, IAS12/3 ???The Torelli Problem: We are giving a finite number of bilinear forms A×B →

C and we’re told that there exists a vector space V and polynomial f ∈ Symd Vsuch that Ri × Rj → Ri+j is isomorphic to A × B → C. Can we completelyrecover f?

Easier question: Given Si×Sj → Si+j isomorphic to A×B → C. Example:i = j = 1, then we have V × V → Sym2 V isomorphic to A × B → C. Picka ∈ A, b ∈ B, and look at a′ ∈ A, b′ ∈ B|a′b′ = ab. There are two possibilities.One is that the factorization is unique, up to C∗, and the other is that thereexist two factorizations.

Geometrically, P(V ) × P(V ) → P(Sym2 V ) has image a segre variety pro-jected to the symmetric product. The image is isomorphic to Sym2 P(V ),(dimV ≥ 3 implies that this is singular)

So bilinear form gives a projected segre variety, which has singular locus P(V )embedded in P(Sym2 V ) by O(2). And so we get C ' Sym2A for dimension≤ 2, replace singular by ”having unique inverse maps”

More talk of Torelli rather than the promised Derived Categories.

8.1 Derived Categories

”Derived Categories for the Working Mathematician” by Thomas. Available onthe Arxiv

The idea is to go from objects to complexes to equivalence categories ofcomplexes.

Input category: Abelian.

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Definition 8.1 (Additive Category). For any A,B objects of A, hom(A,B) isan abelian group and composition is distributive over addition, finite productsexist, and there exists a zero object.

Definition 8.2 (Abelian Category). An abelian category is an additive categorysuch that any morphism has a kernel and a cokernel, f monic implies thatf = ker coker f and f epic implies f = coker ker f .

In an abelian category, we can discuss the following notions:complexes, exact sequences, and cohomology of a complex, morphisms of

complexes, quasi-isomorphism, additive, exact, left and right exact functors.So if A is an abelian category, then C(A) is teh category of complexes.So now D(A), the derived category, has the same objects and the morphisms

are ”invert all quasiisomorphisms”. So a morphism inD(A) is a zigzag...confusinggibberish, will look in book.

Lemma 8.1. Any morphism is Af→ C

g← B.

There are more common flavors than the general derived category.D+, D−, Db are the bounded above, bounded below, and bounded com-

plexes.

9 WEDNESDAY

”Complexes Good, Cohomology Bad”Note that Db(A) is an additive category, but is not Abelian, because kernels

and cokernels do not necessarily exist.The actual structure we have is that of a triangulated category.Let A and B be abelian categories and F : A → B be a left exact functor.

A class R ⊂ ob(A) is adapted if R is stable under direct sums and for anyA ∈ ob(A) injective, there is an element in R, X, such that 0 → A → X isexact, and if F (exact complex of elements of R) is exact.

Lemma 9.1. D+(A) is obtained by inverting quasi-isomorphisms in K∗(A)the category of lower bounded complexes in A with objects in R with morphismshomotopy classes of chain maps.

Example 9.1. Injective objects work for F = Γ.

There exists an analogue for right exact functors.If F is left exact, then we can construct RF : D+(A) → D+(B), the right

derived functor of F .As D+(A) is equivalent to K+(R)/q.i., we can define RF , and do so by

applying F .RF is an exact functor, but what does this mean? It takes ”long exact

sequences” to ”long exact sequences”In general, RF : D+(A) → D+(B) → B with the last one by Hi. These

compositions are RiF , the ith derived functor, because they factor throughD+(A) as functors A→ B.

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Example 9.2. F = Γ :Sheaves→ Ab. Then RiΓ = Hi as above, with Hi thesheaf cohomology we defined. RΓ contains all this information, and a bit moreas well.

Example 9.3. F = ⊗F from sheaves to sheaves. And this is right exact. So

L⊗, which we writeL⊗, the ith derived functor is called Tori

Example 9.4. H om(F ,G ) is covariant in G and contravariant in G , and thefirst is left exact and the second is also left exact. These derived functors arecalled Exti.

Not always, but most of the time, R(F G) ∼= RF RGIn fact, Rn(F G)⇐ RiF (RjG) (this is not quite convergence, but spectral

sequence convergence)This is called the Leray Spectral Sequence.

Example 9.5. Let p : X → Y and F = Γ(Y,−), G = p∗, then we havefunctors Sheaves(X)→ Sheaves(Y )→ Ab, then Leray gives Hi(Y,Rjp∗F )→Hi+j(X,F ).

The Rip∗ are called the higher direct images, and so we have Rp∗ : D+(X)→D+(Y ).

In the C∞ world, we recover the dR, Dolbeault and Cech Theorems.Cones and TrianglesIn topology, take f : X → Y be a continuous map. Construct cone(f) to be

X × [0, 1]∐Y/X × 0, (x, 1) ∼ f(x).

The use is that it gives a sequence on homology, Hi(X) → Hi(Y ) →Hi(Cf )→ Hi−1(X)→ . . .. So Cf acts like (X,Y ). Ci(Cf ) = Ci(Y )⊕Ci−1(X).

So, given a chain map f : A → B, we define Cf to be A[1]⊕ B, and dCfis

a two by two matrix(dA[1] 0f dB

). (check to make sure)

So if we have A∗ c→ B∗i→ C∗f , then C∗(i) = A∗[1]. So this is really like a

triangle, and is the analogue of an exact sequence.

10 Lecture

MISSED

11 Lecture

Previously we looked at varieties and schemes, but last time we started to lookat properties of morphisms f : X → Y

Affine, projective, proper, finitesmooth, flatThe top line is proper in Y , the bottom is local in X.

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Given f : X → Y , what is the expected behavior of the family of coho-mologies Hi(Xy,Fy), where F is a coherent sheaf on X and Xy is the fiberf−1(y).

Example 11.1. Let E be an elliptic curve and Y = E, X = E × E. Thenf : X → Y is a map E × E → E with the map the second projection. Thenlet F = OE×E(∆− E1 − E2) where ∆ is the diagonal, and E1 = E × 0, andsimilarly for E2.

For each y ∈ Y , Xy ' E.Fy = OE(y − 0). This has h1 = h0 1 if y = 0 and 0 else.

Define χ(X,F ) =∑i(−1)ihi(X,F ) to be the Euler characteristic of the

sheaf F .Assume X ⊂ Pn projective. Here we have OX(1). We define the Hilbert

polynomial P(n) = χ(X,F ⊗ O(n)X).Note: for n >> 0, all hi vanish for i > 0.f : X → Y , f∗ is left exact, so we get Rif∗ the higher derived images.What is the behavior of RifxF ⊗k(y) as a function of y ∈ Y vs hi(Xy,Fy)?

Theorem 11.1. Let f : X → Y , F on X, then PFy (n) is independent of y ifand only if F is flat over Y . In particular, POXy

(n) is independent of xy iff Xis flat over Y .

If f : X → Y is any projective morphism of noetherian schemes and F iscoherent over X and flat over Y , then for all i ≥ 0, the function h1(y,F ) =dimH(Xy,Fy) is upper semicontinuous.

Assume also that Y is integral. If hi(Y,F ) is constant, then Rif∗F is locallyfree on Y .

(Rif∗F ⊗ k(y)⇒ Hi(Xy, Fy). (Graouert?)f : X → Y projective morphism of noetherian schemes. F coherent on X

and flat over Y . Then if ρi(y) : Rif∗F ⊗ k(y)→ Hi(Xy,Fy) is surjective at y,then it is an isomorphism on some open set containing y.

Now we have functors Coh(X) → Coh(Y ) → V S or we can do Coh(X) →Coh(Xy) → V S, (the last map is global sections) and the obvious diagramcommutes. And ρi are derived functors of the restriction map

If this holds, then ρi−1(y) is surjective iff Rif∗F is locally free near y.

12 Last Lecture

Another perspective with higher direct images: does taking cohomology com-mute with base change?

Base chance is, given f : X → Y , and u : Y ′ → Y , we have g : X ′ → Y ′,with X ′ = Y ′ ×Y X. This is called the base change, with v : X ′ → X. Thendefine F = v∗F a sheaf on X ′ for a sheaf F on X. Fiber by fiber this ”lookslike” F .

In this situation, we get a natural morphism ∃φi : u∗Rif∗F → Rig∗(v∗F ).Special case: Y ′ = y, y ∈ Y . Then X ′ = Xy is the fiber, and Fy is F ′.

38

In this case, Rig∗G = Hi(Xy,G ).The map from above is φiy : (Rif∗F )y → Hi(Xy,Fy) from Wednesday.Q: If φi an isomorphism?A: If u is a FLAT base change, the φi is an isomorphism. The question is

local on Y ′, so we can take Y = SpecA, Y ′ = SpecA′. So it boils down toHi(X,F )⊗A A′ ∼= Hi(X ′,F ′).

This is a straightforward Cech Cohomology computation.This is also true if u is inclusion of a closed point, then φi is an isomorphism

for the top i.In more detail, φi is surjective at y iff isomorphism at y implies that is is an

isomorphism at y. And then φi−1 is surjective iff Rif∗ is locally free.(Case i = dimX − dimY + 1, and i is the top dimension of fibers. Then φi

is an isomorphism. Rif∗F = 0 is locally free, and so φtop is an isomorphism.)An elliptic fibration is a morphism π : Z → B Choose B,Z smooth and π

flat with the generic fiber an elliptic curve. Take a section u : B → Z. We AREallowing some fibers to be singular.

e.g, f, g ∈ Γ(P2,O(3)), then for all [s, t] ∈ P1, sf + tg is a plane cubic curve,it is elliptic if smooth. If it is smooth, it is an elliptic curve. We want somethinglike f/g : P2 → P1 by [x : y : z] 7→ [f(x, y, z) : g(x, y, z)].

Let Z = ((x, y, z), (s, t)) ∈ P2 × P1 such that sf(x, y, z) + tg(x, y, z) = 0.This is a variety. The inverse image of the first projection gives P1 if f(x, y, z) =g(x, y, z) = 0 and is a point otherwise. Then Z\ nine copies of P1 is isomorphismto P2\ 9 points.

So if we take B = P1, each of the removed P1’s gives a section, and so wehave an elliptic fibration.

This surface is a del Pezzo surface (not quite, actually...del Pezzo only goesup to 8, this is ALMOST del Pezzo)

General construction. Start with B and a line bundle L . Then P2 →PB(L2 ⊕ L3 ⊕ O) = P is a P2 bundle over B. A plane cubic can be written inWeierstrass form y2 = x2−g2x−g3. Choose g2 ∈ Γ(B,L 4) and g3 ∈ Γ(B,L 6).Then Z is the solution space for the homogeneous weierstrass equation.

Recall: On one elliptic curve, E, we have a Poincare sheaf P = OE×E(δ −E1−E2). It’s a line bundle. It’s restriction to E×a is the line bundle OE(a−0).This gives an isomorphism E → Pic0(E). This works verbatum for ellipticfibrations.

P ′ is a sheaf on Z ×B Z and is OZ×BZ(∆− Z1 − Z2).We define P = OZ×BZ(∆ − Z1 − Z2) ⊗ π∗(L −1) (where π : Z ×B Z → B

is the composition of projection and structure map)

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