Zero-cycles on K3 surfaces

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MSc Mathematics

Master Thesis

Zero-cycles on K3 surfaces

Author: Supervisor:

Ties Laarakker

dr. M. Shen

Examination date:

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Abstract

The Chow ring of an algebraic K3 surface S over C is an enormous object. It is infinite dimensional in some sense, and therefore cannot be parametrised by a variety. This follows from a theorem by Mumford, which we will discuss in the second chapter of this thesis. Whereas the Picard group of a K3 surface is well-understood, less is known about the group of 0-cycles. It is possible, however, to find some interesting structure. We will use the distinguished class cS,

introduced by Beauville and Voisin in [1], to describe a filtration of CH0(C) due to O’Grady

([10]). Finally, we will point out the difficulties that arise when one tries to generalise our results to S[2], the Hilbert scheme of pairs of points on S.

Title: Zero-cycles on K3 surfaces

Author: Ties Laarakker, ties.laarakker@student.uva.nl, 5682002 Supervisor: dr. M. Shen

Second Examiner: prof. dr. L.D.J. Taelman Examination date: Februari 17, 2016

Korteweg-de Vries Institute for Mathematics University of Amsterdam

Science Park 105-107, 1098 XG Amsterdam http://kdvi.uva.nl

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Introduction

The Chow ring of an algebraic variety is the formal realisation of the notion of intersection of subvarieties of the variety. A good theory of intersection became available after Chow proved his Moving Lemma, which states that any two subvarieties can be ‘moved’ in some sense in such a way that they intersect nicely. The Chow ring of a variety consists of formal sums of its subvarieties, the so-called algebraic cycles, modulo a relation called rational equivalence, together with a multiplicative structure, given by the intersection product. Rational equivalence can be considered as a degree of ‘freedom’ in which we allow ourselves to move subvarieties, in the sense of Chow’s Moving Lemma.

In the case of a smooth variety over the complex numbers, the Chow ring is closely related to the singular cohomology groups, together with topological intersection, or the cup product. This relation is realised by the cycle class map, which associates to an algebraic cycles its fundamental class in the appropriate cohomology group. The Chow ring, however, is often very large and hard to calculate. This holds in particular for the Chow ring of a complex algebraic K3 surface, which will be the subject of study of this thesis.

Definition 0.1. A surface S, i.e. a complete smooth variety of dimension 2 over some field k, is called a K3 surface if H1(S, OS) = 0 and ωS = Ω2S ∼= OS.

Most of the time we will assume K3 surfaces to be defined over the complex numbers. Algebraic cycles of dimension 1 on a K3 surface are well-understood. They form the part of the Chow ring graded by 1, which is just the classical Picard group. So the difficulty lies in the computation of the group of zero-cycles, the algebraic cycles of dimension 0. By a result of Mumford, proved in his paper [9] from 1968, it is infinite dimensional in some sense, and therefore to big to be parametrized by a variety.

Nevertheless, it is possible to find some interesting structures on this Chow group. In their paper [1], Beauville and Voisin constructed a canonical class of 0-cycles in the Chow ring of a K3 surface. This class has some remarkable properties. For example it is easily shown that intersection of any two divisors lies in the 1-dimensional space generated by this class. This completely describes intersection on a K3 surface. Another remarkable application of the canonical class that was pointed out in the paper, is the decomposition of the so called ‘small diagonal’.

Yet another feature of the distinguished class, which was pointed out by a later paper of Voisin ([12]), is the observation that the canonical class controls the ‘moving parts’ of 0-cycles. It turns out that a 0-cycle can be uniquely decomposed into a ‘fixed part’ and a number of copies of the distinguished class. This phenomenon will be the main topic of study of this thesis.

We will start our discussion with a short review of intersection theory. The canonical reference for the subject is the book Intersection Theory by Fulton ([5]). However, for an introduction, I refer to Appendix A in Hartshorne’s Algebraic Geometry ([6]), which we will follow closely in our exposition. We will soon turn to the complex case, in which we will emphasise the relation

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between the Chow ring and singular cohomology. Here our main reference will be the books by Voisin on Hodge theory and complex algebraic geometry ([13] and [14]). After that, we will mention some results from Hodge theory. Although this is a large and beautiful subject, we will restrict our efforts to stating the results that will be needed later in the text. We will finish the chapter by mentioning the Albanese variety, which we will use to show that the group CH0(S)

is torsion free. The first chapter contains hardly any proofs, and merely serves as an overview of preliminary results.

Chapter 2 will be devoted to Mumford’s theorem. The proof leans heavily on the Bloch-Srinivas decomposition, a powerful result that is interesting in its own right. It was proved by Bloch and Srinivas in their paper from 1983 ([2]), and concerns the decomposition of the algebraic cycle given by the diagonal of the product of a variety with itself in the corresponding Chow ring. We will construct the decomposition by means of an important technique in the theory of algebraic cycles called the spreading principle. We follow the treatment of Voisin given in [14].

After a short discussion of the Picard group of a (complex algebraic) K3 surface, we will turn our attention to 0-cycles. A theorem due to Bogomolov and Mumford, which states that any ample linear system on a complex K3 surface contains a rational curve, serves as the foundation of the theory developed in the third chapter. The theorem is used for the construction of the canonical class introduced by Beauville and Voisin, and for the proof of the properties of this class mentioned above. Along the way, we will mention the conjectural density of rational curves on a K3 surface, and their relation with the so-called constant cycle class curves. The latter will be used to describe the decomposition of 0-cycles, and a filtration due to O’Grady.

In the last chapter of this thesis, we will look at possible generalisations of the theory in a higher dimensional case. We will look at the 0-cycles on the Hilbert space of pairs of points on a K3 surface, and try to give a description in terms of a class that should serve as an analogue of the canonical class on the K3 surface. However, as will be discussed, it turns out that we will soon run into some uniqueness problems, which obstruct nice results as we have in the case of a K3 surface.

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1 The Chow ring and Hodge decomposition

The main object of study of this thesis is the Chow group CH0(S) of a K3 surface S over the

complex numbers. In the first section, we will discuss Chow groups in a more general setting. Our goal is to introduce the notion of the Chow ring, and to give an overview of some of the basic results from intersection theory that we will need in the proof of Mumford’s theorem, discussed in Chapter 2. Another technique that will be used in the proof is the Hodge-decomposition which will be introduced in Section 1.2. We will give hardly any proofs but merely references for the important result. The main references will be the books Hartshorne ([6]) and Voisin ([13], [14]).

1.1 The Chow ring

We will start with a discussion of the notion of rational equivalence. Geometrically, it could be interpreted as a degree of freedom in which we allow ourselves to ‘move around’ algebraic cycles in a variety.

1.1.1 Rational equivalence

Let X be a variety over a field k.

Definition 1.1. Z = Z(X) denotes the free abelian group generated by integral subvarieties. The elements of Z are called (algebraic) cycles. The group Z is graded by the codimension of its generators. Thus an element of Zr _{is a formal sum Z =} _{P c}

iZi, with ci integers and Zi

integral subvarieties of codimension r. We also write Zr:= Zdim X−r for the group of cycles of

dimension r. We refer to an element of Zr as an r-cycle.

Note that the cycles of codimension 1 are simply divisors. We will introduce an equivalence relation on the cycles of codimension r that generalises the notion of linear equivalence. First we show how we can push forward cycles.

Definition 1.2. Let f : X → Y be a morphism of varieties over k. Define f∗ : Z(X) → Z(Y )

by setting for an integral subvariety Z of X,

f∗(Z) = [K(Z) : K(f (Z))] · f (Z)

if dim f (Z) = dim Z and f∗(Z) = 0 otherwise.

Note that the pushforward homomorphism has degree dim(Y ) − dim(X) with respect to the grading by codimension.

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For a closed subvariety W ⊂ X, let j : fW → X be its normalisation. A divisor D on fW gives an element j∗D ∈ Z. Let Ztriv be the subgroup of Z generated by all such elements j∗D, with

D = (f ) a principal divisor in Div(fW ), where W runs over all closed subvarieties of X.

Definition 1.3. We call cycles Z and Z0 on X rationally equivalent, if they differ by an element of Ztriv. We write Z ∼rat Z0 in this case. The group of cycles modulo rational equivalence is

denoted by CH(X) = Z(X)/Ztriv(X). The r-graded component CHr(X) = Zr(X)/Ztrivr (X)

called the rth Chow group of X. We also use the subscript notation CHr(X) := CHdim X−r(X)

For later use we introduce the following terminology.

Definition 1.4. A cycle Z =P ciZi is called effective if ci ≥ 0 for all i. A class in CH(X) is

called effective if it contains an effective cycle. We say that a class α in CH(X) has support in a subvariety i : Y ,→ X if α ∈ i∗CH(Y ).

1.1.2 The degree map

Let X be a complete variety. We can define a degree map on the group of cycles of dimension 0: deg : Z0→ Z X ciZi7→ X ci.

We denote the kernel of this map by Z0,hom. For a complete curve C, a principal divisor on C

has degree 0. Hence it follows from the definitions that we have the inclusion Z0,triv ⊂ Z0,hom.

Hence the degree map induces a degree map on the Chow group of dimension-0 cycles: deg : CH0(X) → Z.

Again, its kernel is denoted by CH0(X)hom. As an illustration of the techniques just introduced,

let us prove the first result on the structure of 0-cycles on a K3 surface over an algebraically closed field. We state it in a much more general setting.

Proposition 1.5. Let X be a complete variety over k = ¯k. Then the group CH0(X)hom is

divisible.

Proof. It is clearly generated by elements x − y for x, y ∈ X. Let C be a curve on X with x, y ∈ C. Let f : eC → X be its normalisation, and let x0, y0 ∈ eC map to x and y respectively. Then CH0( eC)hom is just the Jacobian of eC, and hence divisible as it is an abelian variety over

the algebraically closed k. Hence, for any integer N , it contains an element (x0− y0)/N (only defined up to N -torsion). But this element is mapped to an element (x − y)/N by f∗.

1.1.3 The intersection product

The next step is to introduce a graded ring structure on CH(X). The multiplication will be called the intersection product, and will be denoted by (Z, Z0) 7→ Z.Z0. We follow the approach taken in [6]. The idea is to define the intersection product simultaneously for a class of ‘nice’ varieties, and show that this notion of intersection satisfies, and is in fact uniquely determined by a list of properties. In particular it is well-behaved with respect to pushforward and the following notion of pullback of cycles.

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Definition 1.6. Let f : X → Y be a morphism of varieties, and assume that we have chosen an intersection product on CH(X × Y ). Then the pullback of f is the homomorphism of graded groups

f∗ : CH(Y ) → CH(X) given for a subvariety Z ⊂ Y by

f∗(Z) = prX∗(Γf.prY−1(Z))

in which Γf is the graph of f in X × Y .

We are now ready to formulate the main theorem of this section. It is taken directly from [6], to which we refer for a discussion of the proof.

Theorem 1.7. Let B be the class of non-singular quasi-projective varieties over a fixed field k. There exist products CH(X) × CH(X) → CH(X), (Z, Z0) 7→ Z.Z0 for each of the members X of B, such that the following axioms are satisfied:

A1 The product makes CH(X) into a commutative graded ring for every X ∈ B.

A2 For f a morphism of varieties in B, the map f∗ is a homomorphism of graded rings, and the assignment f 7→ f∗ is functorial.

A3 For f a proper morphism of varieties in B, the map f∗ is a morphims of graded groups,

and the assignment f 7→ f∗ is functorial.

A4 For f : X → Y a proper morphism of varieties in B, x ∈ CH(X) and y ∈ CH(Y ), we have f∗(x.f∗y) = f∗(x).y. This equation will be referred to as the projection formula.

A5 We have the identity Z.Z0 = (∆X)∗(Z ×Z0) for Z, Z0cycles on X and ∆X : X → X ×X

the diagonal morphism.

A6 The intersection product can be defined locally for properly intersecting subvarieties (see [6]).

A7 For a subvariety Y and an effective Cartier divisor Z, meeting Y properly, we have Y.Z = Y ∩ Z.

Moreover, the products are uniquely determined by the axioms A1-A7. Let X be a variety in the class B of the theorem.

Definition 1.8. The graded ring CH(X) is called the Chow ring of X. The product on CH(X) is called the intersection product on X.

We have the following important proposition.

Proposition 1.9. Let i : Y ,→ X be a non-singular closed subvariety, and let j : U ,→ X be its open complement. Then we have an exact sequence

CH(Y )−i→ CH(X)∗ j

∗

−→ CH(U ) → 0.

1.1.4 Correspondences

The following terminology will prove to be useful, especially in the context of Hodge decompo-sition.

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Definition 1.10. A correspondence between smooth varieties X and Y is a class Γ ∈ CH(X × Y ).

If X is complete, then the projection X × Y → Y is proper, and we can define group homomor-phisms

Γ∗ : CH(X) → CH(Y )

Z 7→ prY ∗(Γ.pr∗X(Z))

Similarly, for Y complete, we define

Γ∗ : CH(Y ) → CH(X)

Z 7→ prX∗(Γ.pr∗Y(Z))

Note that if Γ is a cycle of codimension r, Γ∗ is an homomorphism of graded groups of degree

r − dim(X). Similarly, Γ∗ is an homomorphism of graded groups of degree r − dim(Y ).

1.2 Cohomology classes and the Hodge decomposition

Here we take a different point of view. In the rest of this section, let X be a smooth variety over the complex numbers. The space of closed points of X equipped with the Euclidean topology has the structure of a complex manifold, and is denoted by Xan. The variety X is proper if and

only if the manifold Xan is compact. If X is projective, so is Xan and a fortiori it is K¨ahler. We

will state some basic results from Hodge theory that we will need later. By abuse of notation, we will usually omit the subscript an if no confusion can arise.

1.2.1 The cycle class map

An algebraic cycle defines a cohomology class. Roughly, the construction goes like this. Assume X is complete and of dimension d. Let Z be a subvariety of X of codimension r, and let ˜Z → X be its desingularisation. A class in H_DR2d−2r(X) = H2d−2r(X, R) can be represented by a closed (2d − 2r)-form ω on Xan. We can pull back ω to ˜Z, where it is a closed form of top degree

since Z has real dimension 2d − 2r. In particular, it can be integrated over ˜Z. We obtain a map

Zr(X) → H2d−2r(X, R)∨∼= H2r(X, R),

in which the isomorphism is given by Poincar´e duality. As it turns out, we can work in fact with algebraic cycles modulo rational equivalence, and with coefficients in Z rather than in R, i.e. we have a homomorphism

cl : CHr(X) → H2r(X, Z)

for every r. For a precise definition, see [13]. For an algebraic cycle Z we will denote its class cl(Z) by [Z].

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In the previous section, we defined the intersection product, pullback and pushforward of alge-braic cycles. We have similar structures on the singular cohomology groups. Namely the cup product

^: Hk(X, Z) × Hl(X, Z) → Hk+l(X, Z), and for a morphims f : X → Y the pullback homomorphism

f∗: Hk(Y, Z) → Hk(X, Z) and the Gysin morphism

f∗: Hk(X, Z) → Hk+2r(Y, Z)

in which r = dim Y − dim X. For the definitions, let us just remark that the Gysin morphism is given as the Poincar´e dual of the pushforward homomorphism of singular homology (recall that we assume X to be smooth). We have the following compatabilities. For a proof, we refer again to [13].

Proposition 1.11. As above, let X be complete. Let Z, Z0 ∈ CH(X). We have [Z.Z0] = [Z] ^ [Z0].

Let f : X → Y be a morphism of smooth complete varieties over C. Then cl ◦ f∗ = f∗◦ cl

and

cl ◦ f∗ = f∗◦ cl.

Now consider an element γ ∈ H∗(X × Y, Z), for smooth complete X and Y . Exactly as in the case of correspondences, we can define γ∗ and γ∗:

γ∗ : H∗(Y, Z) → H∗(X, Z)

α 7→ prX∗(γ ^ prY∗α)

and

γ∗: H∗(X, Z) → H∗(Y, Z)

α 7→ prY ∗(γ ^ pr∗Xα)

By the proposition, this is compatible with the morphisms given by correspondences, i.e. for Z ∈ CH(X × Y ), we have

cl ◦ Z∗ = [Z]∗◦ cl and

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1.2.2 Hodge decomposition

Finally, we give a sketch of Hodge decomposition. As usual, we refer to [13] for the details. From now on, we assume X to be smooth projective. The cohomology groups Hk_{(X, Z) come naturally}

with a so-called Hodge structure. This structure is given by the Hodge decomposition Hk(X, Z) ⊗ C = Hk(X, C) = M

i+j=k

Hi,j(X),

in which we have Hi,j _{= H}j,i _{(where the bar denotes complex conjungation). The groups}

Hi,j(X) are isomorphic to Hj(X, Ωi_X) for i, j ≥ 0, and trivial otherwise. The cup product, pullback a pushforward map preserve these decompositions. A more precise statement is given in the following proposition.

Proposition 1.12. Let f : X → Y be a morphism of smooth projective varieties over the complex numbers, and let r = dim(Y ) − dim(X). Then the complexifications of the cup product, pullback and pushforward respectively restrict to homomorphisms

^: Hi,j(X) × Hk,l(X) → Hi+k,j+l(X), f∗ : Hi,j(Y ) → Hi,j(X) and f∗ : Hi,j(X) → Hi+r,j+r(Y ).

In the light of the Hodge decomposition, we take another look at the cycle class map. We have the following result.

Proposition 1.13. The image of the composition

CHk(X)−→ Hcl 2k_{(X, Z) → H}2k_{(X, C)} is contained in the component Hk,k(X) ⊂ H2k(X, C).

At this point it is natural to mention the famous Hodge conjecture. It hypothesises that the integral classes of type (k, k) coincide with classes of algebraic cycles. It turns out that for such statement to be logically possible, we need to work modulo torsion.

Conjecture 1.14 (Hodge conjecture). For X smooth, projective over C, the Q-vector space H2k(X, Q) ∩ Hk,k(X) is spanned by classes of algebraic cycles [Z].

The following example will be used later in the text. Moreover, it gives another illustration of the interaction between algebraic cycles and the Hodge decomposition.

Example 1.15. Let f : X → Y be a morphism of smooth projective varieties over C. Let Γ ⊂ X × Y be the graph of f . Thus the codimension of Γ equals the dimension dY of Y . We

calculate the bi-degree of the morphism Γ∗, or more precisely of [Γ]∗⊗C : H∗(Y, C) → H∗(X, C). By definition, it is given by the composition

H∗(Y, C) pr

∗ Y

−−→ H∗_{(X × Y, C)}−−−→ H^[Γ] ∗_{(X × Y, C)} prX∗

−−−→ H∗_{(X, C).}

Let α be a class in Hi,j(Y ). By Proposition 1.13, [Γ] is a homological class of type (dY, dY).

Proposition 1.12 tells us that α is mapped successively to Hi,j(X × Y ), Hi+dY,j+dY_{(X × Y ) and}

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1.3 The Albanese variety

We have now introduced all language we will need in order to prove Mumford’s theorem. Before we do so, let us still mention the Albanese variety. It will be used to show that the group of 0-cycles on a K3 surface is torsion free. We saw before that it is a divisible group, and therefore we can conclude that it has in fact the structure of a vector space over Q. In order to be able to define the Albanese variety, we still need to work in the analytic setting.

Let X denote a smooth projective variety over C (or more generally, a K¨ahler manifold). Con-sider its Albanese variety

Alb(X) := H0(X, ΩX)∨/H1(X, Z),

in which H1(X, Z) is considered as a subgroup of H0(X, ΩX)∨ via the map α 7→

R

α. The choice

of a point x0 ∈ X gives rise to a morphism

X → Alb(X) x 7→ Z x x0 in whichRx x0 is defined to be R

γ for γ a path from x0 to x. For different paths γ, γ

0 _{from x} 0 to x, we have Z γ0 − Z γ = Z γ0_∗γ− ∈ H₁_{(X, Z) ⊂ H}0_{(X, Ω} X)∨.

We conclude that the assignment x 7→Rx

x0 : X → Alb(X) is well-defined. It turns out that the

map induces a homomorphism

CH0(X)hom→ Alb(X).

Note that this map does not depend on the choice of x0anymore. The following famous theorem

is due to Roitman.

Theorem 1.16 (Roitman). The morphism

CH0(X)hom→ Alb(X)

identifies torsion parts.

Now let S be a K3 surface over C. In Theorem 1.5 we saw that the group of 0-cycles on S is divisible. On the other hand, S has no global differential forms, i.e. H0(S, ΩS) = 0. See for

example [7], Chapter 1, Section 3. Hence, Alb(S) is trivial, and Roitman’s Theorem implies that CH0(S)hom is torsion free. We conclude that CH0(S)hom has the structure of a Q-vector

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2 Mumford’s theorem on 0-cycles

We will discuss Mumford’s theorem on the group CH0(S) of 0-cycles on a K3 surface S. Roughly,

the theorem states that it is ‘too big’ to be parametrised by a variety. We will describe a notion of dimension of the group. Then the theorem tells us that this dimension has to be infinite. A precise version of Mumford’s theorem is the statement given in Theorem 2.1 below. This is the theorem that will be proven in this chapter, following Voisin’s treatment in [14].

2.1 Mumford’s theorem

We constructed the Chow group CH0(S) by considering all 0-cycles, and dividing out a relation

called rational equivalence. Let’s say we want to find a geometric structure on this group. First note that the discrete invariant given by the degree map ‘should’ split the hypothetical space in infinitely many components of constant degree. So we restrict our attention to the component CH0(S)hom of 0-cycles of degree 0. An element of this group can be written as the class of an

algebraic cycle α1 − α2, with α1 and α2 effective 0-cycles of a certain degree n. An effective

0-cycle of degree n on the surface S is just an unordered n-tuple of points of S. Hence the set of such cycles is naturally identified with the closed points of the variety S(n). We obtain the natural maps (of sets):

σn: S(n)× S(n)→ CH0(S)hom

(α1, α2) 7→ α1− α2

Let’s say that we want to determine the dimension of the ‘space’ CH0(S)hom. We will compare

it with the dimension of the spaces S(n)× S(n) _{by means of the maps σ}

n. The images of these

maps, considered as subspaces of CH0(S)hom should have dimension equal to the dimension of

S(n)× S(n) _{minus the dimension of a general fibre of the map. But this number is determined}

by the maps σn, merely as maps of sets. In fact, we have the following theorem:

Theorem 2.1. Let S be a K3 surface over C. The maps σn have countable general fibre.

By this we mean that there is a subset U ⊂ S(n)×S(n)_{that is a countable intersection of (Zariski)}

opens, and that has the property that for any x ∈ U the set σ−1(σ(x)) is countable.

A general fibre of the maps σn is thus 0-dimensional. So the image of σn should have the same

dimension 4n as the space S(n)× S(n)_{. In particular the dimensions of the images of the maps}

σn are not bounded. We conclude that for any ‘reasonable’ geometric structure on CH0(S)hom,

the resulting space would have to be infinite dimensional. In [14] this philosophy is used to prove the following corollary:

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2.2 Bloch-Srinivas decomposition

An essential ingredient of the proof of Mumford’s theorem is a decomposition due to Bloch and Srinivas. In order to formulate the result, we need the notion of a universal domain. A universal domain is an algebraically closed field that is big enough, in some sense. Avoiding the precise definition, we will just state the properties that we need.

Fact 2.3. A universal domain is an uncountable algebraically closed field k. In particular, it has infinite transcendence degree over any countable subfield k0.

The example that we keep in mind is just the field of complex numbers C. In this section we will prove the following theorem.

Theorem 2.4 (Bloch, Srinivas). Let X, Y be smooth, projective, irreducible varieties, over a universal domain k. Let Z ⊂ X × Y be a cycle of codimension equal to the dimension n of Y . Assume that there is a subvariety Y0 ,→ Y such that for closed x ∈ X the Zx vanish in

CH0(Y \Y0). Then there exists a positive integer m and a decomposition

mZ = Z0+ Z00

in CHn(X × Y ), in which Z0 has support in X × Y0 and Z00 in X0× Y for some proper algebraic

subset X0 ⊂ X.

2.2.1 The generic case

Let us first clarify the role of the universal domain. Let X and k be as in Theorem 2.4. As X is of finite type over k, we find that the function field K(X) is finitely generated as a field extension of k. Hence its algebraic closure K(X) has a finite transcendence degree over k. Now let k0 be countable subfield of k. Then k and K(X) have the same transcendence degree over

k0. Moreover, as both are algebraically closed, they are isomorphic as field extensions of k0. In

the proof of the theorem, this principle is used to relate the fibre Zx over a closed point x ∈ X

to the generic fibre Zη. This is made precise in the following lemma.

Let ¯η be a geometric generic point of X, i.e. a morphism Spec K(X) → X that factors through the generic point η of X.

Lemma 2.5. Under the hypothesis of Theorem 2.4, we have Zη¯= 0 in CH0((Y \Y0)η¯).

The proof is based on an argument given by Vial in [11], Lemma 2.1.

Proof. First note that we can find a countable subfield k0 of k, over which we can define X, Y ,

Y0 and Z. More precisely, we find varieties X0, Y0, Y00 and Z0 over k0, such that X, Y , Y0 and

Z are obtained by base change with Spec k → Spec k0. To see this, start with the prime field

of k. Then we obtain k0 by adding the coefficients of the equations defining the (projective)

varieties.

Consider the set

U := \

V ⊂X0

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in which the intersection is taken over the proper closed subsets of X0. As k0 is countable, the

variety X0has only countably many subvarieties over k0. The preimages of these do not exhaust

(the closed points of) X, as k is uncountable. Hence U is non-empty. We call the elements of U very general.

Let x = Spec k ∈ X be a very general closed point. As x does not lie over any proper closed subset of X0, it lies over the generic point ηX0 = Spec K(X0) of X0. Also the geometric generic

point ¯η of X lies over ηX0. As K(X0) is countable, we can find, by the discussion above, an

isomorphism ¯η → x over ηX0. See the following diagram, in which triangle × fails to commute.

¯ η $$ ## ∼ = ## × x //_η_X 0 X //_X₀ Spec k //Spec k0

But now we have

Yx = ((Y0)k)x= ((Y0)η_X0)x

and

Yη¯= ((Y0)k)η¯ = ((Y0)η_X0)η¯.

Hence the isomorphism ¯η → x identifies Yη¯ and Yx, not as varieties over k, but as abstract

schemes! We can apply this trick to Y , Y0 and Z simultaneously. We conclude that via the isomorphism Yη¯ → Yx the subspaces Yη¯0 and Yx0 coincide. Similarly, the classes Zx and Zη¯

are identified. By definition, the Chow group of a variety only depends on the structure of the variety as an abstract scheme. Hence, by the hypothesis, we conclude that Zη¯ = 0 in

CH0((Y \Y0)η¯).

2.2.2 Spread

Before we complete the proof of the Bloch-Srinivas decomposition, let us say a few words about the so-called spreading principle. Let B be a variety with a geometric generic point ¯η → B, so ¯

η = Spec K(B). Now let Xη¯ be a variety over K(B). We want to ‘spread out’ Xη¯ over B, that

is, we want to find scheme ˜B, finite over some open of B and a family X_B˜ → ˜B that restricts

to X¯η over ¯η. (Note that a geometric generic fibre factors through any such ˜B.)

We may assume B = Spec R to be affine. For simplicity, let Xη¯ be projective. Xη¯ can be

defined by homogeneous polynomials fi, with coefficients in Frac(R). Consider the minimum

polynomials of these coefficients over Frac(R). Let q be a common denominator of all the coefficients of these minimum polynomials. Now the minimum polynomials all lie in the ring R[1/q][x]. This means that the coefficients of the fi satisfy integral equations over R0 := R[1/q].

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a projective family X_B˜ → ˜B = Spec S, that restricts to Xη¯ over the point ¯η, and ˜B is indeed

finite over the open B0 := Spec R0 of B.

The spreading principle works in fact much more generally. It can be applied to subvarieties of X¯η, functions on Xη¯, etc. We use the spreading principle to complete the proof of the

theorem.

Proof of Theorem 2.4. By the lemma, and the localisation exact sequence (Proposition 1.9), we find a cycle Z_η0_¯ on Y_¯_η0 such that Zη¯− Zη0¯ vanishes in CH0(Yη¯). This last condition means, by

definition, that we find jη¯ : Wη¯ → Yη¯, (the normalisation of) a reduced curve on Yη¯, and a

function f ∈ K(Wη¯) such that Zη¯− Zη0¯= (j¯η)∗(f ). By the spreading principle, we can find an

open U ⊂ X, a finite cover s : ˜U → U , a curve j : W → Y_U˜, that restricts to j¯η over ¯η and a

cycle Z0 in Y_˜0

U restricting to Z 0 ¯

η. By the same principle, we may assume that f is an element of

K(W ). We have

Z_U˜ − Z0 = j∗(f ),

which is trivial in the Chow ring. Since s∗Z_U˜ = mZ|U, where m is the degree of ˜U → U , we

find

mZ = s∗Z0+ Z00

in which Z00 has support on the complement of U .

2.3 Proof of Mumford’s theorem

A corollary of Theorem 2.4 is the following. We work over C and use the notation hypotheses from Theorem 2.4.

Theorem 2.6. Let k > dim Y0. Then, for ω ∈ H0_{(Y, Ω}k

Y) we have [Z]∗ω = 0.

Proof. By Theorem 2.4, we find m[Z]∗ω = [Z0]∗ω+[Z00]∗ω. Consider i : ˜X0 → X and j : ˜Y0→ Y , the normalisations of X0and Y0respectively. As Z0and Z00are supported in respectively X ×Y0 and X0× Y , we can find cycles ˜Z0∈ CH(X × ˜Y0) and ˜Z00∈ CH( ˜X0× Y ) with Z0= (idX× j)∗Z˜0

and Z0 = (i × idY)∗Z˜00. By the projection formulas for the intersection product, we find

[Z0]∗ = [ ˜Z0]∗◦ j∗ and

[Z00]∗ = i∗◦ [ ˜Z00]∗.

For dimension reasons, we have j∗ω = 0. On the other hand, as the maps i∗ and [Z]∗ are

morphisms of Hodge structures of bidegree s := codim Y0 and 0 respectively. Hence i∗[ ˜Z00]∗ω =

m[Z]∗ω ∈ Hk,0(X), and this forces [ ˜Z00]∗ω ∈ Hk−s,−s(X), which is zero. We conclude that [Z]∗ω = 0.

Proof of theorem 2.1. Assume that the map σn has uncountable general fibre. By a Hilbert

scheme argument, spelled out in [14], the fibres of σn are countable unions of closed algebraic

subsets. In particular, the general fibre has a positive dimensional component. Similarly, consider the first projection S2n× S2n _{→ S}2n_{. We find an algebraic set R ⊂ S}2n_{× S}2n _that

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for all (s, t) ∈ R. In particular dim R > 4n. We will show that this is in contradiction with H2,0_{(S) 6= 0.}

In fact, let Zi be the 2-cocycle of S4n× S, given by πi(s) = t for (s, t) ∈ S4n× S, and πi the

ith projection S4n → S. Now consider the correspondence

Z := Z1+ . . . + Zn− Zn+1− . . . − Z2n− Z2n+1− . . . − Z3n+ Z3n+1+ . . . + Z4n.

We have Z(r,r0₎= Z_∗((r, r0)) = σ_n(r) − σ_n(r0) = 0 ∈ CH₀(S) for (r, r0) ∈ R ⊂ S2n× S2n. Thus

we can apply Theorem 2.6 (in which we take for Y0 the empty variety) and we find Z_|R∗ η = 0 for any ω ∈ Hk,0_{(S), k > 0. In particular let ω be a non-zero 2-form. One can show that}

Z∗ω is in fact a holomorphic form, non-degenerate at the generic point of R. As it vanishes at R, we conclude that we have dim R ≤ dim S4n/2 = 4n, in contradiction with the above. This completes the proof of the theorem.

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3 The Chow ring of a K3 surface

In this chapter we will consider a K3 surface over the complex numbers. We will give a dis-cription of its Chow ring. While the Picard group of a K3 surface is well-understood, not much is known about 2nd Chow group. We will study the 0-cycles by means of orbits, which generalise the notion of a complete linear system. An important tool is the distinguished class cS ∈ CH0(S), introduced by Beauville and Voisin in [1]. We will start this chapter with a short

discussion of the Picard group.

3.1 1-cycles on a K3 surface

Let S denote a K3 surface over C. Write O := OSan. We have the canonical sequence of sheaves

on San:

0 → Z → O → O∗ → 0

By definition of a K3 surface H1_{(S, O) = 0, and hence we have the exact sequence}

0 →H1(S, O∗) → H2(S, Z) → H2(S, O). (3.1) Since H2(S, Z) is torsion free (see for example [7]), we have the inclusion

H2(S, Z) ,→ H2(S, C) = H2(S, O) ⊕ H1(S, ΩS) ⊕ H0(S, ωS),

in which the decomposition is the Hodge decomposition discussed in Chapter 1. For α ∈ H2(S, Z), write α = α0,2+ α1,1+ α2,0 with respect to this decomposition. It is known that the last map in (3.1) is given by α 7→ α0,2. Let α ∈ H2(S, Z) with α0,2 = 0. As an element of H2_{(S, Z), α is invariant under complex conjugation, and we find that also α}2,0 _{= 0, since we}

have H2,0(S) = H0,2_{(S) in the Hodge decomposition. Hence, we have the following series of}

equalities. CH1(S) ∼= Pic(S) ∼ = Im(H1(S, O∗) → H2(S, Z)) = Ker(H2(S, Z) → H2(S, O)) = H2(S, Z) ∩ H1(S, ΩS).

This is known as the Lefschetz theorem on (1,1)-classes.

3.2 0-cycles on a K3 surface

For the 2nd Chow group, the picture is a bit more complicated. However, as we will see in this section, it is possible to find some interesting structures on the group of 0-cycles.

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3.2.1 The distinguished class

First recall that we have a degree map CH0(S) → Z, the kernel of which is denoted by

CH0(S)hom. So we have the short exact sequence

0 → CH0(S)hom→ CH0(S) → Z → 0 (3.2)

A splitting of this sequence corresponds to the choice of a 0-cycle class of degree 1. In particular, every point of S gives rise to a splitting. It is a natural question whether we can find a canonical splitting. This turns out to be the case ([1]). It will be given by a distinguished class cS ∈ CH0(S). For its construction, we use a result of Bogomolov and Mumford. The theorem

is basically folklore, but a proof is lined out in [7].

Theorem 3.1 (Bogomolov, Mumford). Let H be an ample divisor on a K3 surface S over the complex numbers. Then the linear system |H| contains a connected curve C with rational irreducible components. In particular S contains a rational curve.

Corollary 3.2. Choose a rational curve R on S. Choose a closed point x ∈ R. The class of x in CH0(S) is independent of the choices.

Proof. Let R be a rational curve on S. Let ˜R ∼= P1 be its normalisation. Since the degree map gives an isomorphism DivCl( ˜R) → Z, we see that any two points x, y ∈ ˜R have the same class in DivCl( ˜R). By definition of rational equivalence, this means that any two 0-cycles x, y ∈ R have the same class in S.

Now let R0 be another rational curve on S. Let C be given as in Theorem 3.1. As C is ample, both R ∩ C and R0∩ C are non-empty. Since C is connected, we can find integral components Ci for i = 1, . . . , n such that R ∩ C1, R0∩ Cn and Ci∩ Ci+1for i = 1, . . . , n − 1 are non-empty.

Now let x be a point on R. By the above we can ‘move’ x along R, without changing its class. Hence x is rationally equivalent to point x1 in C1. Since the Ci are rational curves, we can

repeat this argument, and we conclude that x is rationally equivalent to a point x0 in R0. But R is rational, so x is in fact rationally equivalent to any point on R0. This gives the result. Definition 3.3. We call the class from the corollary the distinguished class, and denote it by cS.

Lemma 3.4. Let D1, D2 be divisors on S. Then D1.D2 = deg(D1.D2)cS

Proof. If D2 = H is ample, we may assume by Theorem 3.1 that the irreducible components

of H are rational curves. Then the intersection is a sum of classes of points on rational curves. For general D2, use the fact that any divisor can be written as the difference of two ample

divisors.

Lemma 3.5. The class cS is supported on any curve on S.

Proof. Let C ∈ |H| be as in the theorem. Let D be any curve. As H is ample, it has non-trivial intersection with D. Now any point in the intersection C ∩ D has class cS in CH0(S).

For a (reduced) curve C, we write g(C) for its genus, i.e. g(C) := dim H1( ˜C, O_C˜), in which ˜C

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Corollary 3.6. Let C be a reduced curve on S, not necessarily irreducible. Let α be a cycle supported on C with deg(α) ≥ g(C). Then α is effective.

Proof. If C is irreducible, this follows from Riemann-Roch for curves. If C has components C1, . . . , Cn, we have g(C) = dim H1( ˜C, O_C˜) = dim H1(a ˜Ci, O`_˜ Ci) = X dim H1( ˜Ci, O_C˜_i) = X g(Ci)

where ˜C and ˜Ci denote the normalisations of C and Ci. We can write α =P α0i, in which each

α_i0 is supported on Ci. As the class cS is supported on each curve on S, for any i the cycle

αi := αi0 + (g(Ci) − deg(α0i))cS is supported on Ci. By the preliminary case, these cycles are

effective. By construction we have

α =Xαi+ (deg(α) − g(C))cS.

Now deg(α) ≥ g(C) implies that α is effective.

3.2.2 Constant cycle class curves

Theorem 3.1 asserts the existence of a rational curve on a K3 surface. It is known ([3]) that for a general K3 surface S over the complex numbers, the rational curves lie dense in S. In particular, S contains infinitely many rational curves. A famous conjecture states that this holds for all K3 surfaces.

Conjecture 3.7. A K3 surface over an algebraically closed field contains infinitely many ra-tional curves.

As the density of rational curves is only conjectural, we will work with the more flexible notion of a constant cycle class curve, which has properties similar to the ones of rational curves. We will show that the union of the constant cycle class curves on a K3 surface is a dense subset. Definition 3.8. A curve f : C ,→ S on S is called a constant cycle class curve if f∗CH0(C) ⊂

CH0(S) equals Z · cS.

For a thorough treatment of constant cycle class curves, see [8]. It follows from Corollary 3.2 that a rational curve on S is a constant cycles class curve. Moreover, we have the following lemma. As usual, S denotes a K3 surface over the complex numbers.

Lemma 3.9. Let f : C ,→ S be a constant cycle class curve. Then f∗([x]) = cS for any closed

x ∈ C.

Proof. This follows directly from Lemma 3.5.

As we will see, a K3 surface contains infinitely many constant cycle class curves. For the proof we need the following incarnation of the Bogomolov-Mumford Theorem. The statement is taken directly from [7].

Theorem 3.10 (Bogomolov, Mumford). A K3 surface S over C is dominated by a family of smooth elliptic curves. More precisely, there exist a smooth surface E, a surjective morphism π : E → S and a fibration p : E → B such that the generic fibre Eb is smooth elliptic and the

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Proposition 3.11. The union of all constant cycle class curves of S is a Zariski dense subset. Proof. Let π : E → S be as in the theorem. We can find a curve C on E that maps to a rational curve in S, and is finite over B. By base change along C → B, we replace E by a family of elliptic curves p : E → C. It comes with a section e : C → E. Now E is a commutative group scheme over C with unit e. Let CN be the kernel of multiplication by N . It is a subscheme of

E, finite over C, and its underlying variety is mapped to a (possibly reducible) curve in S. We will show that the images of the curves CN are constant cycle class curves.

A closed point x on CN is an N -torsion point on the elliptic curve Et, with t = p(x) ∈ C. We

find N · (x − et) = 0 in CH0(Et). Therefore the difference of the cycles π∗(x) and π∗(et) in

CH0(S) is torsion, and hence they coincide, as CH0(S) is torsion free. As π(e(C)) is rational,

we have π∗(x) = π∗(et) = cS. We conclude that π(CN) is a constant cycle class curve. As

elliptic curves have infinitely many torsion points, the curves CN lie dense in E, and hence their

images lie dense in S.

3.2.3 A decomposition due to O’Grady and Voisin

Let α ∈ CH0(S). Let C be a reduced curve on S such that α is supported on C. By Lemma

3.5, the canonical class cS is supported on C. By Corollary 3.6, the cycle α − l · cS is effective

for l 0. We conclude that we can write

α = α0+ lα· cS

with α0 effective and lα ∈ Z. We can choose lα to be maximal, so the decomposition becomes

unique. In the rest of this section, we will give geometric interpretations of the cycle α0, its degree d and the number lα.

Proposition 3.12.

d = min

α⊂Cg(C)

where the minimum is taken over curves C on S that support α, and g(C) denotes the genus of C, as usual.

Proof. Let C be a curve such that α has support on C. Let g = g(C) denote its genus. By Lemma 3.5, the cycle α + (g − deg(α))cS has support on C. Corollary 3.6 then asserts that it

is effective. By definition of lα, we have deg(α) − g ≤ lα. We conclude that

d = deg(α) − lα ≤ g.

On the other hand, by Theorem 3.10, α0 is supported on d curves of genus ≤ 1. Their union is a curve of genus at most d, supporting α0. Again by Lemma 3.5, it also supports α. This completes the proof.

3.2.4 Orbits of effective 0-cycles

Let X be be a non-singular quasi-projective variety over a field k. Generalising the notion of the complete linear systems, we can define the orbit of a cycle.

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Definition 3.13. Let α be a cycle on X. The orbit Oα of α is given by

Oα := {α0 ∈ Z(X) : α0 ≥ 0, α0 ∼ratα}.

Thus Oα consists of the effective cycles in the class of α in CH(X).

A priori, the orbit of a cycle is merely a set. For a cycle of codimension 1, the orbit is simply the complete linear system of the divisor, and hence has the structure of projective space. At the other extreme, we have the following result for 0-cycles.

Proposition 3.14. Let α be a 0-cycle on X of degree d ≥ 0. Then Oα is a countable union of

closed subvarieties of X(d), the dth symmetric power of X.

Proof. As effective cycles of degree d are just unordered d-tuples of points of X, we can identify them with the closed points of the variety X(d). Now an orbit is just a fibre of the map X(d) → CH₀(X). By a Hilbert scheme argument like the one discussed in the second chapter, these fibres are countable unions of closed subvarieties.

As a result we can define the dimension of the orbit of a 0-cycle.

Notation 3.15. For α a cocycle on X the dimension of its orbit Oα is given by

dim(Oα) := max Y ⊂Oα

dim(Y )

in which the maximum is taken over the irreducible components of algebraic subsets of Oα.

The following theorem shows that in the decomposition α = α0+ lα· cS, we can view α0 as the

fixed part of α. The integer lαthen measures the degree of freedom in which the cycle α ‘moves’

up to rational equivalence.

Theorem 3.16. Let α ∈ CH0(S) be effective. In the decomposition α = α0 + lα· cS, we have

dim(Oα0) = 0 and l_α= dim(O_α).

Proof. Let d = deg(α0). Assume that dim(Oα0) > 0. Let Y ⊂ Sd be an irreducible algebraic

subset of the inverse image of O_α0 in Sd of positive dimension. The image of Y along one of the projections Sd→ S is positive dimensional. By symmetry, we may assume that this holds for the first projection p : Sd → S. Since Y is proper, its image p(Y ) in S is either a curve or the surface S. In either case, p(Y ) contains a closed point c representing the class cX. A

closed point of Y mapping to c is an element α00+ c ∈ X(d), with α00 ∈ X(d−1)_{, that maps to}

α0 in the Chow group CH0(S). But this contradicts the maximality of lα. We conclude that

dim(Oα0) = 0. The equality l_α= dim(O_α) now follows from the following lemma.

Lemma 3.17. For α ∈ CH0(S) effective, we have dim(Oα+cS) = dim(Oα) + 1.

Proof. As the inverse image of Oα+cS in S

d _{contains the product of the inverse images of O} α

and OcS, we have dim(Oα+cS) ≥ dim(Oα) + dim(OcS) = dim(Oα) + 1. For the other inequality,

let Y be a irreducible subvariety of the inverse image of Oα+cS in S

d _{of maximal dimension.}

Assume that e := dim(Y ) ≥ dim(Oα) + 2. We will deduce a contradiction. Again, at least one

of the projections of Y to S is positive dimensional. As above, we may assume that p(Y ) ⊂ S contains a curve C, for p : Sd→ S the first projection. We distinguish two cases:

i) p(Y) = C. Let c ∈ C represent the distinguished class cS. Then the fibre of Y over c is a

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S contained in the class to α + cS, and for which the coefficient of the point c is positive. Hence

the family {β − c} lies in Oα and has dimension e − 1 > dim(Oα), which is a contradiction.

ii) p(Y) = S. Let C ⊂ S be a constant cycle class curve. Then the YC = p−1(C) ∩ Y is a

hypersurface in Y . Write q : Sd → Sd−1 _{for the projection to the components other than}

the first. YC parametrises effective cycles in the class α + cS. As YC maps to the class cS

along the first projection, the variety q(YC) parametrises effective cycles in the class α. Hence,

dim(q(YC)) ≤ dim(Oα) < e − 1 = dim(YC). As the constant cycle class curves lie dense in S by

Proposition 3.11, we conclude that dim(q(Y )) < dim(Y ). But a fibre of q|Y, are mapped to a

positive dimensional subvariety of S by p, the point of which have a constant class in CH0(S),

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4 More on 0-cycles

In this chapter we study the space S[2], the Hilbert scheme of pairs of points on a K3 surface S, and we will try to find results analogous to the ones in the previous chapter. The text will be less rigorous than before. For example, we will not explicitly construct the several maps given in the first section.

4.1 The Hilbert scheme of pairs of points

Let S be a K3 surface over C as usual. Consider the following Hilbert functor. T 7→ {Subschemes Z → ST, flat and finite of rank 2 over T }

It is represented by a smooth projective variety S[2]. In this chapter, we study 0-cycles on this variety. In particular, we will try to find an analogue of the decomposition of the previous chapter.

The (closed) points of S[2] are easily described. A finite rank-2 subscheme of S is either the reduced subscheme given by two distinct points, or it is given by a skyscraper R sheaf at a point x ∈ S, that is a quotient of OS. Consider the latter case. Let m = (f, g) ⊂ OS,x be the

maximal ideal at x. Now R = OS,x/I for some ideal I ⊂ OS,x. As R has rank 2, we must have

m2⊂ I. Hence, R is a quotient of O_S,x/m2, which has rank 3. The ideals of OS,x/m2 of rank 1

are given by the non-zero elements of m/m2. Any two of such elements define the same ideal if they differ by a scalar. Hence we have

Proposition 4.1. The rank-2 subschemes of S with support in a closed point x are in 1 − 1 correspondence with the closed points of P(mx/m2x) ∼= P1.

It turns out ([4], Chapter 7) that the map that sends a closed subscheme of rank 2 of S to its support, in fact defines a morphism of varieties.

c : S[2] → S(2)_.

The morphism c is called the Hilbert-Chow morphism. It can be shown that c (in the rank-2 case) is in fact the blow-up S(2) _{along the image of the diagonal S → S × S. In particular, we}

have the following commuting diagram:

^ S × S p // f S[2] S × S //S(2)

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where ^S × S is the blow-up of S × S along the diagonal. Let pr : S × S → S be the projection onto the second factor. We have a homomorphism of groups

CH0(S) → CH2(S[2])

given by p∗f∗pr∗. By [5], Corollary 6.7.2, it is given by y 7→ Sy, in which Sy is the class in

CH2(S[2]) of the strict transform of the subvariety y + S ⊂ S(2). For x, y ∈ S distinct, the intersection Sx.Sy is the class of the point of S[2] that is mapped to x + y ∈ S(2) by the

Hilbert-Chow morphism. It is denoted by [x, y]. For a point x ∈ S, we saw that the fibre of c over x + x ∈ S(2) has the structure of the projective line. Hence, any point of the fibre has the same class in the Chow group CH0(S[2]). We will denote this class by [x, x]. By a limit argument it

can be shown that Sx.Sx = [x, x]. The homomorphism y 7→ Sy, together with the intersection

product gives the bilinear map

i : CH0(S) × CH0(S) → CH0(S[2])

(x, y) 7→ [x, y]

With a bit more work, we can also construct a ‘diagonal’ homomorphism.

d: CH0(S) → CH0(S[2])

x 7→ [x, x]

We denote the classes i(cS, y) by [c, y] and the class d(cS) by [c, c]. Equivalently, we could have

given these classes by choosing c to be any point representing the class cS, but this is only

well-defined by the above.

Finally, we remark that the following rule also gives a well-defined homomorphism of Chow groups:

a: CH0(S[2]) → CH0(S)

[x, y] 7→ x + y

4.2 The structure of effective cycles

As before, we can consider orbits of effective cycles on S[2]. They are realised as fibres of the maps

(S[2])(n)→ CH0(S[2])

and hence are countable unions of algebraic varieties (cf. Proposition 3.14). Hence, as above, we can define the dimension of an orbit as the maximal dimension of an algebraic component. As an example, consider the simplest case, in which α = [x, y] is effective of degree 1. Let Oα ⊂ S[2] be its obit, and let Y ⊂ Oα be an algebraic component.

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Case 1. Assume Y is a curve. Now there are two possibilities. If c(Y ) ⊂ S(2) is again a curve, it will contain an element x + c, in which c has class cS in CH0(S). This implies that α can

be written as [x, c]. Otherwise, if c(Y ) consists of a single point, we can apply the following lemma.

Lemma 4.2. Let x ∈ S. Then the class [x, x] is equal to a class [c, y], with c representing cS

in CH0(S).

Proof. Choose an elliptic curve E on S containing x. Find a double cover E → P1, ramified at x. E → S × P1 is a finite subscheme of rank 2. After possibly restricting E → P1 to its smooth locus, it corresponds to an a priori only rational map φ : P1 → S[2]_{, which can be extended to}

a morphism by smoothness of P1. By definition of rational equivalence, the points on the curve φ have the same class in CH0(S[2]). On the other hand, they correspond to the fibres of the

covering E → P1. Let c ∈ E represent cS. Let {c, y} the fibre of E → P1 containing c (possibly

c = y). It follows that [x, x] = [c, y] in CH0(S[2]).

Case 2. Now assume Y is a variety of dimension 2. Again, there are two possibilities. If c(Y ) has dimension 2, it will contain a point representing cS + cS ∈ CH0(S). In this case,

[x, y] = [c, c]. Alternatively, Y is contained in the exceptional divisor. But then it contains a positive dimensional family of points [x, x], parametrised by a curve in S. But any curve in S contains a point c representing cS. We conclude that [x, y] = [c, c].

To summarise, we have the following possibilities: dim(Oα) description of α

0 α is fixed

≥ 1 α is of the form [x, c] ≥ 2 α is of the form [c, c]

By similar techniques, we get the following table for a degree-2 cycle α. dim(Oα) description of α

0 α is fixed

≥ 1 α is of the form [x, y] + [z, c]

≥ 2 α is of the form [x, c] + [y, c] or [x, y] + [c, c] ≥ 3 α is of the form [x, c] + [c, c]

≥ 4 α is of the form [c, c] + [c, c]

4.2.1 Uniqueness?

In the case of an effective cycle of degree 2, we see that this description of the Chow group is not as clean as in the case of a K3 surface. For α with dim(Oα) = 2, we have two possible

descriptions of α. The following proposition describes their relation.

Proposition 4.3. Assume that we have a relation [x, y] + [c, c] = [x0, c] + [y0, c] in CH0(S[2]).

Then [x, y] = [x0, y0].

Proof. Applying the homomorphism a to the relation [x, y] + [c, c] = [x0, c] + [y0, c], we obtain x + y = x0+ y0 in CH0(S). The homomorphism CH0(S) → CH2(S[2]) now gives

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Taking self-intersection on both sides, we obtain

[x, x] + 2[x, y] + [y, y] = [x0, x0] + 2[x0, y0] + [y, y]. (4.1) On the other hand, applying the homomorphism d to the equation x + y = x0+ y0, we get the relation

[x, x] + [y, y] = [x0, x0] = [y0, y0] (4.2) in CH0(S[2]). Subtracting 4.2 from 4.1, we obtain

2[x, y] = 2[x0, y0].

But S[2] has trivial Albanese variety, so as in the case of a K3 surface, Roitman’s theorem (Theorem 1.16) implies that CH0(S[2]) is torsion free. The result follows.

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Popular summary

Algebraic geometry is the study of geometrical objects by means of algebra. One instance of this is the so-called Chow ring. For a surface, i.e. a two dimensional space, the Chow ring consists of all the points and curves that can be drawn on this surface, together with some additional algebraic structure. This extra structure is given by intersection. For example, the intersection of two curves consists, by definition, of the points where the curves meet. Chow rings are in general hard to calculate.

In this thesis, we study the Chow ring of a so called K3 surface. The first result that is discussed is a theorem by David Mumford, which tells us that it is an enormous object. In contrast, results of Arnaud Beauville and Claire Voisin include that any two curves on a K3 surface intersect within a certain distinguished class of points. This allows us to give a more precise description of the Chow ring of a K3 surface. In the final chapter of this thesis we turn to a 4-dimensional case, which turns out to be significantly more complicated.

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Bibliography

[1] Arnaud Beauville and Claire Voisin. On the Chow ring of a K3 surface. J. Algebraic Geom., 13(3):417–426, 2004.

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Zero-cycles on K3 surfaces

MSc Mathematics

Master Thesis