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algebraic cycle cycle, algebraic codimension algebraic cycle prime cycle cycle, prime effective surface codimension curve effective

Introduction

The main goal of these notes is to describe the geometry of a smooth projective surface over an arbitrary field k. As a first approximation, we have to look at algebraic cycles. To see why, think about the intrinsic structure of a scheme S. Points in S correspond in a natural way to irreducible closed subschemes of S. If you embed S in a projective space and intersect it with a generic linear subspace, you get a closed subscheme of S, all of whose irreducible components have the same codimension. Thus, the study of the geometry of a scheme, even at the most elementary levels, requires you to consider its closed subschemes, and to group them by codimension.

We need to fix some terminology. Let S be a smooth k-scheme of finite type. If s ∈ S is any point, we view its closure {s}^{−} as a reduced, irreducible closed subscheme. Both S and {s}^{−} have some dimension, and we define the codimension of s in S to be

codim_{S}(s) = dim(S) − dim({s}^{−}).

For any nonnegative integer i, write Z^{i}(S) for the free abelian group generated by the points in S of codimension i. An element of Z^{i}(S) is called an algebraic cycle. A generator of Z^{i}(S) (which is already known both as a point and as an irreducible closed subscheme) will also be called a prime cycle. Every algebraic cycle z can be represented as a finite sum

z = ∑_{j=1}^{m} n_{j}s_{j},

where the n_{j} are integers and the s_{j} are prime cycles of codimension i. If all the integers n_{j} are nonnegative, then we say that z is an effective algebraic cycle, and we write z ≻ 0.

Let X be a smooth projective surface over the field k. (By a surface, we mean a geometrically irreducible k-scheme of dimension 2.) On X, we need to consider algebraic cycles of three possible codimensions:

• In codimension zero, the only prime cycle is X itself. So, Z^{0}(X) = Z is a free abelian group of rank one.

• In codimension one, the prime cycles are irreducible curves. (By a curve, we mean a nonzero effective cycle of codimension one on a surface.) An

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divisor prime cycle cycle, prime zero cycle family universal flat

algebraic cycle in codimension one is known classically as a divisor.

• In codimension two, the prime cycles are closed points, and so an algebraic cycle is usually referred to—because of its dimension—as a zero cycle.

The basic problem in understanding the geometry of a surface X is to describe all the effective algebraic cycles on X. Let’s start by looking at some examples.

Example 1.1. Let P^{2} be the projective plane, and let P^{2∗} be its dual (which is another copy of the plane). Taking projective coordinates (X_{0} : X_{1} : X_{2}) in P^{2} and (A_{0} : A_{1} : A_{2}) in P^{2∗}, we can consider the subvariety Γ of P^{2∗} × P^{2} defined by the single bilinear polynomial equation

∑_{i=0}^{2} A_{i}X_{i} = 0.

Projection on the first factor defines a morphism π : Γ → P^{2∗}. Fix a point α = (α_{0} : α_{1} : α_{2}) ∈ P^{2∗}. The fibre of π over α is the line in P^{2} defined by the linear equation

∑_{i=0}^{2} α_{i}X_{i} = 0.

Each line in P^{2} occurs exactly once as a fibre of π. In this way, we can identify the set of lines in P^{2} with the points of the algebraic variety P^{2∗}.

There are several important points to make about this example. First, we can replace 2 by n everywhere to get a description of the family of hyperplanes in projective n-space. Next, we can replace the analytic definition of Γ (using an explicit equation) with a synthetic description:

Γ = {(H, p) ∈ P^{2∗} × P^{2} : p ∈ H}.

Finally, suppose that S → P^{2∗} is any morphism of k-schemes. Taking fibre products

V  ⟶  Γ
↓  ↓
S × P^{2} ⟶ P^{2∗} × P^{2}
↓  ↓
S  ⟶  P^{2∗},

one gets a variety V that can be described as a “family of lines in P^{2} parametrized by the scheme S”. The family Γ → P^{2∗} is universal for flat families of lines, in that every flat family of lines arises as a pullback under a uniquely determined morphism. Because flatness is an essential hypothesis in this statement of universality, one of our first technical steps will be to investigate flat morphisms.

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Example 1.2. We continue to study the projective plane. Every plane curve is defined by a homogeneous form. So, we can assign to each curve a positive integer called its degree, which is just the degree of the homogeneous form. Moreover, two forms define the same curve if and only if one is a nonzero constant multiple of the other. So, we can parametrize the curves of a fixed degree d > 0 by a projective space of dimension

N = \frac{(d + 1)(d + 2)}{2} − 1.

As before, we can give a concrete picture of this parametrization: Take the coordinates in P^{N} to be A_{i,j,k}, where the indices range through nonnegative triples of integers such that i + j + k = d. Then the universal curve of degree d on P^{2} is the closed subscheme Γ of P^{N} × P^{2} defined by the equation

∑_{i+j+k=d} A_{i,j,k}X_{0}^{i}X_{1}^{j}X_{2}^{k} = 0.

A k-point of P^{N} is a morphism α : Spec(k) → P^{N}, which corresponds in more elementary terms to choosing values α_{i,j,k} ∈ k for the coordinates A_{i,j,k}. There is a fibre product diagram

Γ_{α}  ⟶  Γ
↓  ↓
P^{2}  ⟶ P^{N} × P^{2}
↓  ↓
Spec(k) ⟶^{α}  P^{N}.

Once again, we can recover any particular curve of degree d in P^{2} as a fibre of the universal curve.

Example 1.3. Let X = P^{1} × P^{1}. In this case, every curve is defined by a bihomogeneous form, so we can associate to it a bidegree consisting of a pair (d, e) of nonnegative integers (at least one of which is positive). The set of curves with fixed bidegree (d, e) is again parametrized by a projective space, this time of dimension

(d + 1)(e + 1) − 1.

As in the previous case, we can realize this parametrization concretely by writing down a generic bihomogeneous form of bidegree (d, e).

Remark 1.4. In these examples, the collection of all curves on X has at least four interesting properties:

(i) Each curve is part of a continuous family of curves. (ii) There is an infinite discrete set of such families. (iii) Each family is parametrized by a projective space.

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(iv) There is a single formula that computes the dimension of a family.

An obvious question is: which of these properties holds more generally? (A better question might be: what hypotheses are necessary to make each of these properties true more generally? But we’ll save that for later.)

Example 1.5. Let’s consider the case of zero cycles on any smooth projective surface X. We can assign a degree to a zero cycle by counting the number of points in it. To parametrize the zero cycles of fixed degree d > 0, we start with X^{d}, the dfold fibre product of X with itself over k. A point of X^{d} corresponds to an ordered d-tuple of points on X. The symmetric group S_{d} acts on X^{d} by permuting the factors. Taking the quotient under this group action provides a scheme X^{(d)}, which is usually called the d^{th} symmetric product of X. This is a scheme of dimension 2d whose points correspond to unordered d-tuples of points on X; that is, to effective zero cycles made up of d-points.

In this example, we obtain three of the four properties mentioned above: Each zero cycle is a member of one family out of an infinite discrete set of families, with a common formula for the dimension of the parameter space. However, the parametrizing family is not a projective space. After all, when d = 1, the parameter space is X itself.

We give a special name to a family of algebraic cycles that is parametrized by a copy of projective space: we call it a linear family. The most important example of a linear family of divisors is called a linear system. These arise as follows. If f ∈ k(X)^{∗} is a nonzero function on X, then we can associate to f its divisor (f). We say that two divisors D, E ∈ Z^{1}(X) are linearly equivalent if there exists a rational function f ∈ k(X)^{∗} such that

(f) = D − E.

Now if D is any divisor on X, we can consider the k vector space

L(D) = {f ∈ k(X)^{∗} : D + (f) ≻ 0} ∪ {0}.

(This space can be identified with the space of global sections of the invertible sheaf O_{X}(D) associated to D on X.) The projectivization of this vector space is a projective space |D| that parametrizes the effective divisors on X that are linearly equivalent to D. Any linear subspace of |D| is called a linear system; the entire space |D| is called a complete linear system of divisors on X.

Example 1.6. Let E be an elliptic curve, and let X = E × P^{1}. Given a curve C on X, consider the natural projections

C → P^{1},  and  C → E.

We can associate to C a pair of degrees d and e by taking the degrees of these projections. There are two cases.

◯^{c}Coombes 1996 family superabundance linear equivalence equivalence, linear rational equivalence equivalence, rational Chow group Picard group

(i) If d = 0, then C projects to a finite set of points in P^{1}. So, C must consist solely of fibres of this map. Thus, to give such a curve is to pick a divisor of degree e in P^{1}, and such divisors are parametrized by a projective space of dimension e + 1.

(ii) Suppose d > 0. The set of all curves of type (d, e) forms a continuous irreducible family of dimension d(e + 1). However, it is not a linear system. Instead, it is fibred in a natural way by linear subfamilies of dimension d(e + 1) − 1.

In this example, two of our four properties fail to hold. As we have just seen, the space parametrizing a familiy of curves on X need not be a projective space. Moreover, the formula for the dimension of the parameter space in the general case (ii) does not specialize to the formula in case (i); this phenomenon is known as superabundance.

Remark 1.7. For those who prefer geometry to algebra, we can avoid the appeal to homogeneous forms when defining the degrees that appear in our examples so far. In the case of P^{2}, we can realize the degree of a curve as the number of points in which it intersects a generic line. In the case of P^{1} × P^{1}, there are two natural classes of lines, D and E, corresponding to the fibres of the two natural projections onto P^{1}. The bidegree of a curve C is just the pair of integers that counts the number of points of intersection with D and with E. In particular, all the discrete families of curves that we have been able to identify are determined by intersection numbers.

Remark 1.8. The relation of linear equivalence on divisors extends to a relation on all algebraic cycles usually known as rational equivalence. If S is any smooth scheme, we let S^{<i>} denote the set of points of codimension i in S. If s ∈ S^{<i−1>} is any point of codimension i−1 and if f ∈ k(s)^{∗} is a nonzero rational function on the corresponding irreducible closed subscheme, then the divisor of f is an algebraic cycle on S of codimension i. The mapping that takes such a function to its divisor is a group homomorphism, and we define its cokernel to be the i^{th} Chow group. Consequently, we have an exact sequence

∐_{s∈S^{<i−1>}} k(s)^{∗} → Z^{i}(S) → CH^{i}(S) → 0.

In the case of codimension 1, we have CH^{1}(S) = Pic(S), the Picard group. In our last example, we have Pic(E × P^{1}) = E × Z × Z. The bidegree corresponds to the two copies of the integers in the Picard group. Once you fix the bidegree, you can fix a curve C_{0} with that bidegree, and map any other such curve C to the class in the Picard group of C − C_{0}. This gives a map from the family of all such curves to the elliptic curve E, and the claim in (ii) reduces to the statement that each fibre of this map consists of a complete linear system. In one sense, then, the

◯^{c}Coombes 1996 linear system Picard group zero cycle Chow group geometric genus genus, geometric

essential result about families of curves comes from combining a study of linear systems with the fact that some part of the Picard group can be represented by an algebraic variety. This is the main result we will try to understand in the first part of these notes. The treatment presented here is based on Mumford @@ 464 @@.

Remark 1.9. One of the more remarkable facts in algebraic geometry is that this extremely nice property of divisors does not extend to zero cycles. We saw above that it is easy to represent the zero cycles of fixed degree by an algebraic variety. One could ask, therefore, if the Chow group of zero cycles, CH_{0}(X) = CH^{2}(X), decomposes similarly. Mumford @@ 349 @@ has shown that if the geometric genus of a surface is nonzero, then it is not possible to represent the “continuous” part of the Chow group of zero cycles by any algebraic variety. This is one of the few places in algebraic geometry where a naturally arising classification problem does not have an algebraic variety as its solution. A second goal of these notes will be to explain this result of Mumford.

Remark 1.10. The Chow groups are important invariants of individual algebraic surfaces. They not only provide information about the fundamental geometry of a surface, but they also provide subtle arithmetic information about the structure of a surface over an arbitrary field. (For some examples, see @@ 370, 371, 232, 376, 397, 417 @@.) As we try to understand these invariants, we will naturally be lead to consider many other invariants of surfaces. The only sensible way to organize this information is to discuss the Enriques-Kodaira classification of algebraic surfaces over an arbitrary field, which is the third goal of these notes. For a description of the classification over the complex numbers, you can consult @@ @@ or @@ @@. Bombieri and Mumford @@ 402, 373, 374 @@ worked out this classification over algebraically closed fields in any characteristic. In these notes, we intend to carry out the classification over an arbitrary field.

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