Surfaces

Report on Meeting 2

Proposition 1. (Genus Formula) Let C be a smooth curve of genus g on a smooth projective surface X over a field k. Let K be the canonical divisor on X. Then

2g − 2 = C.(C + K).

Theorem 2. (Riemann-Roch for Surfaces) Let D be a divisor on a smooth surface X over a field k. Then one has

l(D) − s(D) + l(K − D) = \frac{1}{2}D.(D − K) + 1 + p_{a}(X).

Remark 3. The proof requires us to write D as a difference of smooth curves, write down some exact sequences, and then apply the Genus Formula to the curves. As a consequence, however, we recover the following more general version of the Genus Formula: Let C be any irreducible curve on X. Then

p_{a}(C) = h^{1}(O_{C}) = 1 + \frac{1}{2}C.(C + K).

Remark 4. The Riemann-Roch Theorem for surfaces actually includes another statement. To understand it, we need to talk about algebraic cycles.

Definition 5. Let X be a nonsingular, noetherian, integral, separated scheme. Write X^{(r)} for the set of points of X of codimensions r. (Note: a point x ∈ X has codimension r if and only if its closure {x}¯ is an irreducible closed subscheme of codimension r.) We define the group Z^{r}(X) of algebraic cycles of codimension r on X to be the free abelian group generated by X^{(r)}. If f is a nonzero rational function defined on the closure of a point of codimension r − 1, then its divisor can naturally be viewed as an algebraic cycle of codimension r on X. We say that such cycles are rationally equivalent to zero, and we take the r^{th} Chow group on X to be the quotient of Z^{r}(X) by the subgroup generated by all cycles that are rationally equivalent to zero. In other words, we have an exact sequence

∐_{x∈X^{(r−1)}} k(x)^{∗} → ∐_{x∈X^{(r)}} Z → CH^{r}(X) → 0.

Remark 6. When r = 1, we have CH^{1}(X) = P ic(X). So, the Chow groups generalize the notion of the Picard group to arbitrary codimension. When r = 0, we have CH^{0}(X) = Z, generated by the class of the (irreducible) scheme X itself.

Remark 7. When r = dim(X), the group CH^{r}(X) consists of zero cycles. On a projective scheme, there is a natural degree map d : CH^{r}(X) → Z; we denote the kernel of this map by A_{0}(X). Next, there exists an abelian variety Alb_{X}, called the Albanese variety of X, and a morphism α : X → Alb_{X} that is universal for morphisms of X into abelian varieties. Choosing a base point P (which can be assumed to satisfy α(P) = 0), one can use α to define a homomorphism A_{0}(X) → Alb_{X}. When the base field is algebraically closed, this map is surjective (by universality); its kernel is called the Mumford group and denoted by M_{0}(X). Mumford has shown that if X is a surface and p_{g} > 0, then M_{0}(X) cannot be represented as the points of any finite dimensional algebraic variety.

Remark 8. Suppose Y (resp., Z) is an irreducible closed subscheme of codimension r (resp., s) on X. Assume further that Y and Z are in “general position”. Then the irreducible components of the intersecion Y ∩ Z all have codimension equal to the sum r + s. (As a special case, two curves on a surface intersect infinitely many points.) This suggests that we can define an intersection pairing CH^{r}(X) × CH^{s}(X) → CH^{r+s}(X).

Theorem 9. There exists a unique intersection pairing on the class of nonsingular varieties over a field k satisfying the following properties:

(CH1) The pairing makes CH(X) = ⨁_{r=0}^{dim(X)} CH^{r}(X) into a graded ring with identity.

(CH2) For any morphism f : X → Y, there is a natural ring homomorphism f^{∗} : CH(Y) → CH(X). (Thus, CH is a contravariant functor from the category of nonsingular varieties to the category of rings.)

(CH3) For every proper morphism f : X → Y, there is a natural morphism f_{∗} : CH(X) → CH(Y) of graded groups (that shifts degrees). (Thus, CH can also be considered as a covariant functor from the category of varieties with proper morphisms to the category of graded groups.)

(CH4) If f : X → Y is proper, then for all x ∈ CH(X) and all y ∈ CH(Y), one has the

projection formula

f_{∗}(x.f_{∗}y) = f_{∗}(X).y.

(CH5) If Y and Z are cycles on X and if Δ : X → X × X is the diagonal morphism, then

Y.Z = Δ^{∗}(Y × Z).

(CH6) If Y and Z are cycles on X that meet properly (in that the codimension of the components of their intersection is equal to the sum of their codimensions), then there are local intersection multiplicities (that depend only on a neighborhood of the generic point of each component) such that

Y.Z = ∑ i(Y, Z; W_{i})W_{i}.

(CH7) If Y is a subvariety of X and if Z is an effective Cartier divisor on X meeting Y properly, then Y.Z is the cycle associated to the effective Cartier divisor Y ∩ Z on Y.

Remark 10. The maps f_{∗} and f^{∗} associated with a morphism f : X → Y are defined as follows. First, let Z be an irreducible closed subscheme of X. If dim(f(Z)) < dim(Z), we set f_{∗}(Z) = 0. Otherwise, we define

f_{∗}(Z) = [k(Z) : k(f(Z))]\overline{f(Z)},

which makes sense in this case because the extension of function fields is finite.

Next, let Z be an irreducible closed subscheme of Y. Let Γ_{f} denote the graph of f, considered as an algebraic cycle on the product X ×Y. Let Z̃ = p_{Y}^{−1}(Z) = X ×Z ⊂ X ×Y also be viewed as a cycle on the product. Now define

f^{∗}(Z) = (p_{X})_{∗}(Γ_{f}.Z̃).

Remark 11. The two key ingredients of the proof are (i) a correct definition of the local intersection multiplicities and (ii) Chow’s Moving Lemma, which says that given two cycles Y and Z, you can find a cycle Z′ rationally equivalent to Z that meets Y properly.