Surfaces

Report on Meeting 1

Theorem 1. (Serre) Let X be a projective scheme over a noetherian ring A, let O(1) be a very ample invertible sheaf on X, and let F be a coherent O_{X}-module. Then there exists an integer n_{0} such that for all integers n ≥ n_{0}, the sheaf F(n) is finitely generated by global sections.

Corollary 2. Let X be a nonsingular projective variety over a field k. Then every divisor D on X is linearly equivalent to a difference of effective very ample divisors.

Proof: Take F = O_{X}(D) to be the invertible sheaf associated to the divisor D, and let H be a fixed effective very ample divisor. For large enough n, the sheaf F(n) = O_{X}(D + nH) is finitely generated by global sections. Taking n even larger if necessary, we know that the global sections define a projective embedding, and we can take an effective very ample divisor E that is linearly equivalent to D + nH. But then D ∼ E − nH gives the desired representation.

Remark 3. By combining this theorem with Bertini’s Theorem, we can write D as a difference of smooth effective divisors. Adding Zariski’s Main Theorem to the mix allows us to write D as difference of smooth irreducible effective divisors, provided dim(X) ≥ 2.

Theorem 4. Let X be an irreducible separtated scheme of finite type over an algebraically closed field k. Then Ω_{X/k} is locally free of rank equal to dim(X) if and only if X is a nonsingular variety.

Remark 5. The proof of this theorem (see [Hartshorne, II.8.15]) only uses the hypothesis that k is algebraically closed as part of a local computation. It asserts that at each closed point x ∈ X, the local ring O_{X,x} contains a copy of its perfect residue field k, and that its field of fractions is a separably generated extension of k.

Definition 6. A morphism f : X → Y between schemes of finite type over a field k is called smooth of relative dimension n if

(i) f is flat; (ii) if X′ ⊂ X and Y′ ⊂ Y are irreducible components such that f(X′) ⊂ Y′, then

dim(X′) = dim(Y′) + n; and (iii) for all x ∈ X, one has dim(Ω_{X/Y} ⊗ k(x)) = n.

Remark 7. If X is integral, then the last condition is equivalent to asserting that Ω_{X/Y} is locally free of rank equal to the dimension. Thus, the previous theorem holds over any field with “nonsingular” replaced by “smooth”.

Remark 8. For varieties over an algebraically closed field, “smooth” and “nonsingular” are synonyms. Over general fields, however, “smooth” is a harder condition to satisfy. For example, purely inseparable field extensions are nonsingular, but not smooth.

Theorem 9. Let X be a nonsingular variety over an algebraically closed field k. let Y ⊂ X be a closed subscheme defined by the ideal sheaf I. Then Y is nonsingular if and only if (i) Ω_{Y/k} is locally free, and (ii) there is an exact sequence

0 → I/I^{2} → Ω_{X/k} ⊗ O_{Y} → Ω_{Y/k} → 0.

Remark 10. The proof of this theorem (see [Hartshorne, II.8.17]) only uses algebraically closed to apply the previous theorem. Thus, it remains true over any field by replacing “nonsingular” with “smooth”.

Proposition 11. Let Y be a smooth subvariety of codimension r in a smooth variety X over a field k. Then ω_{Y} ≈ ω_{X} ⊗ ∧^{r}N_{Y/X}. In particular, if Y is a smooth divisor, then ω_{Y} ≈ ω_{X} ⊗ O_{X}(Y) ⊗ O_{Y}.

Theorem 12. (Serre Duality) Let X be a smooth projective scheme of dimension n over a field k. Then ω_{X} is a dualizing sheaf on X. Moreover, for every coherent sheaf F, the natural functorial pairing

Ext^{i}(F, ω_{X}) ⊗ H^{n−i}(X, F) → H^{n}(X, ω_{X})⟶^{tr}k

is a perfect pairing.

Corollary 13. Let E be a locally free sheaf on a smooth projective scheme X over a field k. Then there are natural isomorphisms

H^{i}(X, E) ≈ H^{n−i}(X, E^{v} ⊗ ω_{X}).

Definition 14. Let X be a smooth projective variety over a field k, and let F be a coherent sheaf on X.

(i) Define its Euler characteristic to be

χ(F) = ∑(−1)^{i} dim_{k}(H^{i}(X, F)).

(ii) Define its arithmetic genus to be

p_{a}(X) = (−1)^{dim(X)}(χ(O_{X}) − 1).

(iii) Define its geometric genus to be

p_{g}(X) = dim_{k}(H^{0}(X, ω_{X})).

(iv) Define the Hodge numbers of X to be

h^{p,q} = dim_{k}(H^{q}(X, Ω_{X}^{p})).

(v) Let K be a canonical divisor on X. Define the n^{th} plurigenera to be P_{n} = dim_{k}(H^{0}(X, O_{X}(nK))).

Remark 15. There is some redundancy in these definitions. For example, if dim(X) = n, then one has Ω_{X}^{n} = ω_{X} = O_{X}(K). Thus, h^{n,0} = p_{g} = P_{1}.

Remark 16. By Serre duality, one always has h^{p,q} = h^{n−p,n−q}.

Theorem 17. Let X be a smooth projective variety over a field k. Let X̄ be the corresponding variety obtained by base extension to the algebraic closure k̄ of k. Let π : X̄ → X be the natural map. If F is a coherent sheaf on X, then dim_{k}(H^{i}(X, F)) = dimk̄(H^{i}(X̄, f^{∗}(F))).

Proof: We can choose a finite affine cover on which to compute the derived functor cohomology in terms of a Cech complex. Use the fact that f^{∗} is exact to get the result.

Remark 18. Since the structure sheaf and the sheaf of differentials behave properly with respect to base extension, this theorem implies that all the invariants defined above only depend on the variety X̄ over the algebraic closure.

Remark 19. If X is a smooth projective curve, then the definitions together with Serre duality yields

p_{a}(X) = p_{g}(X) = dim_{k}(H^{1}(X, O_{X})).

In this case, we write g for the (common) genus.

Theorem 20. (Riemann-Roch for Curves) Let X be a smooth projective curve of genus g over a field k. For every divisor D on X, one has

l(D) − l(K − D) = deg(D) + 1 − g.

Remark 21. If you follow the proof in [Hartshorne, IV.1.3], you see that Serre Duality shows that the left hand side is just the Euler characteristic of O_{X}(D), so it only depends on the linear equivalence class. Also, the result is true when D = 0. Now you need to see that both sides change properly when you replace D by D +P for any closed point P. The left hand side changes by

dim_{k}(k(P)) = [k(P) : k],

where k(P), being the residue field of the point, is a finite field extension of k. So, we get the right-hand side to behave correctly by taking deg(P) = [k(P) : k].