Divisors, Linear Systems and Projective Embeddings

18.726: Algebraic Geometry (K.S. Kedlaya, MIT, Spring 2009) Divisors, linear systems, and projective embeddings (updated 1 Apr 09) We conclude the first half of the course by translating into the language of schemes some classical notions related to the concept of a divisor. This will serve to explain (in part) why we will be interested in the cohomology of quasicoherent sheaves. In order to facilitate giving examples, I will mostly restrict to locally noetherian schemes. See Hartshorne II.6 for divisors, and IV.1 for Riemann-Roch. 1 Weil divisors Introduce Hartshorne’s hypothesis (*): let X be a scheme which is noetherian, integral, separated, and regular in codimension 1. The latter means that for each point x ∈ X whose local ring OX,x has Krull dimension 1, that local ring must be regular. Lemma. Let A be a noetherian local ring of dimension 1. Then the following are equivalent. (a) A is regular. (b) A is normal. (c) A is a discrete valuation ring. (This is why normalizing a one-dimensional noetherian ring produces a regular ring.) Warning: for a noetherian integral domain, normal implies regular in codimension 1 but not conversely. You have to add Serre’s condition S2: for a ∈ A, every associated prime of the principal ideal (a) has codimension 1 when a is not a zerodivisor, and has codimension 0 when a = 0. A prime (Weil) divisor on X is a closed integral (irreducible and reduced) subscheme of codimension 1. A formal Z-linear combination of prime divisors is called a Weil divisor. If only nonnegative coefficients are used, we say the divisor is effective. For example, let K(X) be the function field of X, i.e., the local ring of X at its generic point. (This equals Frac(O(U)) for any nonempty open affine subscheme U of X.) For f ∈ K(X) nonzero, we can define a principal divisor associated to f as follows. For each prime divisor Z on X, let ηZ be the generic point of Z. Then OX,ηZ is a discrete valuation ring; let vZ be the valuation. Now define the divisor � (f ) = vZ (f )Z; Z this makes sense because only finitely many vZ (f ) are nonzero. (That’s because f restricts to an invertible regular function on some nonempty open subscheme U of X, and vZ (f ) = 0 whenever Z 6⊆ X − U.) Let Div X be the group of Weil divisors of X. The principal divisors form a subgroup (since (f ) + (g) = (f g)); the quotient by this subgroup is called the divisor class group of 1 X, denoted Cl X. For example, if X = Spec(A) with A a Dedekind domain, then Div X is the group of fractional ideals, and Cl X is the ideal class group. We say two divisors which differ by a principal divisor are linearly equivalent. There are a number of examples in Hartshorne. One of my favorites is that of an elliptic curve; here is a summary. Let k be an algebraically closed field (for starters). Let P (x, y, z) ∈ k[x, y, z] be a homogeneous polynomial of degree 3 defining a nonsingular subvariety C of P2k . Pick a point O ∈ C(k). There is a surjective map Div X → Z mapping each prime divisor P to 1, called the degree. This map factors through Cl X because each principal divisor has degree 0. The kernel of the degree map Cl X → Z is generated by (P ) − (O) for P ∈ C(k). In fact it is equal to the set of such elements: given P, Q ∈ C, we first draw the line through P, Q in P2k and find its third intersection point R with C. We then draw the line through R and O in P2k and find its third intersection point S with C. Then (P ) + (Q) + (R) ∼ (R) + (S) + (O), so (P ) − (O) + (Q) − (O) ∼ (S) − (O). 2 Cartier divisors When the scheme X is not regular, there is a more restrictive notion of divisors that turns out to be more useful in many cases. Let K be the locally constant sheaf associated to the function field K(X). A Cartier divisor on X is a section of the sheaf K(X)/O× . Using the construction of principal divisors, we obtain a map from Cartier divisors to Weil divisors: if the Cartier divisor is represented on some open subset U of X by the rational function f ∈ K(X), then the Weil divisor we get should agree with (f ) when restricted to U (i.e., only keep the components of those prime divisors meeting U). This map is injective if X is normal, because an integrally closed noetherian domain is the intersections of its localizations at minimal prime ideals. Proposition (Hartshorne, Proposition II.6.11). Suppose X is locally factorial (i.e., each local ring OX,x is a unique factorization domain). Then the previous map is an isomorphism. (In particular, this holds if X is regular, because a regular local ring is factorial by a not-so­ easy theorem of commutative algebra.) Example: if X = Spec k[x, y, z]/(xy − z 2 ), the ideal (x, z) defines a Weil divisor which is not a Cartier divisor. Again, there is an obvious notion of a principal Cartier divisor, namely one defined by a single element of K(X). The group of Cartier divisors modulo principal divisors is called the Cartier class group of X, denoted CaCl X. 2 3 The Picard group The Cartier class group is “usually” the same as the Picard group, namely the group of invertible sheaves on X under the tensor product. Namely, if D is a Cartier divisor on X, let L(D) be the subsheaf of K such that L(D)(U) = {f ∈ K(X) : ((f ) + (D))|U ≥ 0}. Assuming that X is normal, this is locally free of rank 1, hence an invertible sheaf. This gives a homomorphism from Cartier divisors to the Picard group, which we see kills the principal divisors. The resulting homomorphism is always injective, even without any hypotheses on X (Hartshorne, Corollary II.6.14) but may not be surjective; however, it is surjective if X is integral (Hartshorne, Proposition II.6.15). Note that if D is effective, then the function 1 defines a global section of L(D). Since L is locally principal, we can locally identify L with OX ; when we do so, the subsheaf of L(D) generated by 1 goes into correspondence with an ideal sheaf of OX , which doesn’t depend on any choices. This ideal sheaf defines D as a closed subscheme. In other words, D is the zero locus of a certain section of L(D). More generally, even if D is effective, we can view D as the zero locus of a meromorphic section of L(D) (meaning a zero locus of L(D) ⊗OX KX ), and indeed the zero locus of any meromorphic section of L(D) is linearly equivalent to D. 4 Linear systems Suppose X is an integral separated scheme of finite type over a field k (which need not be algebraically closed). Let L be an invertible sheaf on X. A linear system defined by L is the set of zero loci of some k-linear subspace H of H 0 (X, L). If we take the entire space, that is called the complete linear system defined by L. We can attempt to use the elements of H to define a map X → Pnk , where n = dimk (H)−1. This might fail to give a morphism because H may have a base point, i.e., a point in the intersection of all of the divisors in the linear system. In fact, we get a morphism X → Pnk if and only if H has no base points. Suppose now that k is algebraically closed, and that X is one-dimensional, projective, irreducible, and nonsingular (i.e., a “curve”). Consider the complete linear system associated to L(D) for some divisor D. (a) We get a map X → Pnk if and only if for each closed point x ∈ X, we have dimk H 0 (X, L(D− x) = dimk H 0 (X, L(D)) − 1. (In other words, there must be a section of L(D) not vanishing at x.) (b) The map in (a) is injective as a map of sets if and only if for each pair of distinct closed points x, y ∈ X, we have dimk H 0 (X, L(D − x − y)) = dimk H 0 (X, L(D)) − 2. (In other words, there must be a section of L(D) vanishing at x but not at y, and vice versa.) 3 (c) The map in (b) is a closed immersion if and only if for each closed point x ∈ X, we have dim H 0 (X, L(D − 2x)) = dimk H 0 (X, L(D)) − 2. (In other words, there must be a section of L(D) not vanishing at x, and a section vanishing to exact order 1 at x.) (Condition (c) is needed to ensure that the tangent space at x embeds into the tangent space at the image of x. See Remark 7.8.2.) Since we would like to know under what circumstances X embeds into a projective space, we would like to be able to compute at least the dimension of H 0 (X, L(D)) for each divisor D. This quest is greatly abetted by the Riemann-Roch theorem, more on which next time. 4