Divisor (algebraic geometry)
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumford). Both are derived from the notion of divisibility in the integers and algebraic number fields.
Globally, every codimension-1 subvariety of projective space is defined by the vanishing of one homogeneous polynomial; by contrast, a codimension-r subvariety need not be definable by only r equations when r is greater than 1. (That is, not every subvariety of projective space is a complete intersection.) Locally, every codimension-1 subvariety of a smooth variety can be defined by one equation in a neighborhood of each point. Again, the analogous statement fails for higher-codimension subvarieties. As a result of this property, much of algebraic geometry studies an arbitrary variety by analysing its codimension-1 subvarieties and the corresponding line bundles.
On singular varieties, this property can also fail, and so one has to distinguish between codimension-1 subvarieties and varieties which can locally be defined by one equation. The former are Weil divisors while the latter are Cartier divisors.
Topologically, Weil divisors play the role of homology classes, while Cartier divisors represent cohomology classes. On a smooth variety (or more generally a regular scheme), a result analogous to Poincaré duality says that Weil and Cartier divisors are the same.
The name "divisor" goes back to the work of Dedekind and Weber, who showed the relevance of Dedekind domains to the study of algebraic curves.[1] The group of divisors on a curve (the free abelian group generated by all divisors) is closely related to the group of fractional ideals for a Dedekind domain.
An algebraic cycle is a higher codimension generalization of a divisor; by definition, a Weil divisor is a cycle of codimension 1.
Functoriality[edit]
Let φ : X → Y be a morphism of integral locally Noetherian schemes. It is often—but not always—possible to use φ to transfer a divisor D from one scheme to the other. Whether this is possible depends on whether the divisor is a Weil or Cartier divisor, whether the divisor is to be moved from X to Y or vice versa, and what additional properties φ might have.
If Z is a prime Weil divisor on X, then is a closed irreducible subscheme of Y. Depending on φ, it may or may not be a prime Weil divisor. For example, if φ is the blow up of a point in the plane and Z is the exceptional divisor, then its image is not a Weil divisor. Therefore, φ*Z is defined to be if that subscheme is a prime divisor and is defined to be the zero divisor otherwise. Extending this by linearity will, assuming X is quasi-compact, define a homomorphism Div(X) → Div(Y) called the pushforward. (If X is not quasi-compact, then the pushforward may fail to be a locally finite sum.) This is a special case of the pushforward on Chow groups.
If Z is a Cartier divisor, then under mild hypotheses on φ, there is a pullback . Sheaf-theoretically, when there is a pullback map , then this pullback can be used to define pullback of Cartier divisors. In terms of local sections, the pullback of is defined to be . Pullback is always defined if φ is dominant, but it cannot be defined in general. For example, if X = Z and φ is the inclusion of Z into Y, then φ*Z is undefined because the corresponding local sections would be everywhere zero. (The pullback of the corresponding line bundle, however, is defined.)
If φ is flat, then pullback of Weil divisors is defined. In this case, the pullback of Z is φ*Z = φ−1(Z). The flatness of φ ensures that the inverse image of Z continues to have codimension one. This can fail for morphisms which are not flat, for example, for a small contraction.
Global sections of line bundles and linear systems[edit]
A Cartier divisor is effective if its local defining functions fi are regular (not just rational functions). In that case, the Cartier divisor can be identified with a closed subscheme of codimension 1 in X, the subscheme defined locally by fi = 0. A Cartier divisor D is linearly equivalent to an effective divisor if and only if its associated line bundle has a nonzero global section s; then D is linearly equivalent to the zero locus of s.
Let X be a projective variety over a field k. Then multiplying a global section of by a nonzero scalar in k does not change its zero locus. As a result, the projective space of lines in the k-vector space of global sections H0(X, O(D)) can be identified with the set of effective divisors linearly equivalent to D, called the complete linear system of D. A projective linear subspace of this projective space is called a linear system of divisors.
One reason to study the space of global sections of a line bundle is to understand the possible maps from a given variety to projective space. This is essential for the classification of algebraic varieties. Explicitly, a morphism from a variety X to projective space Pn over a field k determines a line bundle L on X, the pullback of the standard line bundle on Pn. Moreover, L comes with n+1 sections whose base locus (the intersection of their zero sets) is empty. Conversely, any line bundle L with n+1 global sections whose common base locus is empty determines a morphism X → Pn.[19] These observations lead to several notions of positivity for Cartier divisors (or line bundles), such as ample divisors and nef divisors.[20]
For a divisor D on a projective variety X over a field k, the k-vector space H0(X, O(D)) has finite dimension. The Riemann–Roch theorem is a fundamental tool for computing the dimension of this vector space when X is a projective curve. Successive generalizations, the Hirzebruch–Riemann–Roch theorem and the Grothendieck–Riemann–Roch theorem, give some information about the dimension of H0(X, O(D)) for a projective variety X of any dimension over a field.
Because the canonical divisor is intrinsically associated to a variety, a key role in the classification of varieties is played by the maps to projective space given by KX and its positive multiples. The Kodaira dimension of X is a key birational invariant, measuring the growth of the vector spaces H0(X, mKX) (meaning H0(X, O(mKX))) as m increases. The Kodaira dimension divides all n-dimensional varieties into n+2 classes, which (very roughly) go from positive curvature to negative curvature.
The Grothendieck–Lefschetz hyperplane theorem[edit]
The Lefschetz hyperplane theorem implies that for a smooth complex projective variety X of dimension at least 4 and a smooth ample divisor Y in X, the restriction Pic(X) → Pic(Y) is an isomorphism. For example, if Y is a smooth complete intersection variety of dimension at least 3 in complex projective space, then the Picard group of Y is isomorphic to Z, generated by the restriction of the line bundle O(1) on projective space.
Grothendieck generalized Lefschetz's theorem in several directions, involving arbitrary base fields, singular varieties, and results on local rings rather than projective varieties. In particular, if R is a complete intersection local ring which is factorial in codimension at most 3 (for example, if the non-regular locus of R has codimension at least 4), then R is a unique factorization domain (and hence every Weil divisor on Spec(R) is Cartier).[21] The dimension bound here is optimal, as shown by the example of the 3-dimensional quadric cone, above.