Divisor (algebraic geometry)

Locally, every codimension-1 subvariety of a smooth variety can be defined by one equation in a neighborhood of each point.

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 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 group of divisors on a compact Riemann surface X is the free abelian group on the points of X. Equivalently, a divisor on a compact Riemann surface X is a finite linear combination of points of X with integer coefficients.

For any nonzero meromorphic function f on X, one can define the order of vanishing of f at a point p in X, ordp(f).

It is an integer, negative if f has a pole at p. The divisor of a nonzero meromorphic function f on the compact Riemann surface X is defined as which is a finite sum.

If D has positive degree, then the dimension of H0(X, O(mD)) grows linearly in m for m sufficiently large.

For example, a divisor on an algebraic curve over a field is a formal sum of finitely many closed points.

A divisor on Spec Z is a formal sum of prime numbers with integer coefficients and therefore corresponds to a non-zero fractional ideal in Q.

If f is a regular function, then its principal Weil divisor is effective, but in general this is not true.

The additivity of the order of vanishing function implies that Consequently div is a homomorphism, and in particular its image is a subgroup of the group of all Weil divisors.

Assume that X is a normal integral separated scheme of finite type over a field.

On a normal integral Noetherian scheme X, two Weil divisors D, E are linearly equivalent if and only if

For example, one can use this isomorphism to define the canonical divisor KX of X: it is the Weil divisor (up to linear equivalence) corresponding to the line bundle of differential forms of top degree on U. Equivalently, the sheaf

A fractional ideal sheaf J is invertible if, for each x in X, there exists an open neighborhood U of x on which the restriction of J to U is equal to

and conversely, invertible fractional ideal sheaves define Cartier divisors.

Every line bundle L on an integral Noetherian scheme X is the class of some Cartier divisor.

This is well-defined because the only choices involved were of the covering and of the isomorphism, neither of which change the Cartier divisor.

on an integral Noetherian scheme X determines a Weil divisor on X in a natural way, by applying

A Noetherian scheme X is called factorial if all local rings of X are unique factorization domains.

[7] In general, however, a Weil divisor on a normal scheme need not be locally principal; see the examples of quadric cones above.

An effective Cartier divisor on X is an ideal sheaf I which is invertible and such that for every point x in X, the stalk Ix is principal.

The sum of two effective Cartier divisors corresponds to multiplication of ideal sheaves.

For a line bundle L on an integral Noetherian scheme X, let s be a nonzero rational section of L (that is, a section on some nonempty open subset of L), which exists by local triviality of L. Define the Weil divisor (s) on X by analogy with the divisor of a rational function.

For a complex variety X of dimension n, not necessarily smooth or proper over C, there is a natural homomorphism, the cycle map, from the divisor class group to Borel–Moore homology: The latter group is defined using the space X(C) of complex points of X, with its classical (Euclidean) topology.

Conversely, any line bundle L with n+1 global sections whose common base locus is empty determines a morphism X → Pn.

[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.

A (Weil) Q-divisor is a finite formal linear combination of irreducible codimension-1 subvarieties of X with rational coefficients.

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.