divisor

In algebraic geometry, divisors are a generalization of codimension one subvarieties of algebraic varieties; two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil). The concepts agree on non-singular varieties over algebraically closed fields.

Weil divisor

A Weil divisor is a locally finite linear combination (with integral coefficients) of irreducible subvarieties of codimension one. The set of Weil divisors forms an abelian group under addition. In the classical theory, where locally finite is automatic, the group of Weil divisors on a variety of dimension n is therefore the free abelian group on the (irreducible) subvarieties of dimension (n − 1). For example, a divisor on an algebraic curve is a formal sum of its closed points. An effective Weil divisor is then one in which all the coefficients of the formal sum are non-negative.

Cartier divisor

A Cartier divisor can be represented by an open cover U_i, and a collection of rational functions f_i defined on U_i. The functions must be compatible in this sense: on the intersection of two sets in the cover, the quotient of the corresponding rational functions should be regular and invertible. A Cartier divisor is said to be effective if these f_i can be chosen to be regular functions, and in this case the Cartier divisor defines an associated subvariety of codimension 1. More precisely, a Cartier divisor is a section of the sheaf K^*/O^*, where K is the sheaf with K(U) the total quotient ring of O(U).

To every Cartier divisor D there is an associated line bundle (strictly, invertible sheaf) commonly denoted by O_X(D), and the sum of divisors corresponds to the tensor product of line bundles. Isomorphism of bundles corresponds precisely to linear equivalence of Cartier divisors, and so the divisor classes give rise to elements in the Picard group. In otherwords, this defines a group morphism from the group of Cartier divisors modulo linear equivalence to the Picard group. This morphism is injective but is not always surjective.

There is a natural map from Cartier divisors to Weil divisors, which is an isomorphism for integral separated Noetherian schemes provided that all local rings are unique factorization domains. In general Cartier behave better than Weil divisors when the variety has singular points.

Example

An example of a surface on which the two concepts differ is a cone, i.e. a singular quadric. At the (unique) singular point, the vertex of the cone, a single line drawn on the cone is a Weil divisor — but is not a Cartier divisor.

The divisor appellation is part of the history of the subject, going back to the Dedekind-Weber work which in effect showed the relevance of Dedekind domains to the case of algebraic curves.^[1] In that case the free abelian group on the points of the curve is closely related to the fractional ideal theory.

See also

• nef divisor
• ample divisor
• Theta-divisor

Notes

1. ^ Section VI.6 of Dieudonné 1985

References

• Dieudonné, Jean (1985), History of algebraic geometry. An outline of the history and development of algebraic geometry (Translated from the French by Judith D. Sally), Wadsworth Mathematics Series, Belmont, CA: Wadsworth International Group, MR0780183, ISBN 0-534-03723-2 
• Section II.6 of Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York, Heildelberg, MR0463157, ISBN 0-387-90244-9 Springer-Verlag