Divisor (algebraic geometry)
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In algebraic geometry, divisors are a generalization of 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.
A Weil divisor is a locally finite linear combination 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 points. An effective Weil divisor is then one in which all the coefficients of the formal sum are non-negative.
A Cartier divisor consists of an open cover Ui, and a collection of rational functions fi defined on Ui. 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 fi can be chosen to be regular functions, and in this case the Cartier divisor defines an associated subvariety of codimension 1.
To every Cartier divisor D there is an associated line bundle (strictly, invertible sheaf) denoted by L[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 the Picard group. Following the general conceptual clue that sheaves reveal the 'correct' geometry, Cartier divisors, introduced in the 1950s where Weil divisors are classical, are more appropriate to deal with singular points.
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. In that case the free abelian group on the points of the curve is closely related to the fractional ideal theory.
