|
|
|
|
|
Divisor (Algebraic Geometry)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 Andr Weil) The concepts agree on non-singular varieties over algebraically closed fields. Any Weil divisor is a locally finite linear combination of irreducible subvarieties of codimension one. In the classical theory, where locally finite is automatic, the Weil divisors on a variety of dimension n are therefore the free abelian group on the (irreducible) subvarieties of dimension n − 1. The definition of a Cartier divisor is that locally it is defined by a single equation. To every Cartier divisor D there is an associated line bundle (strictly, invertible sheaf) denoted by LD, and the sum of divisors corresponds to 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.
|
 |
|
| Copyright 2005-2009 OnPedia.com. All Rights Reserved |
|
|