Divisor (algebraic geometry)

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). Both are ultimately derived from the notion of divisibility in the integers and algebraic number fields.Cartier divisors and Weil divisors are parallel notions. Weil divisors are codimension one objects, while Cartier divisors are locally described by a single equation. On non-singular varieties, these two are identical, but when the variety has singular points, the two can differ. 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 (since it is not locally principal).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.An algebraic cycle is a higher-dimensional generalization of a divisor; by definition, a Weil divisor is a cycle of codimension one.