Divisor (algebraic Geometry) - From Cartier Divisors To Line Bundles

From Cartier Divisors To Line Bundles

The notion of transition map associates naturally to every Cartier divisor D a line bundle (strictly, invertible sheaf) commonly denoted by O_X(D).

The line bundle associated to the Cartier divisor D is the sub-bundle of the sheaf M_X of rational fractions described above whose stalk at is given by viewed as a line on the stalk at x of in the stalk at x of . The subsheaf thus described is tautologically locally freely monogenous over the structure sheaf .

The application is a group morphism: the sum of divisors corresponds to the tensor product of line bundles, and isomorphism of bundles corresponds precisely to linear equivalence of Cartier divisors. The group of divisors classes modulo linear equivalence therefore injects into the Picard group. The mapping is not always surjective.