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 OX(D).
The line bundle associated to the Cartier divisor D is the sub-bundle of the sheaf MX 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.
Read more about this topic: Divisor (algebraic Geometry)
Famous quotes containing the words line and/or bundles:
“Gascoigne, Ben Jonson, Greville, Raleigh, Donne,
Poets who wrote great poems, one by one,
And spaced by many years, each line an act
Through which few labor, which no men retract.
This passion is the scholar’s heritage,”
—Yvor Winters (1900–1968)
“He bundles every forkful in its place,
And tags and numbers it for future reference,
So he can find and easily dislodge it
In the unloading. Silas does that well.
He takes it out in bunches like birds’ nests.”
—Robert Frost (1874–1963)