Big Line Bundles
A line bundle is big if it is of maximal Iitaka dimension, that is, if its Iitaka dimension is equal to the dimension of the underlying variety. Bigness is a birational invariant: If f : Y → X is a birational morphism of varieties, and if L is a big line bundle on X, then f*L is a big line bundle on Y.
All ample line bundles are big.
Big line bundles need not determine birational isomorphisms of X with its image. For example, if C is a curve of genus 2, then the canonical bundle KC is big, but it determines a two-to-one covering, a morphism C to its image as canonical curve, which here is a rational normal curve.
Read more about this topic: Iitaka Dimension
Famous quotes containing the words big, line and/or bundles:
“The man who promised to reinforce American families is now eager to pull the plug on Big Bird and Barney.”
—Leslie Harris, U.S. political activist. As quoted in Newsweek magazine, p. 23 (December 19, 1994)
“There’s a line between love and fascination that’s hard to see on an evening such as this.”
—Ned Washington (1901–1976)
“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)