InfoIn algebraic geometry, a very ample line bundle is one with enough global sections to set up an embedding of its base variety or manifold M into projective space.more...An ample line bundle is one such that some positive power is very ample. Globally generated sheaves are those with enough sections to define a morphism to projective space.

Koizumi's theorem states the third power of an ample line bundle is normally generated.The Mumford-Kempf theorem states that the fourth power of an ample line bundle is quadratically presented.…

One of the basic constructions in commutative algebraic geometry is the Proj of a graded commutative ring.This construction builds a projective algebraic variety together with a very ample line bundle whose homogeneous coordinate ring is the original ring.Building the underlying topological space of the variety requires localizing the ring, but building sheaves on that space does not.…

For any projective variety X, Kleiman showed that a divisor is ample if and only if its numerical equivalence class lies in the interior of the nef cone.In particular, every ample line bundle is nef.The cone of curves is defined to be the convex cone of linear combinations of curves with nonnegative real coefficients in the real vector space N 1 (X) of 1-cycles modulo numerical equivalence.…

Koizumi's theorem states the third power of an ample line bundle is normally generated.The Mumford-Kempf theorem states that the fourth power of an ample line bundle is quadratically presented.…

In this definition, one could assume that X is smooth over a field, but the minimal model program has also led to the study of Fano varieties with various types of singularities, such as terminal or klt singularities.The existence of some ample line bundle on X is equivalent to X being a projective variety, so a Fano variety is always projective.…

In a series of papers starting in 2005 he has obtained positivity results for the curvature of holomorphic vector bundles naturally associated to holomorphic fibrations.These vector bundles arise as the zeroth direct images of the adjoint of an ample line bundle over the fibration.The case of a trivial line bundle was considered in earlier work by Phillip Griffiths in connection to variations of Hodge structures and by Fujita, Kawamata and Eckart Viehweg in algebraic geometry.…

To decide in practice when a Cartier divisor D corresponds to an ample line bundle, there are some geometric criteria.

In algebraic geometry, a Seshadri constant is an invariant of an ample line bundle L at a point P on an algebraic variety.It was introduced by Demailly to measure a certain rate of growth, of the tensor powers of L, in terms of the jets of the sections of the L k, The object was the study of the Fujita conjecture.…

In algebraic geometry, given an ample line bundle L on a compact complex manifold X, Matsusaka's big theorem gives an integer m such that the tensor power L^m is very ample.

Projective normality may similarly be translated, by using enough Veronese mappings to reduce it to conditions of linear normality.Looking at the issue from the point of view of a given very ample line bundle giving rise to the projective embedding of V, such a line bundle (invertible sheaf) is said to be normally generated if V as embedded is projectively normal.Projective normality is the first condition N 0 of a sequence of conditions defined by Green and Lazarsfeld.…