Ample line bundle

In algebraic geometry, a very ample line bundle is one with enough global sections to set up an embedding of its base variety or manifold into projective space. 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.

Introduction

Inverse image of line bundle and hyperplane divisors

Given a morphism , any vector bundle on Y, or more generally any sheaf in modules, e.g. a coherent sheaf, can be pulled back to X, (see Inverse image functor). This construction preserves the condition of being a line bundle, and more generally the rank.

The notions described in this article are related to this construction in the case of morphisms to projective spaces

the line bundle corresponding to the hyperplane divisor, whose sections are the 1-homogeneous regular functions. See Algebraic geometry of projective spaces#Divisors and twisting sheaves.

Sheaves generated by their global sections