of that is not zero at x. In short, basepoint-free line bundles are exactly those that can be expressed as the pullback of O(1) by some morphism to projective space. Nef, globally generated, semi-ample The degree of a line bundle L on a proper curve C over k is defined as the degree of the divisor (s) of any nonzero rational section s of L. The coefficients of this divisor are positive at points where s vanishes and negative where s has a pole. Therefore, any line bundle L on a curve C such that has nonnegative degree (because sections of L over C, as opposed to rational sections, have no poles). In particular, every basepoint-free line bundle on a curve has nonnegative degree. As a result, a basepoint-free line bundle L on any proper scheme X over a field is nef, meaning that L has nonnegative degree on every (irreducible) curve in X. More generally, a sheaf F of -modules on a scheme X is said to be globally generated if there is a set I of global sections such that the corresponding morphism of sheaves is surjective. A line bundle is globally generated if and only if it is basepoint-free. For example, every quasi-coherent sheaf on an affine scheme is globally generated. Analogously, in complex geometry, Cartan's theorem A says that every coherent sheaf on a Stein manifold is globally generated. A line bundle L on a proper scheme over a field is semi-ample if there is a positive integer r such that the tensor power is basepoint-free. A semi-ample line bundle is nef (by the corresponding fact for basepoint-free line bundles). Very ample line bundles A line bundle L on a proper scheme X over a field k is said to be very ample if it is basepoint-free and the associated morphism is a closed immersion. Here . Equivalently, L is very ample if X can be embedded into projective space of some dimension over k in such a way that L is the restriction of the line bundle O(1) to X. The latter definition is used to define very ampleness for a line bundle on a proper scheme over any commutative ring. The name "very ample" was introduced by Alexander Grothendieck in 1961. Various names had been used earlier in the context of linear systems of divisors. For a very ample line bundle L on a