ring) is ample if and only if for every coherent sheaf F on X, there is an integer s such that for all and all . In particular, high powers of an ample line bundle kill cohomology in positive degrees. This implication is called the Serre vanishing theorem, proved by Jean-Pierre Serre in his 1955 paper Faisceaux algébriques cohérents. Examples/Non-examples The trivial line bundle on a projective variety X of positive dimension is basepoint-free but not ample. More generally, for any morphism f from a projective variety X to some projective space over a field, the pullback line bundle is always basepoint-free, whereas L is ample if and only if the morphism f is finite (that is, all fibers of f have dimension 0 or are empty). For an integer d, the space of sections of the line bundle O(d) over is the complex vector space of homogeneous polynomials of degree d in variables x,y. In particular, this space is zero for d < 0. For , the morphism to projective space given by O(d) is by This is a closed immersion for , with image a rational normal curve of degree d in . Therefore, O(d) is basepoint-free if and only if , and very ample if and only if . It follows that O(d) is ample if and only if . For an example where "ample" and "very ample" are different, let X be a smooth projective curve of genus 1 (an elliptic curve) over C, and let p be a complex point of X. Let O(p) be the associated line bundle of degree 1 on X. Then the complex vector space of global sections of O(p) has dimension 1, spanned by a section that vanishes at p. So the base locus of O(p) is equal to p. On the other hand, O(2p) is basepoint-free, and O(dp) is very ample for (giving an embedding of X as an elliptic curve of degree d in ). Therefore, O(p) is ample but not very ample. Also, O(2p) is ample and basepoint-free but not very ample; the associated morphism to projective space is a ramified double cover . On curves of higher genus, there are ample line bundles L for which every global section is zero. (But high multiples of L have many sections, by definition.) For example, let X be a smooth plane quartic curve (of degree 4 in )