proper scheme X over a field with associated morphism f, the degree of L on a curve C in X is the degree of f(C) as a curve in . So L has positive degree on every curve in X (because every subvariety of projective space has positive degree). Definitions A line bundle L on a proper scheme X over a commutative ring R is said to be ample if there is a positive integer r such that the tensor power is very ample. In particular, a proper scheme over R has an ample line bundle if and only if it is projective over R. An ample line bundle on a proper scheme X over a field has positive degree on every curve in X, by the corresponding statement for very ample line bundles. A Cartier divisor D on a proper scheme X over a field k is said to be ample if the corresponding line bundle O(D) is ample. (For example, if X is smooth over k, then a Cartier divisor can be identified with a finite linear combination of closed codimension-1 subvarieties of X with integer coefficients.) On an arbitrary scheme X, Grothendieck defined a line bundle L to be ample if X is quasi-compact and for every point x in X there is a positive integer r and a section such that s is nonzero at x and the open subscheme is affine. For example, the trivial line bundle is ample if and only if X is quasi-affine. The rest of this article will concentrate on ampleness on proper schemes over a field. Weakening the notion of "very ample" to "ample" gives a flexible concept with a wide variety of different characterizations. A first point is that tensoring high powers of an ample line bundle with any coherent sheaf whatsoever gives a sheaf with many global sections. More precisely, a line bundle L on a proper scheme X over a field (or more generally over a Noetherian ring) is ample if and only if for every coherent sheaf F on X, there is an integer s such that the sheaf is globally generated for all . Here s may depend on F. Another characterization of ampleness, known as the Cartan–Serre–Grothendieck theorem, is in terms of coherent sheaf cohomology. Namely, a line bundle L on a proper scheme X over a field (or more generally over a Noetherian