A useful weakening of ampleness, notably in birational geometry, is the notion of a big line bundle. A line bundle L on a projective variety X of dimension n over a field is said to be big if there is a positive real number a and a positive integer such that for all . This is the maximum possible growth rate for the spaces of sections of powers of L, in the sense that for every line bundle L on X there is a positive number b with for all j > 0. There are several other characterizations of big line bundles. First, a line bundle is big if and only if there is a positive integer r such that the rational map from X to given by the sections of is birational onto its image. Also, a line bundle L is big if and only if it has a positive tensor power which is the tensor product of an ample line bundle A and an effective line bundle B (meaning that ). Finally, a line bundle is big if and only if its class in is in the interior of the cone of effective divisors. Bigness can be viewed as a birationally invariant analog of ampleness. For example, if is a dominant rational map between smooth projective varieties of the same dimension, then the pullback of a big line bundle on Y is big on X. (At first sight, the pullback is only a line bundle on the open subset of X where f is a morphism, but this extends uniquely to a line bundle on all of X.) For ample line bundles, one can only say that the pullback of an ample line bundle by a finite morphism is ample. Example: Let X be the blow-up of the projective plane at a point over the complex numbers. Let H be the pullback to X of a line on , and let E be the exceptional curve of the blow-up . Then the divisor H + E is big but not ample (or even nef) on X, because This negativity also implies that the base locus of H + E (or of any positive multiple) contains the curve E. In fact, this base locus is equal to E. Relative ampleness Given a quasi-compact morphism of schemes , an invertible sheaf L on X is said to be ample relative