theory then tell us that we must have . The conclusion is that to check nefness of a divisor class, it suffices to compute the intersection number with curves contained in the base locus of the class. So, roughly speaking, the 'smaller' the base locus, the 'more likely' it is that the class is nef. In the modern formulation of algebraic geometry, a complete linear system of (Cartier) divisors on a variety is viewed as a line bundle on . From this viewpoint, the base locus is the set of common zeroes of all sections of . A simple consequence is that the bundle is globally generated if and only if the base locus is empty. The notion of the base locus still makes sense for a non-complete linear system as well: the base locus of it is still the intersection of the supports of all the effective divisors in the system. Example Consider the Lefschetz pencil given by two generic sections , so given by the schemeThis has an associated linear system of divisors since each polynomial, for a fixed is a divisor in . Then, the base locus of this system of divisors is the scheme given by the vanishing locus of , so A map determined by a linear system Each linear system on an algebraic variety determines a morphism from the complement of the base locus to a projective space of dimension of the system, as follows. (In a sense, the converse is also true; see the section below) Let L be a line bundle on an algebraic variety X and a finite-dimensional vector subspace. For the sake of clarity, we first consider the case when V is base-point-free; in other words, the natural map is surjective (here, k = the base field). Or equivalently, is surjective. Hence, writing for the trivial vector bundle and passing the surjection to the relative Proj, there is a closed immersion: where on the right is the invariance of the projective bundle under a twist by a line bundle. Following i by a projection, there results in the map: When the base locus of V is not empty, the above discussion still goes through with in the direct sum replaced by an ideal sheaf defining the base locus and X replaced by the blow-up of it along the (scheme-theoretic) base locus B. Precisely, as above, there is a surjection where