birational geometry In general linear systems became a basic tool of birational geometry as practised by the Italian school of algebraic geometry. The technical demands became quite stringent; later developments clarified a number of issues. The computation of the relevant dimensions — the Riemann–Roch problem as it can be called — can be better phrased in terms of homological algebra. The effect of working on varieties with singular points is to show up a difference between Weil divisors (in the free abelian group generated by codimension-one subvarieties), and Cartier divisors coming from sections of invertible sheaves. The Italian school liked to reduce the geometry on an algebraic surface to that of linear systems cut out by surfaces in three-space; Zariski wrote his celebrated book Algebraic Surfaces to try to pull together the methods, involving linear systems with fixed base points. There was a controversy, one of the final issues in the conflict between 'old' and 'new' points of view in algebraic geometry, over Henri Poincaré's characteristic linear system of an algebraic family of curves on an algebraic surface. Base locus The base locus of a linear system of divisors on a variety refers to the subvariety of points 'common' to all divisors in the linear system. Geometrically, this corresponds to the common intersection of the varieties. Linear systems may or may not have a base locus – for example, the pencil of affine lines has no common intersection, but given two (nondegenerate) conics in the complex projective plane, they intersect in four points (counting with multiplicity) and thus the pencil they define has these points as base locus. More precisely, suppose that is a complete linear system of divisors on some variety . Consider the intersection where denotes the support of a divisor, and the intersection is taken over all effective divisors in the linear system. This is the base locus of (as a set, at least: there may be more subtle scheme-theoretic considerations as to what the structure sheaf of should be). One application of the notion of base locus is to nefness of a Cartier divisor class (i.e. complete linear system). Suppose is such a class on a variety , and an irreducible curve on . If is not contained in the base locus of , then there exists some divisor in the class which does not contain , and so intersects it properly. Basic facts from intersection