In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family. These arose first in the form of a linear system of algebraic curves in the projective plane. It assumed a more general form, through gradual generalisation, so that one could speak of linear equivalence of divisors D on a general scheme or even a ringed space (X, OX). Linear system of dimension 1, 2, or 3 are called a pencil, a net, or a web, respectively. A map determined by a linear system is sometimes called the Kodaira map. Definitions Given a general variety , two divisors are linearly equivalent if for some non-zero rational function on , or in other words a non-zero element of the function field . Here denotes the divisor of zeroes and poles of the function . Note that if has singular points, the notion of 'divisor' is inherently ambiguous (Cartier divisors, Weil divisors: see divisor (algebraic geometry)). The definition in that case is usually said with greater care (using invertible sheaves or holomorphic line bundles); see below. A complete linear system on is defined as the set of all effective divisors linearly equivalent to some given divisor . It is denoted . Let be the line bundle associated to . In the case that is a nonsingular projective variety, the set is in natural bijection with by associating the element of to the set of non-zero multiples of (this is well defined since two non-zero rational functions have the same divisor if and only if they are non-zero multiples of each other). A complete linear system is therefore a projective space. A linear system is then a projective subspace of a complete linear system, so it corresponds to a vector subspace W of The dimension of the linear system is its dimension as a projective space. Hence . Linear systems can also be introduced by means of the line bundle or invertible sheaf language. In those terms, divisors (Cartier divisors, to be precise) correspond to line bundles, and linear equivalence of two divisors means that the corresponding line bundles are isomorphic. Examples Linear equivalence Consider the line bundle on whose sections define quadric surfaces. For the associated divisor , it is linearly equivalent to any other divisor defined