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.
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.
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In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ample line bundle, although there are several related classes of line bundles. Roughly speaking, positivity properties of a line bundle are related to having many global sections. Understanding the ample line bundles on a given variety X amounts to understanding the different ways of mapping X into projective space.
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).
In relation to the history of mathematics, the Italian school of algebraic geometry refers to mathematicians and their work in birational geometry, particularly on algebraic surfaces, centered around Rome roughly from 1885 to 1935. There were 30 to 40 leading mathematicians who made major contributions, about half of those being Italian. The leadership fell to the group in Rome of Guido Castelnuovo, Federigo Enriques and Francesco Severi, who were involved in some of the deepest discoveries, as well as setting the style.