Linear system of divisors

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.

[1] Linear systems of dimension 1, 2, or 3 are called a pencil, a net, or a web, respectively.

The definition in that case is usually said with greater care (using invertible sheaves or holomorphic line bundles); see below.

Linear systems can also be introduced by means of the line bundle or invertible sheaf language.

(Cartier divisors, to be precise) correspond to line bundles, and linear equivalence of two divisors means that the corresponding line bundles are isomorphic.

, it is linearly equivalent to any other divisor defined by the vanishing locus of some

is given by the complete linear system associated with the canonical divisor

This definition follows from proposition II.7.7 of Hartshorne[2] since every effective divisor in the linear system comes from the zeros of some section of

The characteristic linear system of a family of curves on an algebraic surface Y for a curve C in the family is a linear system formed by the curves in the family that are infinitely near C.[4] In modern terms, it is a subsystem of the linear system associated to the normal bundle to

The Cayley–Bacharach theorem is a property of a pencil of cubics, which states that the base locus satisfies an "8 implies 9" property: any cubic containing 8 of the points necessarily contains the 9th.

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.

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.

(as a set, at least: there may be more subtle scheme-theoretic considerations as to what the structure sheaf of

One application of the notion of base locus is to nefness of a Cartier divisor class (i.e. complete linear system).

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

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.

(In a sense, the converse is also true; see the section below) Let L be a line bundle on an algebraic variety X and

For the sake of clarity, we first consider the case when V is base-point-free; in other words, the natural map

for the trivial vector bundle and passing the surjection to the relative Proj, there is a closed immersion: where

Following i by a projection, there results in the map:[5] When the base locus of V is not empty, the above discussion still goes through with

, there results in the map: Finally, when a basis of V is chosen, the above discussion becomes more down-to-earth (and that is the style used in Hartshorne, Algebraic Geometry).

Each morphism from an algebraic variety to a projective space determines a base-point-free linear system on the variety; because of this, a base-point-free linear system and a map to a projective space are often used interchangeably.

has a natural linear system determining a map to projective space from