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Wikipedia / 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. It assumed a more general form, through gradual generalisation, so that one could speak of linear equivalence of divisorss D on a general scheme or even a ringed space.

Linear systems 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

are linearly equivalent if

for some non-zero rational function of the function field.

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 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.

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

define quadric surfaces. Then, there is the equivalence of divisors

Linear systems on curves

of genus. This definition follows from proposition II.7.7 of Hartshorne.

Hyperelliptic curves

One application of linear systems is used in the classification of algebraic curves. A hyperelliptic curve with a 2 morphism. all curves are hyperelliptic: the Riemann–Roch theorem.

gdr

which are called trigonal curves.

Linear systems of hypersurfaces in a projective space

where

.

Linear system of conics

Characteristic linear system of a family of curves

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.

In modern terms, it is a subsystem of the linear system associated to the normal bundle. Note a characteristic system need not to be complete; in fact, the question of completeness is something studied extensively by the Italian school without a satisfactory conclusion; nowadays, the Kodaira–Spencer theory can be used to answer the question of the completeness.

Other examples

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.

Linear systems in 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 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.

. Consider the intersection

in the linear system. This is the base locus (as a set, at least: there may be more subtle scheme-theoretic considerations as to what the structure sheaf should be).

One application of the notion of base locus is to nefness, and so intersects it properly. Basic facts from intersection theory. 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.

. 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 the scheme, 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 a finite-dimensional vector subspace. For the sake of clarity, we first consider the case when V is surjective (here, k for the trivial vector bundle and passing the surjection to the relative Proj, there is a closed immersion:

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 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 is the ideal sheaf of B and that gives rise to

, 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).

Linear system determined by a map to a projective space

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.

(page 158).

O(1) on a projective variety

.

See also

References