Linear system of divisors

Summary

A linear system of divisors algebraicizes the classic geometric notion of a family of curves, as in the Apollonian circles.

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).[1]

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.

Definition

Given the fundamental idea of a rational function on a general variety , or in other words of a function in the function field of , , divisors are linearly equivalent divisors if

where denotes the divisor of zeroes and poles of the function .

Note that if has singular points, '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 [2][further explanation needed] and 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 .

Since a Cartier divisor class is an isomorphism class of a line bundle, linear systems can also be introduced by means of the line bundle or invertible sheaf language, without reference to divisors at all. 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 by the vanishing locus of some using the rational function [2] (Proposition 7.2). For example, the divisor associated to the vanishing locus of is linearly equivalent to the divisor associated to the vanishing locus of . Then, there is the equivalence of divisors

Linear systems on curves

One of the important complete linear systems on an algebraic curve of genus is given by the complete linear system associated with the canonical divisor , denoted . 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 .

Hyperelliptic curves

One application of linear systems is used in the classification of algebraic curves. A hyperelliptic curve is a curve with a degree morphism .[2] For the case all curves are hyperelliptic: the Riemann–Roch theorem then gives the degree of is and , hence there is a degree map to .

grd

A is a linear system on a curve which is of degree and dimension . For example, hyperelliptic curves have a since defines one. In fact, hyperelliptic curves have a unique [2] from proposition 5.3. Another close set of examples are curves with a which are called trigonal curves. In fact, any curve has a for .[3]

Linear systems of hypersurfaces in

Consider the line bundle over . If we take global sections , then we can take its projectivization . This is isomorphic to where

Then, using any embedding we can construct a linear system of dimension .

Linear system of conics

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 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 theory then tell us that we must have . 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.

In the modern formulation of algebraic geometry, a complete linear system of (Cartier) divisors on a variety is viewed as a line bundle on . From this viewpoint, the base locus is the set of common zeroes of all sections of . 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 two generic sections , so given by the scheme

This has an associated linear system of divisors since each polynomial, for a fixed is a divisor in . Then, the base locus of this system of divisors is the scheme given by the vanishing locus of , 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 and a finite-dimensional vector subspace. For the sake of clarity, we first consider the case when V is base-point-free; in other words, the natural map is surjective (here, k = the base field). Or equivalently, is surjective. Hence, writing for the trivial vector bundle and passing the surjection to the relative Proj, there is a closed immersion:

where 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:[4]

When the base locus of V is not empty, the above discussion still goes through with 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. Precisely, as above, there is a surjection where is the ideal sheaf of B and that gives rise to

Since an open subset of , 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.

For a closed immersion of algebraic varieties there is a pullback of a linear system on to , defined as [2] (page 158).

O(1) on a projective variety

A projective variety embedded in has a canonical linear system determining a map to projective space from . This sends a point to its corresponding point .

See also

References

  1. ^ Grothendieck, Alexandre; Dieudonné, Jean. EGA IV, 21.3.
  2. ^ a b c d e f Hartshorne, R. 'Algebraic Geometry', proposition II.7.2, page 151, proposition II.7.7, page 157, page 158, exercise IV.1.7, page 298, proposition IV.5.3, page 342
  3. ^ Kleiman, Steven L.; Laksov, Dan (1974). "Another proof of the existence of special divisors". Acta Mathematica. 132: 163–176. doi:10.1007/BF02392112. ISSN 0001-5962.
  4. ^ Fulton, § 4.4.
  • P. Griffiths; J. Harris (1994). Principles of Algebraic Geometry. Wiley Classics Library. Wiley Interscience. p. 137. ISBN 0-471-05059-8.
  • Hartshorne, R. Algebraic Geometry, Springer-Verlag, 1977; corrected 6th printing, 1993. ISBN 0-387-90244-9.
  • Lazarsfeld, R., Positivity in Algebraic Geometry I, Springer-Verlag, 2004. ISBN 3-540-22533-1.