Linear_system_of_divisors

Linear system of divisors

In mathematics, the concept of a linear system of divisors 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 algebraic variety V.

Definition by means of functions

Given the fundamental idea of a rational function on a general variety V, or in other words of a function F in the function field of V, a divisor D is by definition linearly equivalent to all divisors

D + (F)

where (F) denotes the divisor of zeroes and poles attached to F. Stated otherwise, two divisors are linearly equivalent precisely when their difference is the divisor attached to such a function.

Note that if V 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.

General linear systems

A linear system in general is part of, but not necessarily the whole of, an equivalence class for linear equivalence. Such classes are parametrised by a projective space, and the definition of a linear system is as the divisors corresponding to a linear subspace of that projective space. The reason for having such a definition can be explained geometrically as the need to cut down a complete linear system by constraints in given problems.

Example

For example, the conic sections in the projective plane form a linear system of dimension five, as one sees by counting the constants in the degree two equations. The condition to pass through a given point P imposes a single linear condition, so that conics C through P form a linear system of dimension 4. Other types of condition that are of interest include tangency to a given line L.

In the most elementary treatments a linear system appears in the form of equations

λC + μC′ = 0

with λ and μ unknown scalars, not both zero. Here C and C′ are given conics. Abstractly we can say that this is a projective line in the space of all conics, on which we take

[λ : μ]

as homogeneous coordinates. Geometrically we notice that any point Q common to C and C′ is also on each of the conics of the linear system. According to Bézout's theorem C and C′ will intersect in four points (if counted correctly). Assuming these are in general position, i.e. four distinct intersections, we get another interpretation of the linear system as the conics passing through the four given points (note that the codimension four here matches the dimension, one, in the five-dimensional space of conics).

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 of 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.

Line bundle/invertible sheaf language

Linear systems are still at the heart of contemporary algebraic geometry; but they are typically introduced by means of the ample line bundle language. That is, the language of sheaf theory is considered the most natural starting point, at least to learn the theory. In those terms, divisors D (Cartier divisors, importantly) correspond to line bundles, and linear equivalence of two divisors means that the corresponding line bundles are isomorphic.

References

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