In
algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a
family of curves
In geometry, a family of curves is a set of curves, each of which is given by a function or parametrization in which one or more of the parameters is variable. In general, the parameter(s) influence the shape of the curve in a way that is more ...
; 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 curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
s in the
projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...
. It assumed a more general form, through gradual generalisation, so that one could speak of linear equivalence of
divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s ''D'' on a general
scheme or even a
ringed space (''X'', ''O''
''X'').
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 elements of the set
, which can be written as
, are in natural bijection with
[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] by associating
to