Linear system of divisors

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Type I linear system, (Coffman).

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, O_X).

A linear system of dimension 1, 2, or 3 is called a pencil, a net, or a web.

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, divisors D and E are linearly equivalent if

where (f) denotes the divisor of zeroes and poles of the function f.

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.

A complete linear system on V is defined as the set of all effective divisors linearly equivalent to some given divisor D. It is denoted |D|. Let L(D) be the line bundle associated to D. In the case that V is a nonsingular projective variety the set |D| is in natural bijection with ${\displaystyle (\Gamma (V,L)\setminus \{0\})/k^{\ast },}$ and is therefore a projective space.

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