Linear system of divisors

In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, a linear system of divisors is an algebraic generalization of the geometric notion of a <a href="/facts/Family_of_curves/3bajWkSv">family of curves</a>; 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 <a href="/facts/Algebraic_curve/LnWsPmDP">algebraic curves</a> in the <a href="/facts/Projective_plane/p6ot8oNv">projective plane</a>. It assumed a more general form, through gradual generalisation, so that one could speak of linear equivalence of <a href="/facts/Divisor_(algebraic_geometry)/V0QDypeo">divisors</a> D on a general <a href="/facts/Scheme_(mathematics)/c7uSqGqL">scheme</a> or even a <a href="/facts/Ringed_space/JyeaTMtX">ringed space</a> 
 
 
 
 (
 X
 ,
 
 
 
 O
 
 
 
 X
 
 
 )
 
 
 {\displaystyle (X,{\mathcal {O}}_{X})}
 
.
Linear systems of dimension 1, 2, or 3 are called a <a href="/facts/Pencil_(mathematics)/b75bj3wd">pencil</a>, a net, or a web, respectively.
A map determined by a linear system is sometimes called the Kodaira map.

Linear system of divisors open-in-new

Linear system of divisors