---
tag: GHRPE8CXJ9
kind: concept
name: Linear system of divisors
msc: [14C20]
sources:
  - url: https://en.wikipedia.org/wiki/Linear_system_of_divisors
same_as:
  wikidata: Q17098493
  wikipedia: Linear system of divisors
---

## Summary

A linear system on a projective variety is a family of linearly equivalent effective divisors, parametrized by a projective space.
The complete linear system $\abs{D}$ of a divisor $D$ consists of all effective divisors linearly equivalent to $D$.
It is the projectivization of the space of global sections of $\mathcal{O}_X(D)$.
A linear system without base points gives a morphism to projective space.
Up to a choice of coordinates, every morphism to projective space arises in this way.
They are the basic tool to embed varieties and to study maps between them.

## Explanations

### L1 human

Take all the curves of a fixed kind on a surface, such as the conics in the plane.
Each conic is the zero set of a quadratic polynomial, and two polynomials that differ by a nonzero scalar give the same conic.
So the conics form a projective space, here of dimension $5$, because a quadratic polynomial in three variables has six coefficients.
This family is a linear system: one can add polynomials, so the family is "linear".
The lines in the plane form a smaller linear system of dimension $2$.