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.