Rational map

Formal definition

The importance of rational maps to algebraic geometry is in the connection between such maps and maps between the function fields of ${\displaystyle V}$ and ${\displaystyle W}$. By definition, a rational function is just a rational map whose range is the projective line. Composition of functions then allows us to "pull back" rational functions along a rational map, so that a single rational map ${\displaystyle f\colon V\to W}$ induces a homomorphism of fields ${\displaystyle K(W)\to K(V)}$. In particular, the following theorem is central: the functor from the category of projective varieties with dominant rational maps (over a fixed base field, for example ${\displaystyle \mathbb {C} }$) to the category of finitely generated field extensions of the base field with reverse inclusion of extensions as morphisms, which associates each variety to its function field and each map to the associated map of function fields, is an equivalence of categories.