Rational map of varieties

Definition

Let $X$ and $Y$ be varieties over a field $k$ (which we usually assume to be algebraically closed. A rational map $\phi :X\to Y$ is an equivalence class of:

Pairs $<U,\phi _{U}>$ where $U$ is an open set in $X$ and $\phi _{U}$ is a morphism of varieties from $U$ to $Y$

under the equivalence relation:

$<U,\phi _{U})\sim <V,\phi _{V}>$ if and only if $\phi _{U}$ and $\phi _{V}$ agree on $U\cap V$.