# Birational equivalence between projective varieties is an equivalence relation

The definition for "birational map" I was given was as follows:

Let $V$ and $W$ be irreducible projective varieties. If $\phi$ is a dominant rational map from $V$ to $W$ and $\psi$ is a dominant rational map from $W$ to $V$ and $\phi \circ \psi$ and $\psi \circ \phi$ are identity maps (on the domains they can be defined), the $\phi$ and $\psi$ are called birational maps.

$\phi$ being dominant means that $\phi(\text{dom}\phi)$ is dense in $W$.

Later in the lectures I think it was assumed that birational equivalence between projective varieties was an equivalence relation, but I don't immediately see why--are the composite of dominant rational maps dominant?

• Yes. It suffices to show the image of the composite is dense - try to play with continuity to get that.
– user27126
Commented Feb 17, 2013 at 19:41
• But what are the continuous maps? $\phi$ and $\psi$ are not even everywhere defined? Commented Feb 17, 2013 at 19:46
• they are continuous on the open set they are defined.
– user27126
Commented Feb 17, 2013 at 19:49