Birational map of varieties

Definition
Let $$X$$ and $$Y$$ be varieties over a field $$k$$ (which we assume algebraically closed). A birational map from $$X$$ to $$Y$$ is a rational map $$\phi$$ from $$X$$ to $$Y$$ suchthat there exists a rational map $$\psi: Y \to X$$ so that:


 * $$\phi \circ \psi = id_Y$$ where the equality is as a rational map
 * $$\psi \circ \phi = id_X$$ where the equality is as a rational map

Two algebraic varieties for which there is a birational map from one to the other, are termed birationally equivalent.