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.