Birational geometry

Remarks

Two integral varieties over $k$ are birationally equivalent exactly when their function fields are isomorphic as extensions of $k$. Example: $\mathbb{P}^2$ and $\mathbb{P}^1 \times \mathbb{P}^1$ are birationally equivalent. The map $[x : y : z] \mapsto ([x : z], [y : z])$ is defined where $z \ne 0$, and its inverse $([x : 1], [y : 1]) \mapsto [x : y : 1]$ is defined on the dense open set where both second coordinates are nonzero. Both function fields are $k(s, t)$. Non-example: $\mathbb{P}^1$ and a smooth plane cubic curve $E$ over $\C$ are not birationally equivalent. For smooth projective curves a birational map extends to an isomorphism, and $\mathbb{P}^1$ has genus $0$ while $E$ has genus $1$. The dimension, the geometric genus and the Kodaira dimension of smooth projective varieties are birational invariants.