Birational variety

Irreducible varieties are birational exactly when they have isomorphic function fields.

Over an algebraically closed field, the blowup of a smooth algebraic surface at a closed point replaces the point by an exceptional smooth rational curve $E\cong {\mathbb{P}}^1$, with $E^2=-1$. Locally it is the blowup of the affine plane at the origin. It is a proper birational morphism, an isomorphism off the exceptional curve, and the blown-up surface is again smooth and projective when the original is projective.

For a point blowup of a smooth algebraic surface $\psi :\tilde{X}\to X$ over an algebraically closed field, with exceptional curve $E$, $K_{\tilde{X}}={\psi }^{\ast }{K}_X+E$ after choosing compatible representatives. In local coordinates $y=xu$, the differential $dx\wedge dy=xdx\wedge du$ has one additional zero along $E$. Thus the formula is valid in every characteristic.

A morphism between integral varieties is birational if it induces an isomorphism of their function fields, equivalently an isomorphism on suitable dense open subsets. A proper birational morphism to a normal variety satisfies $f_{\ast }{\mathcal{O}}_X={\mathcal{O}}_Y$. For point blowups of smooth surfaces this identifies sections of pulled-back line bundles by the projection formula for sheaves.

A rational map with a rational inverse restricts to an isomorphism of dense open subsets. It may identify or omit exceptional points outside those subsets.

The blowup of ${\mathbb{A}}^2$ at the origin isIts projection to ${\mathbb{A}}^2$ is an isomorphism away from the origin, while the fiber above the origin is the exceptional curve ${\mathbb{P}}^1$.