Birational morphism

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.