A birational morphism of irreducible algebraic varieties is a morphism that restricts to an isomorphism between dense open subsets. Equivalently, it induces an isomorphism of their fields of rational functions.
The algebraic blow-up of an irreducible variety along a closed subscheme whose underlying closed set is not the whole variety is a birational morphism. Birational morphisms are central to the resolution of singularities, where one replaces a singular variety by a birationally equivalent smooth variety.
