A morphism of schemes that is a birational mapping. The most important examples of birational morphisms include: a normalization, a blowing up and a monoidal transformation. Any proper birational transformation between regular two-dimensional schemes can be decomposed into monoidal transformations with non-singular centres  (cf. Monoidal transformation). This is not true in dimensions higher than two.
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Birational morphism. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Birational_morphism&oldid=23766