Birational morphism

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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 [2] (cf. Monoidal transformation). This is not true in dimensions higher than two.


[1] A. Grothendieck, J. Dieudonné, "Eléments de géometrie algébrique" Publ. Math. IHES , 8 (1960) MR0217083 MR0163908 Zbl 0118.36206
[2] I.R. Shafarevich, "Lectures on minimal models and birational transformations of two-dimensional schemes" , Tata Inst. (1966) MR0217068 Zbl 0164.51704
[3] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[4] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001
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Birational morphism. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by I.V. DolgachevV.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article