# Birational morphism

From Encyclopedia of Mathematics

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.

#### References

[1] | A. Grothendieck, J. Dieudonné, "Eléments de géométrie 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 |

**How to Cite This Entry:**

Birational morphism.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Birational_morphism&oldid=55863

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