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.
|||A. Grothendieck, J. Dieudonné, "Eléments de géometrie algébrique" Publ. Math. IHES , 8 (1960)|
|||I.R. Shafarevich, "Lectures on minimal models and birational transformations of two-dimensional schemes" , Tata Inst. (1966)|
|||I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian)|
|||R. Hartshorne, "Algebraic geometry" , Springer (1977)|
Birational morphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Birational_morphism&oldid=12967