View source for Birational morphism

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Grothendieck,   J. Dieudonné,   "Eléments de géometrie algébrique"  ''Publ. Math. IHES'' , '''8'''  (1960)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.R. Shafarevich,   "Lectures on minimal models and birational transformations of two-dimensional schemes" , Tata Inst.  (1966)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.R. Shafarevich,   "Basic algebraic geometry" , Springer  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R. Hartshorne,   "Algebraic geometry" , Springer  (1977)</TD></TR></table>