Difference between revisions of "Birational morphism"
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| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Grothendieck, J. Dieudonné, "Eléments de géometrie algébrique" ''Publ. Math. IHES'' , '''8''' (1960) {{MR|0217083}} {{MR|0163908}} {{ZBL|0118.36206}} </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) {{MR|0217068}} {{ZBL|0164.51704}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table> |
Revision as of 21:50, 30 March 2012
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 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 |
How to Cite This Entry:
Birational morphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Birational_morphism&oldid=12967
Birational morphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Birational_morphism&oldid=12967
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