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Difference between revisions of "Birational transformation"

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A [[Birational mapping|birational mapping]] of an algebraic variety (or scheme) into itself. Also called sometimes a birational automorphism. The group of all birational transformations of an algebraic variety is canonically isomorphic to the group of automorphisms of its field of rational functions over the field of constants. Examples of birational transformations include the Cremona transformations (cf. [[Cremona transformation|Cremona transformation]]), in particular the standard quadratic transformation of a projective plane, given by the formula
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A
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[[Birational mapping|birational mapping]] of an algebraic variety (or
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016520/b0165201.png" /></td> </tr></table>
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scheme) into itself. Also called sometimes a birational
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automorphism. The group of all birational transformations of an
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016520/b0165202.png" /> are homogeneous coordinates in the projective plane.
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algebraic variety is canonically isomorphic to the group of
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automorphisms of its field of rational functions over the field of
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constants. Examples of birational transformations include the Cremona
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transformations (cf.
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[[Cremona transformation|Cremona transformation]]), in particular the
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standard quadratic transformation of a projective plane, given by the
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formula  
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$$(x_0,x_1,x_2)\mapsto(x_1x_2,x_0x_2,x_0x_1)$$
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where $(x_0,x_1,x_2)$ are homogeneous coordinates in the projective
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plane.
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.A. Iskovskikh,   "Birational automorphisms of three-dimensional algebraic varieties" ''J. Soviet Math.'' , '''13''' : 6 (1960) pp. 815–868 ''Itogi Nauk. i Tekhn. Sovrem. Probl. Mat.'' , '''12''' (1979) pp. 159–239</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD>
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<TD valign="top"> V.A. Iskovskikh, "Birational automorphisms of three-dimensional algebraic varieties" ''J. Soviet Math.'' , '''13''' : 6 (1960) pp. 815–868 ''Itogi Nauk. i Tekhn. Sovrem. Probl. Mat.'' , '''12''' (1979) pp. 159–239</TD>
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</TR></table>

Latest revision as of 22:04, 11 November 2011

A birational mapping of an algebraic variety (or scheme) into itself. Also called sometimes a birational automorphism. The group of all birational transformations of an algebraic variety is canonically isomorphic to the group of automorphisms of its field of rational functions over the field of constants. Examples of birational transformations include the Cremona transformations (cf. Cremona transformation), in particular the standard quadratic transformation of a projective plane, given by the formula $$(x_0,x_1,x_2)\mapsto(x_1x_2,x_0x_2,x_0x_1)$$ where $(x_0,x_1,x_2)$ are homogeneous coordinates in the projective plane.

References

[1] V.A. Iskovskikh, "Birational automorphisms of three-dimensional algebraic varieties" J. Soviet Math. , 13 : 6 (1960) pp. 815–868 Itogi Nauk. i Tekhn. Sovrem. Probl. Mat. , 12 (1979) pp. 159–239
How to Cite This Entry:
Birational transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Birational_transformation&oldid=19617
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
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