View source for Birational transformation

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 
$$(x_0,x_1,x_2)\mapsto(x_1x_2,x_0x_2,x_0x_1)$$
where $(x_0,x_1,x_2)$ are homogeneous coordinates in the projective
plane.

====References====
<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>