Birational transformation

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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.


[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
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Birational transformation. Encyclopedia of Mathematics. URL:
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