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On Birational Maps and Jacobian Matrices

Published online by Cambridge University Press:  04 December 2007

Francesco Russo  and
Aron Simis
Francesco Russo
Affiliation:
Departamento de Matemática, CCEN, Universidade Federal de Pernambuco, Cidade Universitária, 50740-540 Recife, PE, Brazil. E-mail: frusso@dmat.ufpe.br
Aron Simis
Affiliation:
Departamento de Matemática, CCEN, Universidade Federal de Pernambuco, Cidade Universitária, 50740-540 Recife, PE, Brazil. E-mail: aron@dmat.ufpe.br
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Abstract

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One is concerned with Cremona-like transformations, i.e., rational maps from $ P$n to $ P$m that are birational onto the image Y ⊂ $ P$m and, moreover, the inverse map from Y to $ P$n lifts to $ P$m. We establish a handy criterion of birationality in terms of certain syzygies and ranks of appropriate matrices and, moreover, give an effective method to explicitly obtaining the inverse map. A handful of classes of Cremona and Cremona-like transformations follow as applications.

Keywords

birational mapsCremona transformationsalgebraic geometryJacobian matrices
Type
Research Article
Information
Compositio Mathematica , Volume 126 , Issue 3 , May 2001 , pp. 335 - 358
Copyright
© 2001 Kluwer Academic Publishers