Compositio Mathematica



On Birational Maps and Jacobian Matrices


Francesco Russo a1 and Aron Simis a2
a1 Departamento de Matemática, CCEN, Universidade Federal de Pernambuco, Cidade Universitária, 50740-540 Recife, PE, Brazil. E-mail: frusso@dmat.ufpe.br
a2 Departamento de Matemática, CCEN, Universidade Federal de Pernambuco, Cidade Universitária, 50740-540 Recife, PE, Brazil. E-mail: aron@dmat.ufpe.br

Article author query
russo f   [Google Scholar] 
simis a   [Google Scholar] 
 

Abstract

One is concerned with Cremona-like transformations, i.e., rational maps from $ P$n to $ P$m that are birational onto the image Y [subset or is implied by] $ 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.


Key Words: birational maps; Cremona transformations; algebraic geometry; Jacobian matrices.