On Birational Maps and Jacobian Matrices FRANCESCO RUSSO* and ARON SIMIS* Departamento de Matema¤ tica, CCEN, Universidade Federal de Pernambuco, Cidade Universita¤ ria, 50740-540 Recife, PE, Brazil. e-mail: {frusso, aron}@dmat.ufpe.br (Received: 10 December 1999; accepted: 11 May 2000) 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  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. Mathematics Subject Classi¢cations (2000). Primary 14E05, 14E07, 14M05, 13H10; Second- ary 14M12, 14M15, 13B10, 13C14, 13F45. Key words. birational maps, Cremona transformations, algebraic geometry, Jacobian matrices. Introduction Let F : P n P m be a rational map and let Y  P m be its image. We consider the question as when F admits an inverse rational map G de¢ned on Y by the restrictions of forms of the same degree on the ambient P m . Our main result is Theorem 1.4 below which gives a criterion in order that F admit an inverse G and, moreover, tells how to compute it. The criterion relies in a strong way on the notion of Jacobian dual introduced in [14] and gives insight even in the case of ordinary Cremona transformations. Quite a bit of the present work hinges on this criterion as such, although we also give new structure results on representation of rational varieties by projective spaces which invokes a mix of the criterion and some other arguments. A word on the terminology. For convenience, rational maps such as G above will be called liftable (with reference to the ¢xed projective embedding Y  P m ). For a birational map F : P n Y  P m , whose inverse F 1 is liftable, one can de¢ne its type to be the pair k; k 0 , where k (resp. k 0 ) is the degree of the forms de¢ning F (resp. the degree of the forms de¢ning F 1 ). Fixing the embedding Y  P m , in order to have the condition that all rational maps with source Y be liftable, one could express it in terms of sheaves and sections, namely: ¢rst, every locally free O Y -module L ought to be the restriction to Y of O P m d , for some d X 0; second, each individual section de¢ning the rational map ought to be lifted to an element of H 0 O P m d , i.e., the restriction maps --- --- *Partially supported by a CNPq grant. Compositio Mathematica 126: 335^358, 2001. 335 # 2001 Kluwer Academic Publishers. Printed in the Netherlands.