Counting square-free monomial Cremona maps

Abstract

We give a complete list of square-free monomial Cremona maps of \({\mathbb {P}}^{n-1}\), with \(n\le 6\), up to equivalence classes. Also, we present a sufficient condition for all the ideals associated to a square-free Cremona transformation with same degree d in a fixed projective space \({\mathbb {P}}^{n-1}\) to have eight two. Finally, we introduce an algorithm to count them. Using this algorithm, we obtain the complete list of square-free Cremona maps of \({\mathbb {P}}^{n-1}\) when the degree is two for \(n=7,n=8,n=9, n=10, n=11\) and when the degree is three for \(n=7\).