The group of Cremona transformations generated by linear maps and the standard involution

This article studies the group generated by automorphisms of the projective space of dimension $n$ and by the standard birational involution of degree $n$. Every element of this group only contracts rational hypersurfaces, but in odd dimension, there are simple elements having this property which do not belong to the group. Geometric properties of the elements of the group are given, as well as a description of its intersection with monomial transformations

Classification of the monomial Cremona transformations of the plane

We classify all monomial planar Cremona maps by multidegree using recent methods developed by Aluffi. Following the main result, we prove several more properties of the set of these maps, and also extend the results to the more general `r.c. monomial' maps.Comment: Work in progress... comments are welcom

Cremona maps defined by monomials

Cremona maps defined by monomials of degree 2 are thoroughly analyzed and classified via integer arithmetic and graph combinatorics. In particular, the structure of the inverse map to such a monomial Cremona map is made very explicit as is the degree of its monomial defining coordinates. As a special case, one proves that any monomial Cremona map of degree 2 has inverse of degree 2 if and only if it is an involution up to permutation in the source and in the target. This statement is subsumed in a recent result of L. Pirio and F. Russo, but the proof is entirely different and holds in all characteristics. One unveils a close relationship binding together the normality of a monomial ideal, monomial Cremona maps and Hilbert bases of polyhedral cones. The latter suggests that facets of monomial Cremona theory may be NP-hard

Rational surface maps with invariant meromorphic two forms

We consider a rational map f:S->S of a complex projective surface together with an invariant meromorphic two form. Under a mild topological assumption on the map, we show that the zeroes of the invariant form can be eliminated by birational change of coordinate. In this context, when the form has no zeroes, we investigate the notion of algebraic stability for f. We show in particular that algebraic stability is equivalent to a more tractable condition involving the behavior of f on the poles of the form. Finally, we illustrate our results in the particular case where S is the projective plane and the invariant form is dx dy / xy, showing that our criterion for stability translates to whether or not the rotation number for a certain circle homeomorphism is rational.Comment: Maps on irrational surfaces are now discussed in much more detai