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

TL;DR

This paper investigates a specific group of Cremona transformations generated by linear automorphisms and a standard involution, analyzing its properties, elements, and intersections with monomial transformations.

Contribution

It characterizes the geometric properties of this transformation group and identifies elements outside the group in odd dimensions.

Findings

All elements contract only rational hypersurfaces.

In odd dimensions, some simple elements with this property are outside the group.

The intersection with monomial transformations is described.

Abstract

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.