Some effectivity questions for plane Cremona transformations

TL;DR

This paper provides methods to estimate the hyperbolic length of plane Cremona transformations, explores their subgroup structures, and establishes rigidity and tightness criteria, advancing understanding of their algebraic and geometric properties.

Contribution

It introduces a quick estimation technique for hyperbolic lengths, demonstrates subgroup properties over certain fields, and offers criteria for rigidity and tightness of Cremona transformations.

Findings

Estimation method for hyperbolic lengths of transformations

Normal closure of high powers forms proper subgroups

Hyperbolic Cremona transformations are rigid

Abstract

We give a way of estimating quickly the length of the hyperbolic isometry associated to a plane Cremona transformation that is presented as the product of two involutions. We also show, using the ideas of Cantat and Lamy, that, provided that the ground field is either algebraic of characteristic not 2 or of characteristic zero, the normal closure of a high power of such a transformation is a proper subgroup of the Cremona group; this gives an effective instantiation of some of their results. Moreover, we show that any hyperbolic Cremona transformation is rigid and give a simple criterion for it to be tight. This version is thoroughly revised from the previous version. In particular, it corrects an error in an earlier version that was pointed out by Lonjou.