Three chapters on Cremona groups

TL;DR

This paper investigates properties of Cremona groups, answering a question about regularizability of birational maps, analyzing degree sequences of iterates, and studying invariant curve pencils, with implications for conjugacy classes and topology.

Contribution

It provides new insights into the regularizability of birational maps, degree growth sequences, and invariant pencils, advancing understanding of Cremona group structure.

Findings

Not all linear maps make a birational map regularizable.

Degree differences of iterates can be arbitrary for small n.

Degree of invariant pencils is bounded for certain twists.

Abstract

In the first part of this article, we answer a question of I. Dolgachev, which is related to the following problem: given a birational map $f\in\mathrm{Bir}(\mathbb{P}^m_\mathbf{k})$ and a linear projective map $A\in\mathrm{PGL}_{m+1}(\mathbf{k})$, when is $A\circ f$ regularizable? Dolgachev's initial question is whether this may happen for all $A$ in $\mathrm{PGL}_{m+1}(\mathbf{k})$, and the answer is negative. We then look at the sequence $n\mapsto\mathrm{deg}\, f^n$, $f\in\mathrm{Bir}(\mathbb{P}^2_\mathbf{k})$. We show that there is no constraint on the sequence $n\mapsto\mathrm{deg}\, f^n-\mathrm{deg}\, f^{n-1}$ for small values of $n$. Finally we study the degree of pencils of curves which are invariant by a birational map. When f is a Halphen or Jonqui\`eres twist, we prove that this degree is bounded by a function of $\mathrm{deg}\, f$. We derive corollaries on the structure of conjugacy classes, and their properties with respect to the Zariski topology of $\mathrm{Bir}(\mathbb{P}^2_\mathbf{k})$.