Three chapters on Cremona groups

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 $f\in \mathrm{Bir}({\mathbb{P}}_{\mathbf{k}}^m)$ 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 $n\mapsto \mathrm{deg}{f}^n$, $f\in \mathrm{Bir}({\mathbb{P}}_{\mathbf{k}}^2)$. 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 Jonquières 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 $\mathrm{Bir}({\mathbb{P}}_{\mathbf{k}}^2)$.

References in corpus (2)

• The existence of Zariski dense orbits for endomorphisms of projective surfaces (with an appendix in collaboration with Thomas Tucker)
• Regularization of Rational Group Actions

Cited by in corpus (1)

• On regularizable birational maps