On conjugacy classes of the Klein simple group in Cremona group

Abstract

We consider countably many three dimensional ${\mathtt{PSL}}_2({\mathbb{F}}_7)$-del Pezzo surface fibrations over ${\mathbb{P}}^1$. Conjecturally they are all irrational except two families, one of which is the product of a del Pezzo surface with ${\mathbb{P}}^1$. We show that the other model is ${\mathtt{PSL}}_2({\mathbb{F}}_7)$-equivariantly birational to ${\mathbb{P}}^2\times {\mathbb{P}}^1$. Based on a result of Prokhorov, we show that they are non-conjugate as subgroups of the Cremona group ${\mathtt{Cr}}_3(\mathbb{C})$.

arXiv commentThe results and content of the older version has been sharpened. Some parts have been published separately, and this is an update for section 3 in V1. Final version, to appear in Glasgow Math Journal