Three embeddings of the Klein simple group into the Cremona group of rank three

Abstract

We study the action of the Klein simple group PSL2(\( {\mathbb{F}_7} \)) consisting of 168 elements on two rational threefolds: the three-dimensional projective space and a smooth Fano threefold X of anticanonical degree 22 and index 1. We show that the Cremona group of rank three has at least three non-conjugate subgroups isomorphic to PSL2(\( {\mathbb{F}_7} \)). As a by-product, we prove that X admits a Kähler–Einstein metric, and we construct a smooth polarized K3 surface of degree 22 with an action of the group PSL2(\( {\mathbb{F}_7} \)).Unless explicitly stated otherwise, varieties are assumed to be projective, normal and complex.