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

TL;DR

This paper investigates how the Klein simple group acts on certain rational threefolds, revealing multiple non-conjugate embeddings into the Cremona group of rank three, and explores related geometric structures.

Contribution

It introduces three distinct embeddings of the Klein group into the Cremona group of rank three and connects these to geometric properties of specific threefolds and K3 surfaces.

Findings

At least three non-conjugate subgroups of G in the Cremona group of rank three.

X admits a Kähler-Einstein metric.

Construction of a polarized K3 surface of degree 22 with G-action.

Abstract

We study the action of the Klein simple group G 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 G. As a by-product, we prove that X admits a Kahler-Einstein metric, and we construct a smooth polarized K3 surface of degree 22 with an action of the group G.