Transformation Groups

, Volume 17, Issue 2, pp 303–350 | Cite as

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

  • Ivan Cheltsov
  • Constantin Shramov


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.


General Surface Abelian Surface Irreducible Curve Canonical Singularity Minimal Center 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.University of EdinburghEdinburghUK
  2. 2.Steklov Mathematical InstituteMoscowRussia

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