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Finite quasisimple groups acting on rationally connected threefolds

Part of: Surfaces and higher-dimensional varieties Birational geometry Algebraic groups Representation theory of groups Permutation groups

Published online by Cambridge University Press:  06 December 2022

JÉRÉMY BLANC ,
IVAN CHELTSOV ,
ALEXANDER DUNCAN  and
YURI PROKHOROV
JÉRÉMY BLANC
Affiliation:
Departement Mathematik und Informatik Universität Basel, Spiegelgasse 1, 4051 Basel, Switzerland. e-mail: jeremy.blanc@unibas.ch
IVAN CHELTSOV
Affiliation:
School of Mathematics, The University of Edinburgh, Edinburgh EH9 and 3JZ. e-mail: I.Cheltsov@ed.ac.uk
ALEXANDER DUNCAN
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A. e-mail: duncan@math.sc.edu
YURI PROKHOROV
Affiliation:
Steklov Mathematical Institute of Russian Academy of Sciences 8 Gubkina str., Moscow 119991, Russia and AG Laboratory, National Research University Higher School of Economics, 6 Usacheva str., Moscow, 119048, Russia. e-mail: prokhoro@mi-ras.ru
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Abstract

We show that the only finite quasi-simple non-abelian groups that can faithfully act on rationally connected threefolds are the following groups: ${\mathfrak{A}}_5$ , ${\text{PSL}}_2(\textbf{F}_7)$ , ${\mathfrak{A}}_6$ , ${\text{SL}}_2(\textbf{F}_8)$ , ${\mathfrak{A}}_7$ , ${\text{PSp}}_4(\textbf{F}_3)$ , ${\text{SL}}_2(\textbf{F}_{7})$ , $2.{\mathfrak{A}}_5$ , $2.{\mathfrak{A}}_6$ , $3.{\mathfrak{A}}_6$ or $6.{\mathfrak{A}}_6$ . All of these groups with a possible exception of $2.{\mathfrak{A}}_6$ and $6.{\mathfrak{A}}_6$ indeed act on some rationally connected threefolds.

MSC classification

Primary: 14E07: Birational automorphisms, Cremona group and generalizations 14L30: Group actions on varieties or schemes (quotients) 14J45: Fano varieties
Secondary: 20D05: Finite simple groups and their classification 20D99: None of the above, but in this section 14J30: $3$-folds 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 174 , Issue 3 , May 2023 , pp. 531 - 568
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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