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Sporadic simple groups and quotient singularities. (English. Russian original) Zbl 1282.14005

Izv. Math. 77, No. 4, 846-854 (2013); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 77, No. 4, 215-224 (2013).
The present paper is devoted to prove the following statement:
Let \(G\cong 6.\mathrm{Suz}\) be the universal perfect central extension of the Suzuki simple group and \(U\) be a \(12\)-dimensional irreducible representation of \(G\). Then the quotient singularity \(U/G\) is weakly exceptional but not exceptional, in the sense of [V. V. Shokurov, J. Math. Sci., New York 102, No. 2, 3876–3932 (2000; Zbl 1177.14078)] and [Yu. G. Prokhorov, Blow-ups of canonical singularities. A. G. Kurosh, Moscow, Russia, May 25-30, 1998. Berlin: Walter de Gruyter. 301–317 (2000; Zbl 1003.14005)].
As consequence of this theorem, the authors get the following classification result: let \(G\) be a sporadic group or a central extension of one with centre contained in the commutator subgroup and let \(G\hookrightarrow \mathrm{GL}(U)\) be a faithful finite-dimensional complex representation of \(G\). Then the singularity \(U/G\) is:
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exceptional if and only if \(G\cong 2.\mathrm{J}_2\) is a central extension of the Hall-Janko sporadic simple group and \(U\) is a \(6\)-dimensional irreducible representation of \(G\);
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weakly exceptional but not exceptional if and only if \(G\cong 6.\mathrm{Suz}\) and \(U\) is a \(12\)-dimensional irreducible representation of \(G\).
This result shows that among the sporadic simple groups, the groups \(\mathrm{J}_2\) and \(\mathrm{Suz}\) are somehow distinguished from a geometric point of view. This motivates the author to pose the following question: “Is there a group-theoretic property that distinguishes the groups \(\mathrm{J}_2\) and \(\mathrm{Suz}\) among the sporadic simple groups?”

MSC:

14B05 Singularities in algebraic geometry
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
14J17 Singularities of surfaces or higher-dimensional varieties
14E30 Minimal model program (Mori theory, extremal rays)

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References:

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