Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-25T06:11:52.788Z Has data issue: false hasContentIssue false
  • English
  • Français

THREE-DIMENSIONAL ISOLATED QUOTIENT SINGULARITIES IN EVEN CHARACTERISTIC

Part of: General commutative ring theory Algebraic groups

Published online by Cambridge University Press:  30 October 2017

VLADIMIR SHCHIGOLEV  and
DMITRY STEPANOV
VLADIMIR SHCHIGOLEV
Affiliation:
Financial University under the Government of the Russian Federation, 49 Leningradsky Prospekt, Moscow, Russia e-mail: shchigolev_vladimir@yahoo.com
DMITRY STEPANOV
Affiliation:
The Department of Mathematical Modelling Bauman Moscow State Technical University 2-ya Baumanskaya ul. 5, Moscow 105005, Russia e-mail: dstepanov@bmstu.ru
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper is a complement to the work of the second author on modular quotient singularities in odd characteristic. Here, we prove that if V is a three-dimensional vector space over a field of characteristic 2 and G < GL(V) is a finite subgroup generated by pseudoreflections and possessing a two-dimensional invariant subspace W such that the restriction of G to W is isomorphic to the group SL2(𝔽2n), then the quotient V/G is non-singular. This, together with earlier known results on modular quotient singularities, implies first that a theorem of Kemper and Malle on irreducible groups generated by pseudoreflections generalizes to reducible groups in dimension three, and, second, that the classification of three-dimensional isolated singularities that are quotients of a vector space by a linear finite group reduces to Vincent's classification of non-modular isolated quotient singularities.

MSC classification

Primary: 13A50: Actions of groups on commutative rings; invariant theory
Secondary: 14L30: Group actions on varieties or schemes (quotients)
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 60 , Issue 2 , May 2018 , pp. 435 - 445
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

References

REFERENCES

1. Benson, D. J., Polynomial invariants of finite groups, London Mathematical Society Lecture Note Series, vol. 190 (Cambridge University Press, Cambridge, UK, 1993).Google Scholar
2. Bonnafé, C., Representations of SL2(𝔽q), Algebra and Applications, vol. 13 (Springer Verlag, London, 2011).Google Scholar
3. Sah, C.-H., Cohomology of split group extensions, II, J. Algebra 45 (1977), 17–68.Google Scholar
4. Cline, E., Parshall, B. and Scott, L., Cohomology of finite groups of Lie type, I, Publ. Math. l'IHES 45 (1975), 169–191.CrossRefGoogle Scholar
5. Campbell, H. E. A. E. and Wehlau, D. L., Modular invariant theory, Encyclopaedia of Mathematical Sciences, vol. 139, Invariant Theory and Algebraic Transformation Groups VIII (subseries Gamkrelidze, R. V. and Popov V. L., Editors) (Springer, 2011), XIV, 234 p.Google Scholar
6. Kemper, G. and Malle, G., The finite irreducible linear groups with polynomial ring of invariants, Transformation Groups 2 (1) (1997), 57–89.CrossRefGoogle Scholar
7. Stepanov, D. A., Three-dimensional isolated quotient singularities in odd characteristic, Sbornik: Mathematics 207 (6) (2016), 873–887.Google Scholar