Geodesic
Three-dimensional isolated quotient singularities in odd characteristic
Sbornik. Mathematics, Tome 207 (2016) no. 6, pp. 873-887 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let a finite group $G$ act linearly on a finite-dimensional vector space $V$ over an algebraically closed field $k$ of characteristic $p>2$. Suppose that the quotient space $V/G$ has an isolated singularity only. The isolated singularities of the form $V/G$ are completely classified in the case when $p$ does not divide the order of $G$, and their classification reduces to Vincent's classification of isolated quotient singularities over $\mathbb C$. In the present paper we show that, if $\dim V=3$, then the classification of isolated quotient singularities reduces to Vincent's classification in the modular case as well (when $p$ divides $|G|$). Some remarks on quotient singularities in other dimensions and in even characteristic are also given. Bibliography: 14 titles.
Keywords: quotient singularity, modular representation, pseudo-reflection
Mots-clés : transvection. 
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     title = {Three-dimensional isolated quotient singularities in odd characteristic},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2016_207_6_a5/}
}
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D. A. Stepanov. Three-dimensional isolated quotient singularities in odd characteristic. Sbornik. Mathematics, Tome 207 (2016) no. 6, pp. 873-887. http://geodesic.mathdoc.fr/item/SM_2016_207_6_a5/

[1] J. A. Wolf, Spaces of constant curvature, 3rd ed., Publish or Perish, Boston, MA, 1972, xv+408 pp. | MR | MR | Zbl | Zbl

[2] D. A. Stepanov, “Gorenstein isolated quotient singularities over $\mathbb C$”, Proc. Edinb. Math. Soc. (2), 57:3 (2014), 811–839 | DOI | MR | Zbl

[3] D. Lorenzini, “Wild quotient singularities of surfaces”, Math. Z., 275:1-2 (2013), 211–232 | DOI | MR | Zbl

[4] D. J. Benson, Polynomial invariants of finite groups, London Math. Soc. Lecture Note Ser., 190, Cambridge Univ. Press, Cambridge, 1993, x+118 pp. | DOI | MR | Zbl

[5] G. Kemper, G. Malle, “The finite irreducible linear groups with polynomial ring of invariants”, Transform. Groups, 2:1 (1997), 57–89 | DOI | MR | Zbl

[6] H. Nakajima, “Invariants of finite groups generated by pseudoreflections in positive characteristic”, Tsukuba J. Math., 3:1 (1979), 109–122 | MR | Zbl

[7] G. Kemper, “Loci in quotients by finite groups, pointwise stabilizers, and the Buchsbaum property”, J. Reine Angew. Math., 2002:547 (2002), 69–96 | DOI | MR | Zbl

[8] M. F. Atiyah, I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley, Reading–London-Don Mills, 1969, ix+128 pp. | MR | MR | Zbl | Zbl

[9] Ch. W. Curtis, I. Reiner, Representation theory of finite groups and associative algebras, Pure Appl. Math., 11, Interscience Publishers (John Wiley Sons), New York–London, 1962, xiv+685 pp. | MR | MR | Zbl

[10] H. H. Mitchell, “Determination of the ordinary and modular ternary linear groups”, Trans. Amer. Math. Soc., 12:2 (1911), 207–272 | DOI | MR | Zbl

[11] E. Cline, B. Parshall, L. Scott, “Cohomology of finite groups of Lie type. I”, Inst. Hautes Études Sci. Publ. Math., 1975, no. 45, 169–191 | DOI | MR | Zbl

[12] Chih-Han Sah, “Cohomology of split group extensions. II”, J. Algebra, 45:1 (1977), 17–68 | DOI | MR | Zbl

[13] R. M. Guralnik, “Small representations are completely reducible”, J. Algebra, 220:2 (1999), 531–541 | DOI | MR | Zbl

[14] H. E. A. E. Campbell, D. L. Whelau, Modular invariant theory, Encyclopaedia Math. Sci., 139, Springer-Verlag, Berlin, 2011, xiv+233 pp. | DOI | MR | Zbl