Modular quotient varieties and singularities by the cyclic group of order \(2p\). (English) Zbl 1443.13005
Summary: We classify all \(n\)-dimensional reduced Cohen-Macaulay modular quotient varieties \(A^n_\mathbb{F} / C_{2p}\) and study their singularities, where \(p\) is a prime number and \(C_{2p}\) denotes the cyclic group of order \(2p\). In particular, we present an example that demonstrates that the problem proposed by T. Yasuda [“Open problems in the wild McKay correspondence”, Preprint, https://www.math.tohoku.ac.jp/yasuda/notes.html] has a negative answer if the condition that “\(G\) is a small subgroup” was dropped.
MSC:
| 13A50 | Actions of groups on commutative rings; invariant theory |
| 13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |
References:
| [1] | Alperin, J. L. (1986). Local representation theory. Modular representations as an introduction to the local representation theory of finite groups. Cambridge Studies in Advanced Mathematics, Vol. 11. Cambridge: Cambridge University Press. x+178 pp. · Zbl 0593.20003 |
| [2] | Batyrev, V. V., Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs, J. Eur. Math. Soc, 1, 1, 5-33 (1999) · Zbl 0943.14004 · doi:10.1007/PL00011158 |
| [3] | Batyrev, V. V.; Dais, D. I., Strong McKay correspondence, string-theoretic Hodge numbers and Mirror symmetry, Topology., 35, 4, 901-929 (1996) · Zbl 0864.14022 · doi:10.1016/0040-9383(95)00051-8 |
| [4] | Bertin, M., Anneaux d’invariants d’anneaux de polynomes en caractéristique p, C. R. Acad. Sci. Paris Sér. A-B, 264, 653-656 (1967) · Zbl 0147.29503 |
| [5] | Braun, A., On the Gorenstein property for modular invariants, J. Algebra., 345, 1, 81-99 (2011) · Zbl 1243.13003 · doi:10.1016/j.jalgebra.2011.07.030 |
| [6] | Campbell, H. E. A.; Hughes, I.; Kemper, G.; Shank, R. J.; Wehlau, D. L., Depth of modular invariant rings, Transform. Groups., 5, 1, 21-34 (2000) · Zbl 0961.13003 · doi:10.1007/BF01237176 |
| [7] | Campbell, H. E. A., Wehlau, D. L. (2011). Modular invariant theory. EMS. Invariant Theory and Algebraic Transformation Groups, 8, Vol. 139. Berlin: Springer-Verlag. xiv+233 pp. · Zbl 1216.14001 |
| [8] | Chen, Y., On modular invariants of a vector and a covector, Manuscripta Math, 144, 3-4, 341-348 (2014) · Zbl 1303.13008 · doi:10.1007/s00229-013-0648-4 |
| [9] | Chen, Y.; Tang, Z., Vector invariant fields of finite classical groups, J. Algebra, 534, 129-144 (2019) · Zbl 1429.13009 · doi:10.1016/j.jalgebra.2019.05.032 |
| [10] | Chen, Y.; Wehlau, D. L., Modular invariants of a vector and a covector: a proof of a conjecture of Bonnafe and Kemper, J. Algebra, 472, 195-213 (2017) · Zbl 1358.13010 · doi:10.1016/j.jalgebra.2016.09.029 |
| [11] | Chen, Y.; Wehlau, D. L., On invariant fields of vectors and covectors, J. Pure Appl. Algebra., 223, 5, 2246-2257 (2019) · Zbl 1408.13016 · doi:10.1016/j.jpaa.2018.07.015 |
| [12] | Chevalley, C., Invariants of finite groups generated by reflections, Amer. J. Math, 77, 4, 778-782 (1955) · Zbl 0065.26103 · doi:10.2307/2372597 |
| [13] | Denef, J.; Loeser, F., Motivic integration, quotient singularities and the McKay correspondence, Compotisio Math, 131, 3, 267-290 (2002) · Zbl 1080.14001 · doi:10.1023/A:1015565912485 |
| [14] | Derksen, H., Kemper, G. (2015). Computational Invariant Theory. Invariant Theory and Algebraic Transformation Groups, I. EMS, Vol. 130., 2nd ed. Berlin: Springer-Verlag. x+268 pp. · Zbl 1332.13001 |
| [15] | Ellingsrud, G.; Skjelbred, T., Profondeur d’anneaux d’invariants en caractéristique p, Compositio Math, 41, 233-244 (1980) · Zbl 0438.13007 |
| [16] | Gonzalez-Sprinberg, G.; Verdier, J.-L., On McKay’s rule in positive characteristic, C. R. Acad. Sci. Paris Sér. I Math, 301, 585-587 (1985) · Zbl 0589.14003 |
| [17] | Harris, J. C.; Wehlau, D. L., Resolutions of 2 and 3 dimensional rings of invariants for cyclic groups, Comm. Algebra., 41, 11, 4278-4289 (2013) · Zbl 1285.13008 · doi:10.1080/00927872.2012.695834 |
| [18] | Hartmann, A., The quotient map on the equivariant Grothendieck ring of varieties, Manuscripta Math, 151, 3-4, 419-451 (2016) · Zbl 1408.14149 · doi:10.1007/s00229-016-0842-2 |
| [19] | Hochster, M.; Eagon, J. A., Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci, Am. J. Math, 93, 4, 1020-1058 (1971) · Zbl 0244.13012 · doi:10.2307/2373744 |
| [20] | Kemper, G., Calculating invariant rings of finite groups over arbitrary fields, J. Symbolic Comput, 21, 3, 351-366 (1996) · Zbl 0889.13004 · doi:10.1006/jsco.1996.0017 |
| [21] | Kemper, G., On the Cohen-Macaulay property of modular invariant rings, J. Algebra., 215, 1, 330-351 (1999) · Zbl 0934.13003 · doi:10.1006/jabr.1998.7716 |
| [22] | Neusel, M. D., Smith, L. (2002). Invariant Theory of Finite Groups. MSM, Vol. 94. Providence, RI: American Mathematical Society. viii+371. · Zbl 0999.13002 |
| [23] | Reid, M., La correspondance de Mckay, Séminaire Bourbaki, 42, 867, 53-72 (1999) · Zbl 0996.14006 |
| [24] | Richman, D. R., On vector invariants over finite fields, Adv. Math, 81, 1, 30-65 (1990) · Zbl 0715.13002 · doi:10.1016/0001-8708(90)90003-6 |
| [25] | Schröer, S., The Hilbert scheme of points for supersingular abelian surfaces, Ark. Mat, 47, 1, 143-181 (2009) · Zbl 1190.14032 · doi:10.1007/s11512-007-0065-6 |
| [26] | Serre, J.-P. (1968). Groupes finis d’automorphismes d’anneaux locaux réguliers. Colloque D’Algèbre (1967), Secrétariat mathématique, Exp. 8. (French) · Zbl 0200.00002 |
| [27] | Shephard, G. C.; Todd, J. A., Finite unitary reflection groups, Can. J. Math, 6, 274-304 (1954) · Zbl 0055.14305 · doi:10.4153/CJM-1954-028-3 |
| [28] | Smith, L., Some rings of invariants that are Cohen-Macaulay, Can. Math. Bull, 39, 2, 238-240 (1996) · Zbl 0868.13006 · doi:10.4153/CMB-1996-030-2 |
| [29] | Watanabe, K., Certain invariant subrings are Gorenstein I, Osaka J. Math, 11, 1-8 (1974) · Zbl 0281.13007 |
| [30] | Watanabe, K., Certain invariant subrings are Gorenstein II, Osaka J. Math, 11, 379-388 (1974) · Zbl 0292.13008 |
| [31] | Yasuda, T., The p-cyclic McKay correspondence via motivic integration, Compositio Math, 150, 7, 1125-1168 (2014) · Zbl 1310.14023 · doi:10.1112/S0010437X13007781 |
| [32] | Yasuda, T. (2015) |
| [33] | Yasuda, T., Toward motivic integration over wild Deligne-Mumford stacks. Proceedings of the conference “Higher Dimensional Algebraic Geometry - in honour of Yujiro Kawamata’s sixtieth birthday”, Adv. Stud. Pure Math, 74, 407-437 (2017) · Zbl 1388.14053 |
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