$\mu _p$- and $\alpha _p$-actions on K3 surfaces in characteristic $p$
Author:
Yuya Matsumoto
Journal:
J. Algebraic Geom. 32 (2023), 271-322
DOI:
https://doi.org/10.1090/jag/804
Published electronically:
August 4, 2022
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Abstract |
References |
Additional Information
Abstract: We consider $\mu _p$- and $\alpha _p$-actions on RDP K3 surfaces (K3 surfaces with rational double point (RDP) singularities allowed) in characteristic $p > 0$. We study possible characteristics, quotient surfaces, and quotient singularities. It turns out that these properties of $\mu _p$- and $\alpha _p$-actions are analogous to those of $\mathbb {Z}/l\mathbb {Z}$-actions (for primes $l \neq p$) and $\mathbb {Z}/p\mathbb {Z}$-quotients respectively. We also show that conversely an RDP K3 surface with a certain configuration of singularities admits a $\mu _p$- or $\alpha _p$- or $\mathbb {Z}/p\mathbb {Z}$-covering by a “K3-like” surface, which is often an RDP K3 surface but not always, as in the case of the canonical coverings of Enriques surfaces in characteristic $2$.
References
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References
- M. Artin, Coverings of the rational double points in characteristic $p$, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 11–22. MR 0450263
- E. Bombieri and D. Mumford, Enriques’ classification of surfaces in char. $p$. III, Invent. Math. 35 (1976), 197–232. MR 491720, DOI 10.1007/BF01390138
- François R. Cossec and Igor V. Dolgachev, Enriques surfaces. I, Progress in Mathematics, vol. 76, Birkhäuser Boston, Inc., Boston, MA, 1989. MR 986969, DOI 10.1007/978-1-4612-3696-2
- I. Dolgachev and J. Keum, Wild $p$-cyclic actions on $K3$-surfaces, J. Algebraic Geom. 10 (2001), no. 1, 101–131. MR 1795552
- Igor V. Dolgachev and JongHae Keum, Finite groups of symplectic automorphisms of $K3$ surfaces in positive characteristic, Ann. of Math. (2) 169 (2009), no. 1, 269–313. MR 2480606, DOI 10.4007/annals.2009.169.269
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- T. Ekedahl, J. M. E. Hyland, and N. I. Shepherd-Barron, Moduli and periods of simply connected Enriques surfaces, arXiv:1210.0342 (2012).
- Toshiyuki Katsura, On Kummer surfaces in characteristic $2$, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) Kinokuniya Book Store, Tokyo, 1978, pp. 525–542. MR 578870
- Toshiyuki Katsura, Generalized Kummer surfaces and their unirationality in characteristic $p$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 1, 1–41. MR 882121
- Toshiyuki Katsura and Shigeyuki Kondō, A $1$-dimensional family of Enriques surfaces in characteristic $2$ covered by the supersingular $K3$ surface with Artin invariant $1$, Pure Appl. Math. Q. 11 (2015), no. 4, 683–709. MR 3613126, DOI 10.4310/PAMQ.2015.v11.n4.a6
- T. Katsura and Y. Takeda, Quotients of abelian and hyperelliptic surfaces by rational vector fields, J. Algebra 124 (1989), no. 2, 472–492. MR 1011608, DOI 10.1016/0021-8693(89)90144-0
- JongHae Keum, Orders of automorphisms of K3 surfaces, Adv. Math. 303 (2016), 39–87. MR 3552520, DOI 10.1016/j.aim.2016.08.014
- Christian Liedtke, Arithmetic moduli and lifting of Enriques surfaces, J. Reine Angew. Math. 706 (2015), 35–65. MR 3393362, DOI 10.1515/crelle-2013-0068
*18pt
- Joseph Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195–279. MR 276239
- Yuya Matsumoto, Canonical coverings of Enriques surfaces in characteristic $2$, J. Math. Soc. Japan 74 (2022), no. 3, 849–872., DOI 10.2969/jmsj/86318631
- Yuya Matsumoto, Inseparable maps on $W_n$-valued Ext groups of non-taut rational double point singularities and the height of $K3$ surfaces, J. Commut. Algebra (2021), to appear. arXiv:1907.04686v3.
- Yuya Matsumoto, $\mu _{n}$-actions on K3 surfaces in positive characteristic, Nagoya Math. J. (2022), to appear. arXiv:1710.07158v4.
- Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461
- Shigeru Mukai, Finite groups of automorphisms of $K3$ surfaces and the Mathieu group, Invent. Math. 94 (1988), no. 1, 183–221. MR 958597, DOI 10.1007/BF01394352
- V. V. Nikulin, Finite groups of automorphisms of Kählerian $K3$ surfaces, Trudy Moskov. Mat. Obshch. 38 (1979), 75–137 (Russian). MR 544937
- A. N. Rudakov and I. R. Šafarevič, Inseparable morphisms of algebraic surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 6, 1269–1307, 1439 (Russian). MR 0460344
- Stefan Schröer, Enriques surfaces with normal K3-like coverings, J. Math. Soc. Japan 73 (2021), no. 2, 433–496. MR 4255083, DOI 10.2969/jmsj/83728372
- Conjeerveram Srirangachari Seshadri, L’opération de Cartier. Applications, Séminaire C. Chevalley, 3ième année: 1958/59. Variétés de Picard, École Normale Supérieure, Paris, 1960, pp. 1–26 (French).
- Philippe Gille and Patrick Polo (eds.), Schémas en groupes (SGA 3). Tome I. Propriétés générales des schémas en groupes, Documents Mathématiques (Paris) [Mathematical Documents (Paris)], vol. 7, Société Mathématique de France, Paris, 2011 (French). Séminaire de Géométrie Algébrique du Bois Marie 1962–64 [Algebraic Geometry Seminar of Bois Marie 1962–64]; A seminar directed by M. Demazure and A. Grothendieck with the collaboration of M. Artin, J.-E. Bertin, P. Gabriel, M. Raynaud and J-P. Serre; Revised and annotated edition of the 1970 French original.
- Nikolaos Tziolas, Quotients of schemes by $\alpha _p$ or $\mu _p$ actions in characteristic $p>0$, Manuscripta Math. 152 (2017), no. 1-2, 247–279. MR 3595379, DOI 10.1007/s00229-016-0854-y
Additional Information
Yuya Matsumoto
Affiliation:
Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, 2641 Yamazaki, Noda, Chiba 278-8510, Japan
MR Author ID:
1079656
ORCID:
0000-0002-7371-7956
Email:
matsumoto.yuya.m@gmail.com, matsumoto_yuya@ma.noda.tus.ac.jp
Received by editor(s):
January 31, 2021
Received by editor(s) in revised form:
January 3, 2022
Published electronically:
August 4, 2022
Additional Notes:
This work was supported by JSPS KAKENHI Grant Numbers 15H05738, 16K17560, and 20K14296.
Article copyright:
© Copyright 2022
University Press, Inc.
