$\mu _p$- 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

Full-text PDF

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
- 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
- T. Ekedahl and N. I. Shepherd-Barron,
*On exceptional Enriques surfaces*, arXiv:0405510 (2004).
- 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 Kond\B{o},
*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
- Joseph Lipman,
*Rational singularities, with applications to algebraic surfaces and unique factorization*, Inst. Hautes Études Sci. Publ. Math. **36** (1969), 195–279. MR **276239**, DOI 10.1007/BF02684604
- 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

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
- T. Ekedahl and N. I. Shepherd-Barron,
*On exceptional Enriques surfaces*, arXiv:0405510 (2004).
- 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.