Enriques surfaces in characteristic $2$ with a finite group of automorphisms
Authors:
Toshiyuki Katsura and Shigeyuki Kondō
Journal:
J. Algebraic Geom. 27 (2018), 173-202
DOI:
https://doi.org/10.1090/jag/697
Published electronically:
August 30, 2017
MathSciNet review:
3722693
Full-text PDF
Abstract |
References |
Additional Information
Abstract: Complex Enriques surfaces with a finite group of automorphisms are classified into seven types. In this paper, we determine which types of such Enriques surfaces exist in characteristic 2. In particular we give a 1-dimensional family of classical and supersingular Enriques surfaces with the automorphism group $\mathrm {Aut}(X)$ isomorphic to the symmetric group $\mathfrak {S}_5$ of degree 5.
References
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- Shigeru Mukai and Yukihiko Namikawa, Automorphisms of Enriques surfaces which act trivially on the cohomology groups, Invent. Math. 77 (1984), no. 3, 383–397. MR 759266, DOI https://doi.org/10.1007/BF01388829
- S. Mukai and H. Ohashi, Finite groups of automorphisms of Enriques surfaces and the Mathieu group $M_{12}$, arXiv:1410.7535
- V. V. Nikulin, Description of automorphism groups of Enriques surfaces, Dokl. Akad. Nauk SSSR 277 (1984), no. 6, 1324–1327 (Russian). MR 760514
- 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
- P. Salomonsson, Equations for some very special Enriques surfaces in characteristic two, arXiv:math/0309210v1.
- Tetsuji Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972), 20–59. MR 429918, DOI https://doi.org/10.2969/jmsj/02410020
- Tetsuji Shioda, An example of unirational surfaces in characteristic $p$, Math. Ann. 211 (1974), 233–236. MR 374149, DOI https://doi.org/10.1007/BF01350715
- Tetsuji Shioda, Kummer surfaces in characteristic $2$, Proc. Japan Acad. 50 (1974), 718–722. MR 491728
- È. B. Vinberg, Some arithmetical discrete groups in Lobačevskiĭ spaces, Discrete subgroups of Lie groups and applications to moduli (Internat. Colloq., Bombay, 1973) Oxford Univ. Press, Bombay, 1975, pp. 323–348. MR 0422505
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
- M. Artin and B. Mazur, Formal groups arising from algebraic varieties, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 1, 87–131. MR 0457458
- Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven, Compact complex surfaces, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 4, Springer-Verlag, Berlin, 2004. MR 2030225
- W. Barth and C. Peters, Automorphisms of Enriques surfaces, Invent. Math. 73 (1983), no. 3, 383–411. MR 718937, DOI https://doi.org/10.1007/BF01388435
- E. Bombieri and D. Mumford, Enriques’ classification of surfaces in char. $p$. III, Invent. Math. 35 (1976), 197–232. MR 0491720, DOI https://doi.org/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
- Richard M. Crew, Etale $p$-covers in characteristic $p$, Compositio Math. 52 (1984), no. 1, 31–45. MR 742696
- Elisa Dardanelli and Bert van Geemen, Hessians and the moduli space of cubic surfaces, Algebraic geometry, Contemp. Math., vol. 422, Amer. Math. Soc., Providence, RI, 2007, pp. 17–36. MR 2296431, DOI https://doi.org/10.1090/conm/422/08054
- I. Dolgachev, On automorphisms of Enriques surfaces, Invent. Math. 76 (1984), no. 1, 163–177. MR 739632, DOI https://doi.org/10.1007/BF01388499
- Igor V. Dolgachev, Numerical trivial automorphisms of Enriques surfaces in arbitrary characteristic, Arithmetic and geometry of K3 surfaces and Calabi-Yau threefolds, Fields Inst. Commun., vol. 67, Springer, New York, 2013, pp. 267–283. MR 3156418, DOI https://doi.org/10.1007/978-1-4614-6403-7_8
- I. Dolgachev and C. Liedtke, Enriques surfaces, manuscript, November 2015.
- T. Ekedahl and N. I. Shepherd-Barron, On exceptional Enriques surfaces, arXiv: math/0405510v1.
- T. Ekedahl, J. M. E. Hyland and N. I. Shepherd-Barron, Moduli and periods of simply connected Enriques surfaces, arXiv:1210.0342.
- G. Fano, Superficie algebriche di genere zero e bigenere uno e loro casi particolari, Rend. Circ. Mat. Palermo 29 (1910), 98–118.
- Hiroyuki Ito, On extremal elliptic surfaces in characteristic 2 and 3, Hiroshima Math. J. 32 (2002), no. 2, 179–188. MR 1925896
- Hiroyuki Ito and Christian Liedtke, Elliptic K3 surfaces with $p^n$-torsion sections, J. Algebraic Geom. 22 (2013), no. 1, 105–139. MR 2993049, DOI https://doi.org/10.1090/S1056-3911-2012-00584-0
- 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 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 https://doi.org/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 https://doi.org/10.1016/0021-8693%2889%2990144-0
- Toshiyuki Katsura and Kenji Ueno, On elliptic surfaces in characteristic $p$, Math. Ann. 272 (1985), no. 3, 291–330. MR 799664, DOI https://doi.org/10.1007/BF01455561
- Shigeyuki Kondō, Enriques surfaces with finite automorphism groups, Japan. J. Math. (N.S.) 12 (1986), no. 2, 191–282. MR 914299
- William E. Lang, Extremal rational elliptic surfaces in characteristic $p$. I. Beauville surfaces, Math. Z. 207 (1991), no. 3, 429–437. MR 1115175, DOI https://doi.org/10.1007/BF02571400
- William E. Lang, Extremal rational elliptic surfaces in characteristic $p$. II. Surfaces with three or fewer singular fibres, Ark. Mat. 32 (1994), no. 2, 423–448. MR 1318540, DOI https://doi.org/10.1007/BF02559579
- Christian Liedtke, Arithmetic moduli and lifting of Enriques surfaces, J. Reine Angew. Math. 706 (2015), 35–65. MR 3393362, DOI https://doi.org/10.1515/crelle-2013-0068
- Qing Liu, Dino Lorenzini, and Michel Raynaud, Néron models, Lie algebras, and reduction of curves of genus one, Invent. Math. 157 (2004), no. 3, 455–518. MR 2092767, DOI https://doi.org/10.1007/s00222-004-0342-y
- Shigeru Mukai, Numerically trivial involutions of Kummer type of an Enriques surface, Kyoto J. Math. 50 (2010), no. 4, 889–902. MR 2740697, DOI https://doi.org/10.1215/0023608X-2010-017
- Shigeru Mukai and Yukihiko Namikawa, Automorphisms of Enriques surfaces which act trivially on the cohomology groups, Invent. Math. 77 (1984), no. 3, 383–397. MR 759266, DOI https://doi.org/10.1007/BF01388829
- S. Mukai and H. Ohashi, Finite groups of automorphisms of Enriques surfaces and the Mathieu group $M_{12}$, arXiv:1410.7535
- V. V. Nikulin, Description of automorphism groups of Enriques surfaces, Dokl. Akad. Nauk SSSR 277 (1984), no. 6, 1324–1327 (Russian). MR 760514
- 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
- P. Salomonsson, Equations for some very special Enriques surfaces in characteristic two, arXiv:math/0309210v1.
- Tetsuji Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972), 20–59. MR 0429918, DOI https://doi.org/10.2969/jmsj/02410020
- Tetsuji Shioda, An example of unirational surfaces in characteristic $p$, Math. Ann. 211 (1974), 233–236. MR 0374149, DOI https://doi.org/10.1007/BF01350715
- Tetsuji Shioda, Kummer surfaces in characteristic $2$, Proc. Japan Acad. 50 (1974), 718–722. MR 0491728
- È. B. Vinberg, Some arithmetical discrete groups in Lobačevskiĭ spaces, Discrete subgroups of Lie groups and applications to moduli (Internat. Colloq., Bombay, 1973) Oxford Univ. Press, Bombay, 1975, pp. 323–348. MR 0422505
Additional Information
Toshiyuki Katsura
Affiliation:
Faculty of Science and Engineering, Hosei University, Koganei-shi, Tokyo 184-8584, Japan
MR Author ID:
99175
Email:
toshiyuki.katsura.tk@hosei.ac.jp
Shigeyuki Kondō
Affiliation:
Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan
Email:
kondo@math.nagoya-u.ac.jp
Received by editor(s):
December 22, 2015
Received by editor(s) in revised form:
May 25, 2016
Published electronically:
August 30, 2017
Additional Notes:
The research of the first author was partially supported by Grant-in-Aid for Scientific Research (B) No. 15H03614, and the second author by (S) No. 15H05738.
Article copyright:
© Copyright 2017
University Press, Inc.