Geodesic
On an Enriques Surface Associated With a Quartic Hessian Surface
Canadian journal of mathematics, Tome 71 (2019) no. 1, pp. 213-246

Voir la notice de l'article provenant de la source Cambridge University Press

Let $Y$ be a complex Enriques surface whose universal cover $X$ is birational to a general quartic Hessian surface. Using the result on the automorphism group of $X$ due to Dolgachev and Keum, we obtain a finite presentation of the automorphism group of $Y$. The list of elliptic fibrations on $Y$ and the list of combinations of rational double points that can appear on a surface birational to $Y$ are presented. As an application, a set of generators of the automorphism group of the generic Enriques surface is calculated explicitly.
DOI : 10.4153/CJM-2018-022-7
Mots-clés : Enriques surface, K3 surface, automorphism, lattice 
Shimada, Ichiro. On an Enriques Surface Associated With a Quartic Hessian Surface. Canadian journal of mathematics, Tome 71 (2019) no. 1, pp. 213-246. doi: 10.4153/CJM-2018-022-7
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[1] Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., and Wilson, R. A., Atlas of finite groups. Oxford University Press, Eynsham, 1985. Google Scholar

[2] Barth, W. and Peters, C., Automorphisms of Enriques surfaces . Invent. Math. 73(1983), no. 3, 383–411. . Google Scholar | DOI

[3] Borcherds, Richard, Automorphism groups of Lorentzian lattices . J. Algebra 111(1987), no. 1, 133–153. . Google Scholar | DOI

[4] Borcherds, Richard, Coxeter groups, Lorentzian lattices, and K3 surfaces . Internat. Math. Res. Notices(1998), no. 19, 1011–1031. . Google Scholar | DOI

[5] Conway, J. H., The automorphism group of the 26-dimensional even unimodular Lorentzian lattice . J. Algebra 80(1983), no. 1, 159–163. . Google Scholar | DOI

[6] Dardanelli, Elisa and Van Geemen, Bert, Hessians and the moduli space of cubic surfaces. In: Algebraic geometry. Contemp. Math., 422. Amer. Math. Soc., Providence, RI, 2007, pp. 17–36. . Google Scholar | DOI

[7] Dolgachev, Igor, Salem numbers and Enriques surfaces . Exp. Math. 27(2018), 287–301. . Google Scholar | DOI

[8] Dolgachev, Igor and Keum, Jonghae, Birational automorphisms of quartic Hessian surfaces . Trans. Amer. Math. Soc. 354(2002), no. 8, 3031–3057. . Google Scholar | DOI

[9] Ebeling, Wolfgang, Lattices and codes. Advanced Lectures in Mathematics. Third edition. Springer Spektrum, Wiesbaden 2013. . Google Scholar | DOI

[10] The GAP Group, GAP - Groups, Algorithms, and Programming. Version 4.7.9; 2015 http://www.gap-system.org. Google Scholar

[11] Katsura, Toshiyuki, Kondo, Shigeyuki, and Shimada, Ichiro, On the supersingular K3 surface in characteristic 5 with Artin invariant 1 . Michigan Math. J. 63(2014), no. 4, 803–844. . Google Scholar | DOI

[12] Keum, Jonghae, Every algebraic Kummer surface is the K3-cover of an Enriques surface . Nagoya Math. J. 118(1990), 99–110. . Google Scholar | DOI

[13] Koike, Kenji, Hessian K3 surfaces of non-Sylvester type . J. Algebra 330(2011), 388–403. . Google Scholar | DOI

[14] Kondō, Shigeyuki, Enriques surfaces with finite automorphism groups . Japan. J. Math. (N.S.) 12(1986), no. 2, 191–282. . Google Scholar | DOI

[15] Kondō, Shigeyuki, The automorphism group of a generic Jacobian Kummer surface . J. Algebraic Geom. 7(1998), no. 3, 589–609. Google Scholar

[16] Kondō, Shigeyuki, The moduli space of Hessian quartic surfaces and automorphic forms . J. Pure Appl. Algebra 216(2012), no. 10, 2233–2240. . Google Scholar | DOI

[17] Magnus, Wilhelm, Karrass, Abraham, and Solitar, Donald, Combinatorial group theory. Second edition. Dover Publications, Mineola, NY, 2004. Google Scholar

[18] Mcmullen, Curtis T., Automorphisms of projective K3 surfaces with minimum entropy . Invent. Math. 203(2016), no. 1, 179–215. Google Scholar

[19] Mukai, Shigeru, Numerically trivial involutions of Kummer type of an Enriques surface . Kyoto J. Math. 50(2010), no. 4, 889–902. . Google Scholar | DOI

[20] Mukai, Shigeru and Namikawa, Yukihiko, Automorphisms of Enriques surfaces which act trivially on the cohomology groups . Invent. Math. 77(1984), no. 3, 383–397. . Google Scholar | DOI

[21] Mukai, S. and Ohashi, H., The automorphism groups of Enriques surfaces covered by symmetric quartic surfaces . In: Recent advances in algebraic geometry. London Math. Soc. Lecture Note Ser., 417. Cambridge Univ. Press, Cambridge, 2015, pp. 307–320. Google Scholar

[22] Nikulin, V. V., Integer symmetric bilinear forms and some of their geometric applications . Izv. Akad. Nauk SSSR Ser. Mat. 43(1979), no. 1, 111–177, 238. Google Scholar

[23] Nikulin, V. V., Description of automorphism groups of Enriques surfaces . Dokl. Akad. Nauk SSSR 277(1984), no. 6, 1324–1327. Google Scholar

[24] Oguiso, Keiji, The third smallest Salem number in automorphisms of K3 surfaces. Adv. Stud. Pure Math., 60. Math. Soc. Japan, Tokyo, 2010, pp. 331–360. . Google Scholar | DOI

[25] Piatetski-Shapiro, I. I. and Shafarevich, I. R., Torelli’s theorem for algebraic surfaces of type K3 . Izv. Akad. Nauk SSSR Ser. Mat. 35(1971), 530–572. Reprinted in I. R. Shafarevich, Collected Mathematical Papers, Springer-Verlag, Berlin, 1989, pp. 516–557. Google Scholar

[26] Shimada, Ichiro, Projective models of the supersingular K3 surface with Artin invariant 1 in characteristic 5 . J. Algebra 403(2014), 273–299. . Google Scholar | DOI

[27] Shimada, Ichiro, An algorithm to compute automorphism groups of K3 surfaces and an application to singular K3 surfaces . Int. Math. Res. Not. IMRN(2015), no. 22, 11961–12014. Google Scholar

[28] Shimada, Ichiro, Holes of the Leech lattice and the projective models of K3 surfaces . Math. Proc. Cambridge Philos. Soc. 163(2017), 125–143. . Google Scholar | DOI

[29] Shimada, Ichiro, The automorphism groups of certain singular K3 surfaces and an Enriques surface . In: K3 surfaces and their moduli. Progr. Math., 315. Birkhäuser/Springer, Basel, 2016, pp. 297–343. . Google Scholar | DOI

[30] Shimada, Ichiro, Rational double points on Enriques surfaces, 2017. arxiv:1710.01461. Google Scholar

[31] Shimada, Ichiro, On an Enriques surface associated with a quartic Hessian surface: computational data, 2016. http://www.math.sci.hiroshima-u.ac.jp/∼shimada/K3andEnriques.html. Google Scholar

[32] Vinberg, È. B., Some arithmetical discrete groups in Lobačevskiıi spaces . In: Discrete subgroups of Lie groups and applications to moduli. Oxford Univ. Press, Bombay, 1975, pp. 323–348. Google Scholar

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