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Holes of the Leech lattice and the projective models of K3 surfaces

Published online by Cambridge University Press:  09 September 2016

ICHIRO SHIMADA
ICHIRO SHIMADA*
Affiliation:
Department of Mathematics, Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526Japan e-mail: [email protected]
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Abstract

Using the theory of holes of the Leech lattice and Borcherds method for the computation of the automorphism group of a K3 surface, we give an effective bound for the set of isomorphism classes of projective models of fixed degree for certain K3 surfaces.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 163 , Issue 1 , July 2017 , pp. 125 - 143
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

REFERENCES

[1] Borcherds, R. Automorphism groups of Lorentzian lattices. J. Algebra 111 (1) (1987), 133–153.Google Scholar
[2] Borcherds, R. E. Coxeter groups, Lorentzian lattices and K3 surfaces. Int. Math. Res. Not. 1998 (19) (1998), 1011–1031.Google Scholar
[3] Conway, J. H. The automorphism group of the 26-dimensional even unimodular Lorentzian lattice. J. Algebra 80 (1) (1983), 159–163.CrossRefGoogle Scholar
[4] Conway, J. H. and Sloane, N. J. A. Sphere packings, lattices and groups. Grundlehren der Mathematischen Wissenschaften, vol. 290 (Springer-Verlag, New York, third edition, 1999).Google Scholar
[5] Dolgachev, I. and Kondō, S. A supersingular K3 surface in characteristic 2 and the Leech lattice. Int. Math. Res. Not. 2003 (1) (2003), 1–23.CrossRefGoogle Scholar
[6] Dolgachev, I. and Keum, J. Birational automorphisms of quartic Hessian surfaces. Trans. Amer. Math. Soc. 354 (8) (2002), 3031–3057 (electronic).Google Scholar
[7] The GAP Group. GAP - Groups, Algorithms and Programming. Version 4.7.9 (2015), (http://www.gap-system.org).Google Scholar
[8] Keum, J. and Kondō, S. The automorphism groups of Kummer surfaces associated with the product of two elliptic curves. Trans. Amer. Math. Soc. 353 (4) (2001), 1469–1487 (electronic).Google Scholar
[9] Knuth, D. E. Estimating the efficiency of backtrack programs. Math. Comp. 29 (1975), 122–136. Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday.CrossRefGoogle Scholar
[10] Kondō, S. The automorphism group of a generic Jacobian Kummer surface. J. Algebraic Geom. 7 (3) (1998), 589–609.Google Scholar
[11] Kondō, S. and Shimada, I.. The automorphism group of a supersingular K3 surface with Artin invariant 1 in characteristic 3. Int. Math. Res. Not. IMRN. 2014 (7) (2014), 1885–1924.Google Scholar
[12] Kumar, A. Elliptic fibrations on a generic Jacobian Kummer surface. J. Algebraic Geom. 23 (4) (2014), 599–667.Google Scholar
[13] Kuwata, M. and Shioda, T. Elliptic parameters and defining equations for elliptic fibrations on a Kummer surface. In Algebraic geometry in East Asia–Hanoi 2005. Adv. Stud. Pure Math. vol. 50 (Math. Soc. Japan, Tokyo, 2008), pages 177–215.CrossRefGoogle Scholar
[14] Lieblich, M. and Maulik, D. A note on the cone conjecture for K3 surfaces in positive characteristic, 2011. preprint, arXiv:1102.3377.Google Scholar
[15] Nikulin, V. V. Integer symmetric bilinear forms and some of their geometric applications. Izv. Akad. Nauk SSSR Ser. Mat. 43 (1) 238 (1979), 111–177. English translation: Math USSR-Izv. 14 (1979), no. 1 (1980), 103–167.Google Scholar
[16] Nishiyama, K.-i.. The Jacobian fibrations on some K3 surfaces and their Mordell-Weil groups. Japan. J. Math. (N.S.) 22 (2) (1996), 293–347.CrossRefGoogle Scholar
[17] Oguiso, K. On Jacobian fibrations on the Kummer surfaces of the product of nonisogenous elliptic curves. J. Math. Soc. Japan. 41 (4) (1989), 651–680.Google Scholar
[18] Ogus, A. Supersingular K3 crystals. In Journées de Géométrie Algébrique de Rennes (Rennes, 1978), Vol. II Astérisque 64 (Soc. Math. France, Paris, 1979), pages 3–86.Google Scholar
[19] Ogus, A. A crystalline Torelli theorem for supersingular K3 surfaces. In Arithmetic and Geometry, Vol. II. Progr. Math. 36 (Birkhäuser Boston, Boston, MA, 1983), pages 361–394.Google Scholar
[20] 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 Shafarevich, I. R. Collected Mathematical Papers (Springer-Verlag, Berlin, 1989), pp. 516–557.Google Scholar
[21] Pless, V. On the uniqueness of the Golay codes. J. Combinatorial Theory 5 (1968), 215–228.CrossRefGoogle Scholar
[22] Sengupta, T. Supersingular K3 Surfaces. ProQuest LLC (Ann Arbor, MI, 2011). Ph.D. Thesis (Brandeis University, 2011).Google Scholar
[23] Shimada, I. An algorithm to compute automorphism groups of K3 surfaces and an application to singular K3 surfaces. Int. Math. Res. Not. IMRN 22 (2015), 11961–12014.Google Scholar
[24] Shimada, I. The list of holes of the Leech lattice (2016). http://www.math.sci.hiroshima-u.ac.jp/~shimada/Leech.html.Google Scholar
[25] Shimada, I. 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), pages 297–343.CrossRefGoogle Scholar
[26] Shioda, T. and Inose, H. On singular K3 surfaces. In Complex analysis and algebraic geometry (Iwanami Shoten, Tokyo, 1977), pages 119–136.Google Scholar
[27] Sterk, H. Finiteness results for algebraic K3 surfaces. Math. Z. 189 (4) (1985), 507–513.CrossRefGoogle Scholar
[28] Ujikawa, M. The automorphism group of the singular K3 surface of discriminant 7. Comment. Math. Univ. St. Pauli 62 (1) (2013), 11–29.Google Scholar
[29] Vinberg, È. B.. The two most algebraic K3 surfaces. Math. Ann. 265 (1) (1983), 1–21.CrossRefGoogle Scholar