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Mathematica Slovaca

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Volume 63, Issue 2


On the computation of the Picard group for certain singular quartic surfaces

Andreas-Stephan Elsenhans / Jörg Jahnel
Published Online: 2013-03-28 | DOI: https://doi.org/10.2478/s12175-012-0094-x


We test the methods for computing the Picard group of a K3 surface in a situation of high rank. The examples chosen are resolutions of quartics in P 3 having 14 singularities of type A 1. Our computations show that the method of R. van Luijk works well when sufficiently large primes are used.

MSC: Primary 14J28; Secondary 14C22, 14J27

Keywords: K3 surface; singular quartic surface; Cayley-Rohn quartic; A1 singularity; Picard rank; van Luijk’s method

  • [1] ARTIN, M.— SWINNERTON-DYER, SIR PETER: The Shafarevich-Tate conjecture for pencils of elliptic curves on K3 surfaces, Invent. Math. 20 (1973), 249–266. http://dx.doi.org/10.1007/BF01394097CrossrefGoogle Scholar

  • [2] BEAUVILLE, A.: Complex Algebraic Surfaces. London Math. Soc. Lecture Note Ser. 68, Cambridge University Press, Cambridge, 1983. Google Scholar

  • [3] ELSENHANS, A.-S.— JAHNEL, J.: K3 surfaces of Picard rank one and degree two. In: Algorithmic Number Theory (ANTS 8). Lecture Notes in Comput. Sci. 5011, Springer, Berlin, 2008, pp. 212–225. http://dx.doi.org/10.1007/978-3-540-79456-1_14CrossrefGoogle Scholar

  • [4] ELSENHANS, A.-S.— JAHNEL, J.: K3 surfaces of Picard rank one which are double covers of the projective plane. In: The Higher-dimensional Geometry over Finite Fields, IOS Press, Amsterdam, 2008, pp. 63–77. Google Scholar

  • [5] ELSENHANS, A.-S.— JAHNEL, J.: On the computation of the Picard group for K3 surfaces, Math. Proc. Cambridge Philos. Soc. 151 (2011), 263–270. http://dx.doi.org/10.1017/S0305004111000326CrossrefGoogle Scholar

  • [6] ELSENHANS, A.-S.— JAHNEL, J.: On Weil polynomials of K3 surfaces, In: Algorithmic Number Theory (ANTS 9). Lecture Notes in Comput. Sci. 6197, Springer, Berlin, 2010, pp. 126–141. http://dx.doi.org/10.1007/978-3-642-14518-6_13CrossrefGoogle Scholar

  • [7] ELSENHANS, A.-S.— JAHNEL, J.: The Picard group of a K3 surface and its reduction modulo p, Algebra & Number Theory 5 (2011), 1027–1040. http://dx.doi.org/10.2140/ant.2011.5.1027CrossrefGoogle Scholar

  • [8] FULTON, W.: Intersection Theory, Springer, Berlin, 1984. Google Scholar

  • [9] GRIFFITHS, P.— HARRIS, J.: Principles of Algebraic Geometry, Wiley-Interscience, New York, 1978. Google Scholar

  • [10] ISHII, Y.— NAKAYAMA, N.: Classification of normal quartic surfaces with irrational singularities, J. Math. Soc. Japan 56 (2004), 941–965. http://dx.doi.org/10.2969/jmsj/1191334093CrossrefGoogle Scholar

  • [11] JESSOP, C. M.: Quartic Surfaces with Singular Points, Cambridge University Press, Cambridge, 1916. Google Scholar

  • [12] LIPMAN, J.: Rational singularities, with applications to algebraic surfaces and unique factorization, Publ. Math. Inst. Hautes Études Sci. 36 (1969), 195–279. http://dx.doi.org/10.1007/BF02684604CrossrefGoogle Scholar

  • [13] VAN LUIJK, R.: K3 surfaces with Picard number one and infinitely many rational points, Algebra & Number Theory 1 (2007), 1–15. http://dx.doi.org/10.2140/ant.2007.1.1CrossrefWeb of ScienceGoogle Scholar

  • [14] MILNE, J. S.: On a conjecture of Artin and Tate, Ann. of Math. 102 (1975), 517–533. http://dx.doi.org/10.2307/1971042CrossrefGoogle Scholar

  • [15] MILNE, J. S.: Étale Cohomology, Princeton University Press, Princeton, 1980. Google Scholar

  • [16] ROHN, K.: Die Flächen vierter Ordnung hinsichtlich ihrer Knotenpunkte und ihrer Gestaltung, Gekrönte Preisschrift, Leipzig, 1886. Google Scholar

About the article

Published Online: 2013-03-28

Published in Print: 2013-04-01

Citation Information: Mathematica Slovaca, Volume 63, Issue 2, Pages 215–228, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.2478/s12175-012-0094-x.

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© 2013 Mathematical Institute, Slovak Academy of Sciences. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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