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A One-Dimensional Family of K3 Surfaces with a ℤ4 Action

Published online by Cambridge University Press:  20 November 2018

Michela Artebani
Michela Artebani*
Affiliation:
Departamento de Matemática, Universidad de Concepción, Concepción, Chile e-mail: martebani@udec.cl
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Abstract

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The minimal resolution of the degree four cyclic cover of the plane branched along a GIT stable quartic is a $K3$ surface with a non symplectic action of ${{\mathbb{Z}}_{4}}$. In this paper we study the geometry of the one-dimensional family of $K3$ surfaces associated to the locus of plane quartics with five nodes.

Keywords

14J2814J5014J10genus three curvesK3 surfaces
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 52 , Issue 4 , 01 December 2009 , pp. 493 - 510
Copyright
Copyright © Canadian Mathematical Society 2009

References

[1] Artebani, M., A compactification of via K3 surfaces. http://arxiv.org/abs/math/0511031.Google Scholar
[2] Artebani, M., Heegner divisors in the moduli space of genus three curves. Trans. Amer. Math. Soc. 360(2008), no. 3, 1581–1599.Google Scholar
[3] Cassels, J. W. S., Lectures on elliptic curves. London Mathematical Society Student Texts 24, Cambridge University Press, Cambridge, 1991.Google Scholar
[4] Cassels, J. W. S. and Flynn, E. V., Prolegomena to a middlebrow arithmetic of curves of genus 2. London Mathematical Society Lecture Note Series 230, Cambridge University Press, Cambridge, 1996.Google Scholar
[5] Dolgachev, I. V., Mirror symmetry for lattice polarized K3 surfaces. J. Math. Sci. 81(1996), no. 3, 2599–2630.Google Scholar
[6] Dolgachev, I. V. and Kondō, S., Moduli of K3 surfaces and complex ball quotients. In: Arithmetic and geometry around hypergeometric functions, Progr. Math. 260, Birkhäuser, Basel, 2007, pp. 43–100.Google Scholar
[7] Keum, J., Oguiso, K., and Zhang, D., The alternating group of degree 6 in the geometry of the Leech lattice and K3 surfaces. Proc. London Math. Soc. (3) 90(2005), no. 2, 371–394.Google Scholar
[8] Keum, J. H. and Kondō, S., The automorphism groups of Kummer surfaces associated with the product of two elliptic curves. Trans. Amer. Math. Soc. 353(2001), no. 4, 1469–1487.Google Scholar
[9] Kondō, S., A complex hyperbolic structure for the moduli space of curves of genus three. J. Reine Angew. Math. 525(2000), 219–232.Google Scholar
[10] Kulikov, V., Degenerations of K3 surfaces and Enriques surfaces. Math. USSR Izv. 11(1977), 957–989.Google Scholar
[11] Mumford, D., Fogarty, J., and Kirwan, F., Geometric invariant theory. Third edition, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) 34, Springer-Verlag, Berlin, 1994.Google Scholar
[12] Nikulin, V. V., Finite groups of automorphisms of Kählerian K3 surfaces. Trudy Moskov. Mat. Obshch. 38(1979), 75–137.Google Scholar
[13] Nikulin, V. V., Factor groups of groups of the automorphisms of hyperbolic forms with respect to subgroups generated by 2-reflections. J. Soviet Math. 22(1983), 1401–1475.Google Scholar
[14] Nikulin, V. V., Integral symmetric bilinear forms and its applications. Math. USSR Izv. 14(1980), 103–167.Google Scholar
[15] Nishiyama, K., The Jacobian fibrations on some K3 surfaces and their Mordell-Weil groups. Japan J. Math. 22(1996), no. 2, 293–347.Google Scholar
[16] Oguiso, K., A characterization of the Fermat quartic K3 surface by means of finite symmetries. Compos. Math. 141(2005), no. 2, 404–424.Google Scholar
[17] Oguiso, K. and Zhang, D., The simple group of order 168 and K3 surfaces. In: Complex geometry (Göttingen, 2000), Springer, Berlin, 2002, pp. 165–184.Google Scholar
[18] Shah, J., Degenerations of K3 surfaces of degree 4 . Trans. Amer. Math. Soc. 263(1981), no. 2, 271–308.Google Scholar
[19] Shioda, T., On elliptic modular surfaces. J. Math. Soc. Japan 24(1972), 20–59.Google Scholar
[20] Shioda, T. and Inose, H., On singular K3 surfaces. In: Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 119–136.Google Scholar
[21] Shioda, T. and Mitani, N., Singular abelian surfaces and binary quadratic forms. In: Classification of algebraic varieties and compact complex manifolds, Lecture Notes in Math. 412, Springer, Berlin, 1974, pp. 259–287.Google Scholar
[22] Silverman, J. H. and Tate, J., Rational points on elliptic curves. Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992.Google Scholar
[23] van Geemen, B., Kuga-Satake varieties and the Hodge conjecture. In: The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), NATO Sci. Ser. C Math. Phys. Sci. 548, Kluwer Acad. Publ., Dordrecht, 2000, pp. 51–82.Google Scholar
[24] Vinberg, È. B., The two most algebraic K3 surfaces. Math. Ann. 265(1983), no. 1, 1–21.Google Scholar