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Koike, Kenji

On Jacobian Kummer surfaces. (English) Zbl 1318.14042

J. Math. Soc. Japan 66, No. 3, 997-1016 (2014).
It is well-known that the Kummer surface of the Jacobian of a curve of genus 2 is a complete intersection of 3 quadric in \(\mathbb{P}^5\). F. Klein [Clebsch Ann. II, 198–226 (1869); Gött. Nachr. 1869, 258–276 (1869; JFM 02.0605.01)] explicit equation for this in terms of the Weierstrass points of the curve.
The present paper gives explicit equations in terms of the 10 even theta constants of the corresponding principally polarized abelian surface. The method of proof consists in applying Riemann’s theta relations. Moreover, explicit equations are given for the total space of the fibration of Kummer surfaces over the Satake compactification of the Siegel modular threefold of level \((2,4)\). This can be regarded as a higher-dimensional analogue of Shioda’s elliptic modular surfaces.
Reviewer: Herbert Lange (Erlangen)

MSC:

14K25 Theta functions and abelian varieties
14J28 \(K3\) surfaces and Enriques surfaces

Keywords:

theta functions; Kummer surfaces

Citations:

JFM 02.0605.01
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Full Text: DOI arXiv Euclid

References:

[1] W. Barth and K. Hulek, Projective models of Shioda modular surfaces, Manuscripta Math., 50 (1985), 73-132. · Zbl 0599.14035 · doi:10.1007/BF01168828
[2] Ch. Birkenhake and H. Lange, Complex Abelian Varieties (2nd ed.), Grundlehren Math. Wiss., 302 , Springer-Verlag, Berlin, 2004. · Zbl 1056.14063
[3] A. B. Coble, Algebraic Geometry and Theta Functions, Amer. Math. Soc. Colloq. Publ., 10 , Amer. Math. Soc., Providence, RI, 1929. · JFM 55.0808.02
[4] S. Cynk and S. Rams, On a map between two \(K3\) surfaces associated to a net of quadrics, Arch. Math. (Basel), 88 (2007), 109-122. · Zbl 1125.14021 · doi:10.1007/s00013-006-1721-y
[5] E. V. Flynn, The Jacobian and formal group of a curve of genus \(2\) over an arbitrary ground field, Math. Proc. Cambridge Philos. Soc., 107 (1990), 425-441. · Zbl 0723.14023 · doi:10.1017/S0305004100068729
[6] B. van Geemen and N. Nygaard, On the geometry and arithmetic of some Siegel modular threefolds, J. Number Theory, 53 (1995), 45-87. · Zbl 0838.11047 · doi:10.1006/jnth.1995.1078
[7] B. van Geemen and D. van Straten, The cusp forms of weight 3 on \(\Gamma_2(2,4,8)\), Math. Comp., 61 (1993), 849-872. · Zbl 0787.11018 · doi:10.2307/2153257
[8] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Pure Appl. Math. (Hoboken), Wiley, Hoboken, NJ, 1978. · Zbl 0408.14001
[9] R. Hudson, Kummer’s Quartic Surface, Cambridge Math. Lib., Cambridge University Press, Cambridge, 1990. · Zbl 0716.14025
[10] J. Igusa, On Siegel modular forms of genus two, Amer. J. Math., 84 (1962), 175-200. · Zbl 0133.33301 · doi:10.2307/2372812
[11] J. Igusa, On Siegel modular forms of genus two. II, Amer. J. Math., 86 (1964), 392-412. · Zbl 0133.33301 · doi:10.2307/2373172
[12] J. Igusa, Theta Functions, Grundlehren Math. Wiss., 194 , Springer-Verlag, New York, Heidelberg, 1972. · Zbl 0251.14016
[13] F. Klein, Zur Theorie der Liniencomplexe des ersten und zweiten Grades, Math. Ann., 2 (1870), 198-226. · JFM 02.0605.01 · doi:10.1007/BF01444020
[14] A. Krazer, Lehrbuch der Thetafunktionen, Teubner, Leipzig, 1903. · JFM 34.0492.08
[15] V. Kulikov, Degenerations of \(K3\) surfaces and Enriques surfaces, Math. USSR Izv., 11 (1977), 957-989. · Zbl 0387.14007 · doi:10.1070/IM1977v011n05ABEH001753
[16] M. Kuwata and T. Shioda, Elliptic parameters and defining equations for elliptic fibrations on a Kummer surface, In: Algebraic Geometry in East Asia - Hanoi 2005, Adv. Stud. Pure Math., 50 , Math. Soc. Japan, Tokyo, 2008, pp.,177-215. · Zbl 1139.14032
[17] A. Kumar, Elliptic fibrations on a generic Jacobian Kummer surface, arXiv: · Zbl 1304.14045
[18] D. Mumford, Tata Lectures on Theta. I, Progr. Math., 28 , Birkhäuser, Boston, MA, 1983.
[19] M. V. Nori, Zariski’s conjecture and related problems, Ann. Sci. École Norm. Sup. (4), 16 (1983), 305-344. · Zbl 0527.14016
[20] U. Persson, On Degenerations of Algebraic Surfaces, Mem. Amer. Math. Soc., 11 , no.,189, Amer. Math. Soc., Providence, RI, 1977. · Zbl 0368.14008 · doi:10.1090/memo/0189
[21] R. Sasaki, Moduli space of hyperelliptic curves of genus two with level \((2,4)\) structure and the special orthogonal group of degree three, Kyushu J. Math., 53 (1999), 333-361. · Zbl 0967.14016 · doi:10.2206/kyushujm.53.333
[22] T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan, 24 (1972), 20-59. · Zbl 0226.14013 · doi:10.2969/jmsj/02410020
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