Elliptic fibrations on a generic Jacobian Kummer surface
Author:
Abhinav Kumar
Journal:
J. Algebraic Geom. 23 (2014), 599-667
DOI:
https://doi.org/10.1090/S1056-3911-2014-00620-2
Published electronically:
May 28, 2014
MathSciNet review:
3263663
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Abstract |
References |
Additional Information
Abstract: We describe all the elliptic fibrations with section on the Kummer surface $X$ of the Jacobian of a very general curve $C$ of genus $2$ over an algebraically closed field of characteristic 0, modulo the automorphism group of $X$ and the symmetric group on the Weierstrass points of $C$. In particular, we compute elliptic parameters and Weierstrass equations for the 25 different fibrations and analyze the reducible fibers and Mordell–Weil lattices. This answers completely a question posed by Kuwata and Shioda in 2008.
References
- Richard Borcherds, Automorphism groups of Lorentzian lattices, J. Algebra 111 (1987), no. 1, 133–153. MR 913200, DOI https://doi.org/10.1016/0021-8693%2887%2990245-6
- J. W. S. Cassels and E. V. Flynn, Prolegomena to a middlebrow arithmetic of curves of genus $2$, London Mathematical Society Lecture Note Series, vol. 230, Cambridge University Press, Cambridge, 1996. MR 1406090
- A. Clebsch, Zur Theorie der binären algebraischen Formen, Verlag von B. G. Teubner, Liepzig, 1872.
- Koichiro Harada and Mong-Lung Lang, Some elliptic curves arising from the Leech lattice, J. Algebra 125 (1989), no. 2, 298–310. MR 1018947, DOI https://doi.org/10.1016/0021-8693%2889%2990166-X
- J. H. Conway, Three lectures on exceptional groups, Finite simple groups (Proc. Instructional Conf., Oxford, 1969) Academic Press, London, 1971, pp. 215–247. MR 0338152
- P. Deligne, Relèvement des surfaces $K3$ en caractéristique nulle, Algebraic surfaces (Orsay, 1976–78) Lecture Notes in Math., vol. 868, Springer, Berlin-New York, 1981, pp. 58–79 (French). Prepared for publication by Luc Illusie. MR 638598
- R. W. H. T. Hudson, Kummer’s quartic surface, Cambridge University Press, Cambridge, 1905.
- J. I. Hutchinson, On some birational transformations of the Kummer surface into itself, Bull. Amer. Math. Soc. 7 (1901), no. 5, 211–217. MR 1557786, DOI https://doi.org/10.1090/S0002-9904-1901-00785-9
- Jun-ichi Igusa, Arithmetic variety of moduli for genus two, Ann. of Math. (2) 72 (1960), 612–649. MR 114819, DOI https://doi.org/10.2307/1970233
- Jong Hae Keum, Two extremal elliptic fibrations on Jacobian Kummer surfaces, Manuscripta Math. 91 (1996), no. 3, 369–377. MR 1416718, DOI https://doi.org/10.1007/BF02567961
- Jong Hae Keum, Automorphisms of Jacobian Kummer surfaces, Compositio Math. 107 (1997), no. 3, 269–288. MR 1458752, DOI https://doi.org/10.1023/A%3A1000148907120
- Jonghae Keum and Shigeyuki Kond\B{o}, The automorphism groups of Kummer surfaces associated with the product of two elliptic curves, Trans. Amer. Math. Soc. 353 (2001), no. 4, 1469–1487. MR 1806732, DOI https://doi.org/10.1090/S0002-9947-00-02631-3
- Felix Klein, Ueber Configurationen, welche der Kummer’schen Fläche zugleich eingeschrieben und umgeschrieben sind, Math. Ann. 27 (1886), no. 1, 106–142 (German). MR 1510367, DOI https://doi.org/10.1007/BF01447306
- Shigeyuki Kond\B{o}, The automorphism group of a generic Jacobian Kummer surface, J. Algebraic Geom. 7 (1998), no. 3, 589–609. MR 1618132
- Abhinav Kumar, $K3$ surfaces associated with curves of genus two, Int. Math. Res. Not. IMRN 6 (2008), Art. ID rnm165, 26. MR 2427457, DOI https://doi.org/10.1093/imrn/rnm165
- Masato Kuwata and Tetsuji Shioda, Elliptic parameters and defining equations for elliptic fibrations on a Kummer surface, Algebraic geometry in East Asia—Hanoi 2005, Adv. Stud. Pure Math., vol. 50, Math. Soc. Japan, Tokyo, 2008, pp. 177–215. MR 2409557, DOI https://doi.org/10.2969/aspm/05010177
- Jean-François Mestre, Construction de courbes de genre $2$ à partir de leurs modules, Effective methods in algebraic geometry (Castiglioncello, 1990) Progr. Math., vol. 94, Birkhäuser Boston, Boston, MA, 1991, pp. 313–334 (French). MR 1106431
- Ken-ichi Nishiyama, The Jacobian fibrations on some $K3$ surfaces and their Mordell-Weil groups, Japan. J. Math. (N.S.) 22 (1996), no. 2, 293–347. MR 1432379, DOI https://doi.org/10.4099/math1924.22.293
- Keiji Oguiso, On Jacobian fibrations on the Kummer surfaces of the product of nonisogenous elliptic curves, J. Math. Soc. Japan 41 (1989), no. 4, 651–680. MR 1013073, DOI https://doi.org/10.2969/jmsj/04140651
- I. Piatetski-Shapiro and I. R. Shafarevich, A Torelli theorem for algebraic surfaces of type K3, Math. USSR Izv. 5 (1971), 547–587.
- T. Rehn and A. Schürmann, C++ Tools for Exploiting Polyhedral Symmetries, Lecture Notes in Computer Science Vol. 6327 (2010), 295–298.
- D. Rusin, Maple input file for reducing a curve $y^2 =$ quartic to normal form, Mathematical Atlas, \verb,http://www.math.niu.edu/ rusin/known-math/96/quartic.maple,.
- Tetsuji Shioda, Classical Kummer surfaces and Mordell-Weil lattices, Algebraic geometry, Contemp. Math., vol. 422, Amer. Math. Soc., Providence, RI, 2007, pp. 213–221. MR 2296439, DOI https://doi.org/10.1090/conm/422/08062
- Tetsuji Shioda, On the Mordell-Weil lattices, Comment. Math. Univ. St. Paul. 39 (1990), no. 2, 211–240. MR 1081832
- Hans Sterk, Finiteness results for algebraic $K3$ surfaces, Math. Z. 189 (1985), no. 4, 507–513. MR 786280, DOI https://doi.org/10.1007/BF01168156
- È. B. Vinberg, Some arithmetical discrete groups in Lobačevskiĭ spaces, Discrete subgroups of Lie groups and applications to moduli (Internat. Colloq., Bombay, 1973) Oxford Univ. Press, Bombay, 1975, pp. 323–348. MR 0422505
References
- Richard Borcherds, Automorphism groups of Lorentzian lattices, J. Algebra 111 (1987), no. 1, 133–153. MR 913200 (89b:20018), DOI https://doi.org/10.1016/0021-8693%2887%2990245-6
- J. W. S. Cassels and E. V. Flynn, Prolegomena to a middlebrow arithmetic of curves of genus $2$, London Mathematical Society Lecture Note Series, vol. 230, Cambridge University Press, Cambridge, 1996. MR 1406090 (97i:11071)
- A. Clebsch, Zur Theorie der binären algebraischen Formen, Verlag von B. G. Teubner, Liepzig, 1872.
- I. Connell, Addendum to a paper of K. Harada and M.-L. Lang: "Some elliptic curves arising from the Leech lattice, J. Algebra 125 (1989) no. 2, 298–310. MR 1018947 (90g:11072)
- J. H. Conway, Three lectures on exceptional groups, Finite simple groups (Proc. Instructional Conf., Oxford, 1969) Academic Press, London, 1971, pp. 215–247. MR 0338152 (49 \#2918)
- P. Deligne, Relèvement des surfaces $K3$ en caractéristique nulle, Algebraic surfaces (Orsay, 1976–78) Lecture Notes in Math., vol. 868, Springer, Berlin, 1981, pp. 58–79 (French). Prepared for publication by Luc Illusie. MR 638598 (83j:14034)
- R. W. H. T. Hudson, Kummer’s quartic surface, Cambridge University Press, Cambridge, 1905.
- J. I. Hutchinson, On some birational transformations of the Kummer surface into itself, Bull. Amer. Math. Soc. 7 (1901), no. 5, 211–217. MR 1557786, DOI https://doi.org/10.1090/S0002-9904-1901-00785-9
- Jun-ichi Igusa, Arithmetic variety of moduli for genus two, Ann. of Math. (2) 72 (1960), 612–649. MR 0114819 (22 \#5637)
- Jong Hae Keum, Two extremal elliptic fibrations on Jacobian Kummer surfaces, Manuscripta Math. 91 (1996), no. 3, 369–377. MR 1416718 (97h:14053), DOI https://doi.org/10.1007/BF02567961
- Jong Hae Keum, Automorphisms of Jacobian Kummer surfaces, Compositio Math. 107 (1997), no. 3, 269–288. MR 1458752 (98e:14039), DOI https://doi.org/10.1023/A%3A1000148907120
- Jonghae Keum and Shigeyuki Kondō, The automorphism groups of Kummer surfaces associated with the product of two elliptic curves, Trans. Amer. Math. Soc. 353 (2001), no. 4, 1469–1487. MR 1806732 (2001k:14075), DOI https://doi.org/10.1090/S0002-9947-00-02631-3
- Felix Klein, Ueber Configurationen, welche der Kummer’schen Fläche zugleich eingeschrieben und umgeschrieben sind, Math. Ann. 27 (1886), no. 1, 106–142 (German). MR 1510367, DOI https://doi.org/10.1007/BF01447306
- Shigeyuki Kondō, The automorphism group of a generic Jacobian Kummer surface, J. Algebraic Geom. 7 (1998), no. 3, 589–609. MR 1618132 (99i:14043)
- Abhinav Kumar, $K3$ surfaces associated with curves of genus two, Int. Math. Res. Not. IMRN 6 (2008), Art. ID rnm165, 26. MR 2427457 (2009d:14044), DOI https://doi.org/10.1093/imrn/rnm165
- Masato Kuwata and Tetsuji Shioda, Elliptic parameters and defining equations for elliptic fibrations on a Kummer surface, Algebraic geometry in East Asia—Hanoi 2005, Adv. Stud. Pure Math., vol. 50, Math. Soc. Japan, Tokyo, 2008, pp. 177–215. MR 2409557 (2009g:14039)
- Jean-François Mestre, Construction de courbes de genre $2$ à partir de leurs modules, Effective methods in algebraic geometry (Castiglioncello, 1990) Progr. Math., vol. 94, Birkhäuser Boston, Boston, MA, 1991, pp. 313–334 (French). MR 1106431 (92g:14022)
- Ken-ichi Nishiyama, The Jacobian fibrations on some $K3$ surfaces and their Mordell-Weil groups, Japan. J. Math. (N.S.) 22 (1996), no. 2, 293–347. MR 1432379 (97m:14037)
- Keiji Oguiso, On Jacobian fibrations on the Kummer surfaces of the product of non-isogenous elliptic curves, J. Math. Soc. Japan 41 (1989), no. 4, 651–680. MR 1013073 (90j:14044), DOI https://doi.org/10.2969/jmsj/04140651
- I. Piatetski-Shapiro and I. R. Shafarevich, A Torelli theorem for algebraic surfaces of type K3, Math. USSR Izv. 5 (1971), 547–587.
- T. Rehn and A. Schürmann, C++ Tools for Exploiting Polyhedral Symmetries, Lecture Notes in Computer Science Vol. 6327 (2010), 295–298.
- D. Rusin, Maple input file for reducing a curve $y^2 =$ quartic to normal form, Mathematical Atlas, \verb,http://www.math.niu.edu/ rusin/known-math/96/quartic.maple,.
- Tetsuji Shioda, Classical Kummer surfaces and Mordell-Weil lattices, Algebraic geometry, Contemp. Math., vol. 422, Amer. Math. Soc., Providence, RI, 2007, pp. 213–221. MR 2296439 (2008h:14036), DOI https://doi.org/10.1090/conm/422/08062
- Tetsuji Shioda, On the Mordell-Weil lattices, Comment. Math. Univ. St. Paul. 39 (1990), no. 2, 211–240. MR 1081832 (91m:14056)
- Hans Sterk, Finiteness results for algebraic $K3$ surfaces, Math. Z. 189 (1985), no. 4, 507–513. MR 786280 (86j:14038), DOI https://doi.org/10.1007/BF01168156
- È. B. Vinberg, Some arithmetical discrete groups in Lobačevskiĭ spaces, Discrete subgroups of Lie groups and applications to moduli (Internat. Colloq., Bombay, 1973), Oxford Univ. Press, Bombay, 1975, pp. 323–348. MR 0422505 (54 \#10492)
Additional Information
Abhinav Kumar
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
MR Author ID:
694441
Email:
abhinav@math.mit.edu
Received by editor(s):
April 20, 2011
Received by editor(s) in revised form:
February 15, 2012
Published electronically:
May 28, 2014
Additional Notes:
The author was supported in part by NSF grants DMS-0757765 and DMS-0952486, and by a grant from the Solomon Buchsbaum Research Fund. He thanks Princeton University for its hospitality during the period when this project was started.
Article copyright:
© Copyright 2014
University Press, Inc.