A note on elliptic K3 surfaces

Author:
JongHae Keum

Journal:
Trans. Amer. Math. Soc. **352** (2000), 2077-2086

MSC (2000):
Primary 14J28, 14J27, 11H31

DOI:
https://doi.org/10.1090/S0002-9947-99-02587-8

Published electronically:
November 17, 1999

MathSciNet review:
1707196

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the relationship between an elliptic fibration on an elliptic K3 surface and its Jacobian surface. We give an explicit description of the Picard lattice of the Jacobian surface. Then we use the description to prove that certain K3 surfaces do not admit a non-Jacobian fibration. Moreover, we obtain an inequality involving the determinant of the Picard lattice and the number of components of reducible fibres, which implies, among others, that if an elliptic K3 surface has Picard lattice with relatively small determinant, then every elliptic fibration on it must have a reducible fibre. Some examples of K3 surfaces are discussed.

**[1]**S. Belcastro,*Picard lattices of families of surfaces*, Ph.D. Thesis, University of Michigan (1997).**[2]**J. Conway and N. Sloane,*Sphere packings, lattices and groups*, Springer-Verlag, 1988.MR**89a:11067****[3]**F. Cossec and I. Dolgachev,*Enriques Surfaces I*, Birkhäuser, Boston, 1989.MR**90h:14052****[4]**D. Cox,*Mordell-Weil groups of elliptic curves over with or*, Duke Math. J.**49**(1982), 677-689.MR**84a:14029****[5]**J. Keum,*Two extremal elliptic fibrations on Jacobian Kummer surfaces*, manuscripta math.**91**(1996), 369-377; erratum, 94 (1997), 543. MR**97h:14053**; MR**98m:14038****[6]**J. Keum,*Automorphisms of Jacobian Kummer surfaces*, Compositio Math.**107**(1997), 269-288.MR**98e:14039****[7]**D. Morrison,*On surfaces with large Picard number*, Invent. Math.**75**(1984), 105-121.MR**85j:14071****[8]**S. Mukai,*On the moduli space of bundles on surfaces, I*, Vector Bundles on Algebraic Varieties, Proc. Bombay Conference, 1984, Tata Inst. Fund. Research Studies**No.11**(1987), 341-413.MR**88i:14036****[9]**V. Nikulin,*Integral symmetric bilinear forms and some of their applications*, Math. USSR Izv.**14**(1980), 103-167.**[10]**K. Nishiyama,*The Jacobian fibrations on some K3 surfaces and their Mordell-Weil groups*, Japan. J. Math.**22**(1996), 293-347. MR**97m:14037****[11]**K. Oguiso,*On Jacobian fibrations on the Kummer surfaces of the product of nonisogenous elliptic curves*, J. Math. Soc. Japan**41**(1989), 651-680.MR**90j:14044****[12]**T. Shioda,*On the Mordell-Weil lattices*, Com. Math. Univ. St. Pauli**39**(1990), 211-240.MR**91m:14056****[13]**T. Shioda,*Theory of Mordell-Weil lattices*, Proc. ICM, Kyoto (1990), 473-489.MR**93k:14046****[14]**E. Vinberg,*The two most algebraic K3 surfaces*, Math. Ann.**265**(1983), 1-21.MR**85k:14020**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
14J28,
14J27,
11H31

Retrieve articles in all journals with MSC (2000): 14J28, 14J27, 11H31

Additional Information

**JongHae Keum**

Affiliation:
Department of Mathematics, Konkuk University, 93-1 Mojin-dong Kwangjin-gu, Seoul 143-701, Korea

Email:
jhkeum@kkucc.konkuk.ac.kr

DOI:
https://doi.org/10.1090/S0002-9947-99-02587-8

Keywords:
Elliptic $K3$ surface,
Jacobian surface,
Picard lattice,
lattice packing

Received by editor(s):
October 22, 1997

Published electronically:
November 17, 1999

Additional Notes:
The research was supported by the Korea Research Foundation (1998) and GARC

Article copyright:
© Copyright 2000
American Mathematical Society