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Elliptic Fibrations on K3 Surfaces

Published online by Cambridge University Press:  19 December 2013

Viacheslav V. Nikulin
Viacheslav V. Nikulin*
Affiliation:
Department of Pure Mathematics, University of Liverpool, Liverpool L69 3BX, UK ([email protected]) and Steklov Mathematical Institute, ul. Gubkina 8, Moscow 117966, GSP-1, Russia ([email protected])
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Abstract

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This paper consists mainly of a review and applications of our old results relating to the title. We discuss how many elliptic fibrations and elliptic fibrations with infinite automorphism groups (or Mordell–Weil groups) an algebraic K3 surface over an algebraically closed field can have. As examples of applications of the same ideas, we also consider K3 surfaces with exotic structures: with a finite number of non-singular rational curves, with a finite number of Enriques involutions, and with naturally arithmetic automorphism groups.

Keywords

K3 surfaceelliptic fibrationPicard latticeautomorphism groupPrimary 14J2814J2714J5014C22
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 57 , Issue 1 , February 2014 , pp. 253 - 267
Copyright
Copyright © Edinburgh Mathematical Society 2014 

References

1.Cantat, S., Dynamique des automorphismes des surfaces K3, Acta Math. 187(1) (2001), 1–57.Google Scholar
2.Cossec, F. R. and Dolgachev, I. V., Enriques surfaces I, Progress in Mathematics, Volume 76 (Birkhäuser, 1989).Google Scholar
3.Kulikov, V. S., Degenerations of K3 surfaces and Enriques surfaces, Math. USSR Izv. 11(5) (1977), 957–989.CrossRefGoogle Scholar
4.Nikulin, V. V., Integral symmetric bilinear forms and some of their geometric applications, Math. USSR Izv. 14(1) (1980), 103–167.Google Scholar
5.Nikulin, V. V., On arithmetic groups generated by reflections in Lobachevsky spaces, Math. USSR Izv. 16(3) (1981), 573–601.CrossRefGoogle Scholar
6.Nikulin, V. V., On the classification of arithmetic groups generated by reflections in Lobachevsky spaces, Math. USSR Izv. 18(1) (1982), 99–123.Google Scholar
7.Nikulin, V. V., On the quotient groups of the automorphism groups of hyperbolic forms by the subgroups generated by 2-reflections, algebraic—geometric applications, J. Sov. Math. 22 (1983), 1401–1476.CrossRefGoogle Scholar
8.Nikulin, V. V., Surfaces of type K3 with finite automorphism group and Picard group of rank 3, Proc. Steklov Math. Inst. 165 (1984), 113–142.Google Scholar
9.Nikulin, V. V., Discrete reflection groups in Lobachevsky spaces and algebraic surfaces, in Proceedings of the International Congress of Mathematicians, Berkeley, CA, Volume 1, pp. 654–669 (Defense Technical Information Center, Fort Belvoir, VA, 1988).Google Scholar
10.Nikulin, V. V., Reflection groups in Lobachevsky spaces and the denominator identity for Lorentzian Kac–Moody algebras, Izv. Math. 60 (1996), 305–334.Google Scholar
11.Nikulin, V. V., K3 surfaces with interesting groups of automorphisms, J. Math. Sci. 95(1) (1999), 2028–2048.Google Scholar
12.Piatetsky-Shapiro, I. I. and Shafarevich, I. R., A Torelli theorem for algebraic surfaces of type K3, Math. USSR Izv. 5(3) (1971), 547–588.Google Scholar
13.Rudakov, A. N. and Shafarevich, I. R., Surfaces of type K3 over fields of finite characteristic, J. Sov. Math. 22 (1983), 1476–1533.Google Scholar
14.Shafarevich, I. R., Algebraic surfaces, Proceedings of the Steklov Institute of Mathematics, Volume 75, pp. 3–215 (American Mathematical Society, Providence, RI, 1965).Google Scholar
15.Totaro, B., Algebraic surfaces and hyperbolic geometry, eprint (arXiv:1008.3825v1, 2010).Google Scholar
16.Vinberg, É. B., Absence of crystallographic groups of reflection in Lobachevskii spaces of large dimension, Funct. Analysis Applic. 15(2) (1981), 128–130.CrossRefGoogle Scholar
17.Vinberg, É. B., Hyperbolic groups of reflections, Russ. Math. Surv. 40(1) (1985), 31–75.Google Scholar
18.Vinberg, É. B., Classification of 2-reflective hyperbolic lattices of rank 4, Trans. Mosc. Math. Soc. 68 (2007), 39–66.Google Scholar