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Enriques surfaces covered by Jacobian Kummer surfaces
Hisanori Ohashi
Source: Nagoya Math. J. Volume 195
(2009), 165-186.
Abstract
This paper classifies Enriques surfaces whose K3-cover is a fixed Picard-general Jacobian Kummer surface. There are exactly 31 such surfaces. We describe the free involutions which give these Enriques surfaces explicitly. As a biproduct, we show that $\operatorname{Aut}(X)$ is generated by elements of order 2, which is an improvement of the theorem of S. Kondo.
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Permanent link to this document: http://projecteuclid.org/euclid.nmj/1252934377
Zentralblatt MATH identifier: 05611435
Mathematical Reviews number (MathSciNet): MR2552959
References
W. Barth and C. Peters, Automorphisms of Enriques surfaces, Invent. math., 73 (1983), 383--411.
Mathematical Reviews (MathSciNet): MR718937
Zentralblatt MATH: 0518.14023
Digital Object Identifier: doi:10.1007/BF01388435
N. Bourbaki, Éléments de Mathématique. Groupes et Algèbres de Lie, Chap. IV, V et VI.
J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, 3rd ed., Grundlehren der Mathematischen Wissenschaften 290, Springer-Verlag, 1999.
Mathematical Reviews (MathSciNet): MR1662447
I. V. Dolgachev and J. H. Keum, Birational automorphisms of quartic hessian surfaces, Trans. Amer. Math. Soc., 354 (2002), 3031--3057.
Mathematical Reviews (MathSciNet): MR1897389
Zentralblatt MATH: 0994.14007
Digital Object Identifier: doi:10.1090/S0002-9947-02-03011-8
JSTOR: links.jstor.org
J. I. Hutchinson, The Hessian of the cubic surface, Bull. Amer. Math. Soc., 5 (1899), 282--292.
J. I. Hutchinson, The Hessian of the cubic surface, II, Bull. Amer. Math. Soc., 6 (1899), 328--337.
Mathematical Reviews (MathSciNet): MR1557720
Digital Object Identifier: doi:10.1090/S0002-9904-1900-00716-6
Project Euclid: euclid.bams/1183416238
J. I. Hutchinson, On some birational transformations of the Kummer surfaces into itself, Bull. Amer. Math. Soc. (2), 7 (1901), 211--217.
Mathematical Reviews (MathSciNet): MR1557786
Digital Object Identifier: doi:10.1090/S0002-9904-1901-00785-9
Project Euclid: euclid.bams/1183416539
F. Klein, Ueber Configurationen, Welche der Kummer'schen Fläche Zugleich Eingeschrieben und Umgeschrieben Sind, Math. Ann., 27 (1886), 106--142.
Mathematical Reviews (MathSciNet): MR1510367
Digital Object Identifier: doi:10.1007/BF01447306
M. R. Gonzalez-Dorrego, $(16, 6)$ configurations and geometry of Kummer surfaces in $\bbP^3$, Mem. Amer. Math. Soc. 107, 1994.
Mathematical Reviews (MathSciNet): MR1182682
Zentralblatt MATH: 0809.14032
J. H. Keum, Every algebraic Kummer surface is the $K3$-cover of an Enriques surface, Nagoya Math. J., 118 (1990), 99--110.
Mathematical Reviews (MathSciNet): MR1060704
Zentralblatt MATH: 0699.14047
Project Euclid: euclid.nmj/1118781595
J. H. Keum, Automorphisms of Jacobian Kummer surfaces, Compositio Math., 107 (1997), 269--288.
Mathematical Reviews (MathSciNet): MR1458752
Zentralblatt MATH: 0891.14013
Digital Object Identifier: doi:10.1023/A:1000148907120
S. Kondo, The automorphism group of a generic Jacobian Kummer surface, J. Algebraic Geometry, 7 (1998), 589--609.
Mathematical Reviews (MathSciNet): MR1618132
Zentralblatt MATH: 0948.14034
S. Mukai, Kummer's quartics and numerically reflective involutions of Enriques surfaces, RIMS Preprint, 1633.
Mathematical Reviews (MathSciNet): MR2516519
V. V. Nikulin, An analogue of the Torelli theorem for Kummer surfaces of Jacobians (English translation), Math. USSR Izv., 8 (1974), 21--41.
Mathematical Reviews (MathSciNet): MR357410
Zentralblatt MATH: 0305.14010
V. V. Nikulin, Integral symmetric bilinear forms and some of their applications (English translation), Math. USSR Izv., 14 (1980), 103--167.
H. Ohashi, On the number of Enriques quotients of a $K3$ surface, Publ. Res. Inst. Math. Sci., 43 (2007), 181--200.
Mathematical Reviews (MathSciNet): MR2319542
Digital Object Identifier: doi:10.2977/prims/1199403814
H. Ohashi, Counting Enriques quotients of a $K3$ surface, RIMS Preprint, 1609.
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