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Kummer surfaces and the computation of the Picard group

Part of: Abelian varieties and schemes (Co)homology theory Surfaces and higher-dimensional varieties Cycles and subschemes

Published online by Cambridge University Press:  01 April 2012

Andreas-Stephan Elsenhans  and
Jörg Jahnel
Andreas-Stephan Elsenhans
Affiliation:
Mathematisches Institut, Universität Bayreuth, Universitätsstr. 30, D-95440 Bayreuth, Germany (email: Stephan.Elsenhans@uni-bayreuth.de)
Jörg Jahnel
Affiliation:
Department Mathematik, Universität Siegen, Walter-Flex-Str. 3, D-57068 Siegen, Germany (email: jahnel@mathematik.uni-siegen.de)

Abstract

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We test R. van Luijk’s method for computing the Picard group of a K3 surface. The examples considered are the resolutions of Kummer quartics in ℙ3. Using the theory of abelian varieties, the Picard group may be computed directly in this case. Our experiments show that the upper bounds provided by van Luijk’s method are sharp when sufficiently many primes are used. In fact, there are a lot of primes that yield a value close to the exact one. However, for many but not all Kummer surfaces V of Picard rank 18, we have for a set of primes of density at least 1/2.

MSC classification

Secondary: 14J28: $K3$ surfaces and Enriques surfaces 14F20: Étale and other Grothendieck topologies and (co)homologies 14K99: None of the above, but in this section 14C22: Picard groups
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 15 , May 2012 , pp. 84 - 100
Copyright
Copyright © London Mathematical Society 2012

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