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K3 surfaces with Picard number three and canonical vector heights

Authors: Arthur Baragar and Ronald van Luijk
Journal: Math. Comp. 76 (2007), 1493-1498
MSC (2000): Primary 14G40, 11G50, 14J28, 14C22
Published electronically: January 24, 2007
MathSciNet review: 2299785
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Abstract: In this paper we construct the first known explicit family of K3 surfaces defined over the rationals that are proved to have geometric Picard number $3$. This family is dense in one of the components of the moduli space of all polarized K3 surfaces with Picard number at least $3$. We also use an example from this family to fill a gap in an earlier paper by the first author. In that paper, an argument for the nonexistence of canonical vector heights on K3 surfaces of Picard number $3$ was given, based on an explicit surface that was not proved to have Picard number $3$. We redo the computations for one of our surfaces and come to the same conclusion.

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Additional Information

Arthur Baragar
Affiliation: Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, Nevada 89154-4020

Ronald van Luijk
Affiliation: Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, California 94720-5070
Address at time of publication: Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC V5A 1S6, Canada

Keywords: K3 surfaces, canonical vector heights, heights, Picard numbers
Received by editor(s): February 22, 2006
Received by editor(s) in revised form: July 14, 2006
Published electronically: January 24, 2007
Additional Notes: The first author is supported in part by NSF grant DMS-0403686.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.