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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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K3 surfaces with Picard number three and canonical vector heights
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by Arthur Baragar and Ronald van Luijk PDF
Math. Comp. 76 (2007), 1493-1498 Request permission

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
  • Email: baragar@unlv.nevada.edu
  • 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
  • Email: rmluijk@gmail.com
  • 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.
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 76 (2007), 1493-1498
  • MSC (2000): Primary 14G40, 11G50, 14J28, 14C22
  • DOI: https://doi.org/10.1090/S0025-5718-07-01962-X
  • MathSciNet review: 2299785