K3 surfaces with Picard number three and canonical vector heights

Baragar, Arthur; van Luijk, Ronald

doi:10.1090/S0025-5718-07-01962-X

K3 surfaces with Picard number three and canonical vector heights
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by Arthur Baragar and Ronald van Luijk;

Math. Comp. 76 (2007), 1493-1498

DOI: https://doi.org/10.1090/S0025-5718-07-01962-X

Published electronically: January 24, 2007

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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.

References

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