Abstract
We construct explicit examples of K3 surfaces over which are of degree 2 and geometric Picard rank 1. We construct, particularly, examples of the form \(w^2 = \det M\) where M is a (3 ×3)-matrix of ternary quadratic forms.
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References
Beauville, A.: Surfaces algébriques complexes. In: Astérisque 54, Société Mathématique de France, Paris (1978)
Elsenhans, A.-S., Jahnel, J.: The Asymptotics of Points of Bounded Height on Diagonal Cubic and Quartic Threefolds. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 317–332. Springer, Heidelberg (2006)
Fulton, W.: Intersection theory. Springer, Berlin (1984)
Lieberman, D.I.: Numerical and homological equivalence of algebraic cycles on Hodge manifolds. Amer. J. Math. 90, 366–374 (1968)
van Luijk, R.: K3 surfaces with Picard number one and infinitely many rational points. Algebra & Number Theory 1, 1–15 (2007)
Milne, J.S.: Étale Cohomology. Princeton University Press, Princeton (1980)
Persson, U.: Double sextics and singular K3 surfaces. In: Algebraic Geometry, Sitges (Barcelona) 1983. Lecture Notes in Math., vol. 1124, pp. 262–328. Springer, Berlin (1985)
Tate, J.: Conjectures on algebraic cycles in l-adic cohomology. In: Motives, Proc. Sympos. Pure Math., vol. 55-1, pp. 71–83. Amer. Math. Soc., Providence (1994)
Zeilberger, D.: A combinatorial proof of Newtons’s identities. Discrete Math. 49, 319 (1984)
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Elsenhans, AS., Jahnel, J. (2008). K3 Surfaces of Picard Rank One and Degree Two. In: van der Poorten, A.J., Stein, A. (eds) Algorithmic Number Theory. ANTS 2008. Lecture Notes in Computer Science, vol 5011. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79456-1_14
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DOI: https://doi.org/10.1007/978-3-540-79456-1_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-79455-4
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