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arXiv · 0709.0663

K3 surfaces, rational curves, and rational points

Arthur Baragar·David McKinnon
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Abstract

We prove that for any of a wide class of elliptic surfaces $X$ defined over a number field $k$, if there is an algebraic point on $X$ that lies on only finitely many rational curves, then there is an algebraic point on $X$ that lies on no rational curves. In particular, our theorem applies to a large class of elliptic $K3$ surfaces, which relates to a question posed by Bogomolov in 1981. We apply our results to construct an explicit algebraic point on a $K3$ surface that does not lie on any smooth rational curves.

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BibTeXRIS

Arthur Baragar, David McKinnon. 2008-07-20. K3 surfaces, rational curves, and rational points. https://arxiv.org/abs/0709.0663

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