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arXiv · 2010.08623

Quartic surface, its bitangents and rational points

Pietro Corvaja·Francesco Zucconi
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Abstract

Let X be a smooth quartic surface not containing lines, defined over a number field K. We prove that there are only finitely many bitangents to X which are defined over K. This result can be interpreted as saying that a certain surface, having vanishing irregularity, contains only finitely many rational points. In our proof, we use the geometry of lines of the quartic double solid associated to X. In a somewhat opposite direction, we show that on any quartic surface X over a number field K, the set of algebraic points in X(\overeline K) which are quadratic over a suitable finite extension K' of K is Zariski-dense.

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Pietro Corvaja, Francesco Zucconi. 2020-10-16. Quartic surface, its bitangents and rational points. https://doi.org/10.46298/epiga.2022.8987

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