Pietro Corvaja ; Francesco Zucconi - Quartic surfaces, their bitangents and rational points

epiga:8987 - Épijournal de Géométrie Algébrique, February 10, 2023, Volume 7 - https://doi.org/10.46298/epiga.2022.8987
Quartic surfaces, their bitangents and rational points

Authors: Pietro Corvaja ; Francesco Zucconi

    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.

    Volume: Volume 7
    Published on: February 10, 2023
    Accepted on: October 6, 2022
    Submitted on: January 20, 2022
    Keywords: Mathematics - Number Theory

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