Quartic surfaces, their bitangents and rational points

Pietro Corvaja; Francesco Zucconi

doi:10.46298/epiga.2022.8987

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 pointsArticle

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.

Comment: This is the final version of the paper

https://doi.org/10.46298/epiga.2022.8987

Source: arXiv.org:2010.08623

Volume: Volume 7

Published on: February 10, 2023

Accepted on: October 6, 2022

Submitted on: January 20, 2022

Keywords: Mathematics - Number Theory

Licence: Attribution-ShareAlike 4.0 International (CC BY-SA 4.0)