Quartic surface, its bitangents and rational points

TL;DR

This paper investigates the finiteness of K-rational bitangents to smooth quartic surfaces without lines over number fields, linking geometric properties to rational point distribution.

Contribution

It proves finiteness of K-rational bitangents to certain quartic surfaces and demonstrates density of quadratic points over finite extensions.

Findings

Finiteness of K-rational bitangents to smooth quartic surfaces without lines.

Zariski-density of quadratic points over finite extensions.

Use of quartic double solid geometry in proofs.

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.