Integral points on the complement of plane quartics

TL;DR

This paper proves the Lang-Vojta conjecture for the complement of a generic smooth plane quartic curve over a number field, showing integral points are mostly contained in a finite union of curves, via reduction to the Shafarevich conjecture for K3 surfaces.

Contribution

It establishes the Lang-Vojta conjecture for complements of generic smooth quartic curves, extending to certain reducible quartics, by linking integral points to K3 surface finiteness.

Findings

Integral points are confined to a finite union of curves for generic smooth quartics.

The method reduces the problem to the Shafarevich conjecture for K3 surfaces.

Variants confirm the conjecture for specific reducible quartic configurations.

Abstract

Let $Y$ be the complement of a plane quartic curve $D$ defined over a number field. Our main theorem confirms the Lang-Vojta conjecture for $Y$ when $D$ is a generic smooth quartic curve, by showing that its integral points are confined in a curve except for a finite number of exceptions. The required finiteness will be obtained by reducing it to the Shafarevich conjecture for K3 surfaces. Some variants of our method confirm the same conjecture when $D$ is a reducible generic quartic curve which consists of four lines, two lines and a conic, or two conics.