# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1702.00735