Geometry and arithmetic of certain log K3 surfaces

TL;DR

This paper classifies certain smooth log K3 surfaces over fields of characteristic zero, showing they can be compactified into degree 5 del Pezzo surfaces with specific divisors, and investigates their rational points and obstructions over the rationals.

Contribution

It provides a classification of log K3 surfaces with trivial geometric Picard group that can be compactified into degree 5 del Pezzo surfaces, linking their structure to Galois actions.

Findings

Such surfaces can be compactified into degree 5 del Pezzo surfaces with a cycle of five (-1)-curves.

The Galois action on the dual graph of the divisor determines the surface.

For rational surfaces with trivial Galois action, integral points are not Zariski dense, and Brauer-Manin obstruction is not the only obstruction.

Abstract

Let $k$ be a field of characteristic $0$. In this paper we describe a classification of smooth log K3 surfaces $X$ over $k$ whose geometric Picard group is trivial and which can be compactified into del Pezzo surfaces. We show that such an $X$ can always be compactified into a del Pezzo surface of degree $5$, with a compactifying divisor $D$ being a cycle of five $(-1)$-curves, and that $X$ is completely determined by the action of the absolute Galois group of $k$ on the dual graph of $D$. When $k=\mathbb{Q}$ and the Galois action is trivial, we prove that for any integral model $\mathcal{X}/\mathbb{Z}$ of $X$, the set of integral points $\mathcal{X}(\mathbb{Z})$ is not Zariski dense. We also show that the Brauer-Manin obstruction is not the only obstruction for the integral Hasse principle on such log K3 surfaces, even when their compactification is "split".