# An Hilbert Irreducibility Theorem for integral points of del Pezzo surfaces

## TL;DR

This paper proves potential density of integral points on certain del Pezzo surfaces, classifies simply connected complements, and confirms conjectures about the thinness of integral points in these contexts.

## Contribution

It establishes a Hilbert irreducibility theorem for integral points on del Pezzo surfaces and classifies cases with simply connected complements.

## Key findings

- Integral points are potentially Zariski dense in most cases.
- Classified simply connected complements with potential non-thinness of integral points.
- Confirmed conjecture of Corvaja and Zannier for specific cases.

## Abstract

We prove that the integral points are potentially Zariski dense in the complement of a reduced effective singular anticanonical divisor in a smooth del Pezzo surface, with the exception of $\mathbb{P}^2$ minus three concurrent lines (for which potential density does not hold). This answers positively a question raised by Hassett and Tschinkel and, combined with previous results, completes the proof of the potential density of integral points for complements of anticanonical divisors in smooth del Pezzo surfaces. We then classify the complements which are simply connected and for these we prove that the set of integral points is potentially not thin, as predicted by a conjecture of Corvaja and Zannier.