The arithmetic puncturing problem and integral points

TL;DR

This paper investigates the density of integral points on algebraic varieties with dense rational points, providing positive results for certain classes like rational curves and K3 surfaces, and exploring stronger notions of integrality.

Contribution

It offers new evidence supporting the potential density of D-integral points on varieties with dense rational points, especially for specific geometric cases.

Findings

Positive answers for varieties with rational curves

Results for certain K3 surfaces

Discussion of stronger integrality notions

Abstract

In this paper, we provide evidence to support a positive answer to a question of Hassett and Tschinkel. In particular, if an algebraic variety V has a dense set of rational points, they ask whether or not the set of D-integral points is potentially dense, where D is a set of codimension at least two. We give a positive answer to this question in many cases, including varieties whose generic linear section is a smooth rational curve, and certain K3 surfaces. We also discuss some stronger notions of integrality of points, and give some positive answers to some cases of the analogous question in the stronger context.