On potential density of integral points on the complement of some subvarieties in the projective space

TL;DR

This paper investigates the potential density of integral points on the complements of certain subvarieties in projective space over number fields, extending previous results to higher dimensions and more complex configurations.

Contribution

It generalizes existing results on integral point density to higher-dimensional projective spaces with specific subvariety configurations.

Findings

Integral points are potentially dense in certain complements.

Results extend Corvaja-Zucconi's work to higher dimensions.

Conditions on intersections influence density outcomes.

Abstract

We study some density results for integral points on the complement of a closed subvariety in the $n$-dimensional projective space over a number field. For instance, we consider a subvariety whose components consist of $n-1$ hyperplanes plus one smooth quadric hypersurface in general position, or four hyperplanes in general position plus a finite number of concurrent straight lines. In these cases, under some conditions on intersection, we show that the integral points on the complements are potentially dense. Our results are generalizations of Corvaja-Zucconi's results for complements of subvarieties in the two or three dimensional projective space.