On integral points of some Fano Threefolds and their Hilbert schemes of lines and conics

TL;DR

This paper investigates the density of integral points on certain Fano threefolds and their Hilbert schemes of lines and conics, demonstrating potential density in various geometric configurations.

Contribution

It establishes potential density of integral points on specific Fano threefolds and their Hilbert schemes, expanding understanding of Diophantine properties in algebraic geometry.

Findings

Integral points are potentially dense on certain Fano threefolds.

Potential density holds for some log-Fano and log-Calabi-Yau threefolds.

Results apply to Hilbert schemes of lines and conics on these varieties.

Abstract

Let $X^o=\mathbb P^3\setminus D$ where $D$ is the union of two quadrics such that their intersection contains a smooth conic, or the union of a smooth quadric surface and two planes, or the union of a smooth cubic surface $V$ and a plane $\Pi$ such that the intersection $V\cap\Pi$ contains a line. In all these cases we show that the set of integral points of $X^o$ is potentially dense. We apply the above results to prove that integral points are potentially dense in some log-Fano or in some log-Calabi-Yau threefold.