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

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.