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

Abstract

AbstractWe prove some density results for integral points on affine open sets of Fano threefolds. For instance, let$$X^o=\mathbb P^3{\setminus } D$$Xo=P3\DwhereDis 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 surfaceVand a plane$$\Pi $$Πsuch that the intersection$$V\cap \Pi $$V∩Πcontains a line. In all these cases we show that the set of integral points of$$X^o$$Xois 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.