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

Abstract

We prove some density results for integral points on affine open sets of Fano threefolds. For instance, let $$X^o=\mathbb P^3{\setminus } D$$ X o = P 3 \ 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 $$\Pi $$ Π such that the intersection $$V\cap \Pi $$ V ∩ Π contains a line. In all these cases we show that the set of integral points of $$X^o$$ 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.