Codimension two integral points on some rationally connected threefolds are potentially dense

Let X X be a smooth, projective, rationally connected variety, defined over a number field k k , and let Z ⊂ X Z\subset X be a closed subset of codimension at least two. In this paper, for certain choices of X X , we prove that the set of Z Z -integral points is potentially Zariski dense, in the sense that there is a finite extension K K of k k such that the set of points P ∈ X ( K ) P\in X(K) that are Z Z -integral is Zariski dense in X X . This gives a positive answer to a question of Hassett and Tschinkel from 2001.