Connectedness Principle for 3-Folds in Characteristic p > 5

A conjecture, known as the Shokurov–Kollár connectedness principle, predicts the following. Let (X, B) be a pair, and let f : X → S be a contraction with −(KX + B) nef over S; then, for any point s ∈ S, the intersection f −1 (s) ∩ Nklt(X, B) has at most two connected components, where Nklt(X, B) denotes the non-klt locus of (X, B). This conjecture has been extensively studied in characteristic zero, and it has been recently settled in that context. In this work, we consider this conjecture in the setup of positive characteristic algebraic geometry. We prove that this conjecture holds for threefolds in positive and mixed characteristic, where the residue fields are assumed to have characteristic p > 5. Under the same assumptions, we characterize the cases in which Nklt(X, B) fails to be connected.