Connectedness principle for $3$-folds in characteristic $p>5$

TL;DR

This paper proves the Shokurov-Kollár connectedness conjecture for threefolds in characteristic p>5, establishing conditions under which the non-klt locus is connected or not, extending known results from characteristic zero.

Contribution

It extends the connectedness principle to threefolds in positive characteristic p>5 and characterizes cases of non-connected non-klt loci.

Findings

The conjecture holds for threefolds in characteristic p>5.

Conditions for non-connected non-klt locus are characterized.

The result bridges a gap between characteristic zero and positive characteristic geometry.

Abstract

A conjecture, known as the Shokurov-Koll\'ar connectedness principle, predicts the following. Let $(X,B)$ be a pair, and let $f \colon X \rightarrow S$ be a contraction with $-(K_X + B)$ nef over $S$; then, for any point $s \in S$, the intersection $f^{-1} (s) \cap \mathrm{Nklt}(X,B)$ has at most two connected components, where $\mathrm{Nklt}(X,B)$ 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 this conjecture holds for threefolds in characteristic $p> 5$, and, under the same assumptions, we characterize the cases in which $\mathrm{Nklt}(X,B)$ fails to be connected.