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

Stefano Filipazzi, Joe Waldron

Abstract

A conjecture, known as the Shokurov-Kollár 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.

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

Abstract

A conjecture, known as the Shokurov-Kollár connectedness principle, predicts the following. Let be a pair, and let be a contraction with nef over ; then, for any point , the intersection has at most two connected components, where denotes the non-klt locus of . 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 , and, under the same assumptions, we characterize the cases in which fails to be connected.

Paper Structure

This paper contains 11 sections, 18 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Terminology and conventions
  4. B-divisors
  5. Generalized pairs and singularities
  6. Generalized dlt pairs and generalized MMP
  7. $\mathbb P^{1}$-links
  8. Proof of the main statement
  9. Birational case
  10. Connectedness of the non-lc locus
  11. Main proof

Key Result

Theorem 1.2

Let $R$ be an excellent domain of finite Krull dimension, which admits a dualizing complex, and all of whose closed points have characteristic zero or $p>5$. Let $f \colon X \rightarrow S$ be a projective morphism between varieties that are quasi-projective over $\operatorname{Spec}(R)$ such that $f Furthermore, if $R$ is a perfect field of characteristic $p>5$, the weak $\mathbb P^{1}$-link in (3

Theorems & Definitions (51)

  • Conjecture 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 41 more