Depth of $F$-singularities and base change of relative canonical sheaves

TL;DR

This paper investigates the depth properties of $F$-singularities in characteristic $p > 0$ varieties and demonstrates that relative canonical sheaves maintain compatibility under base change for specific families with sharply $F$-pure fibers.

Contribution

It establishes conditions for Cohen-Macaulayness of divisorial sheaves in positive characteristic and extends Kollár's characteristic zero results to the $F$-singularities setting.

Findings

Divisorial sheaves can be Cohen-Macaulay under certain $F$-singularity conditions.

Relative canonical sheaves are compatible with arbitrary base change in specific $F$-pure fiber families.

Provides a positive characteristic analogue of Kollár's results on singularities.

Abstract

For a characteristic $p > 0$ variety $X$ with controlled $F$-singularities, we state conditions which imply that a divisorial sheaf is Cohen-Macaulay or at least has depth $\geq 3$ at certain points. This mirrors results of Koll\'ar for varieties in characteristic zero. As an application, we show that that relative canonical sheaves are compatible with arbitrary base change for certain families with sharply $F$-pure fibers.