Local cohomology and Lyubeznik numbers of $F$-pure rings

TL;DR

This paper investigates local cohomology modules and Lyubeznik numbers of $F$-pure rings, providing conditions for their vanishing, formulas for computation, and exploring their role in detecting properties of $F$-split varieties.

Contribution

It introduces new criteria for Lyubeznik number vanishing, derives a formula for their calculation in graded $F$-pure rings, and examines their geometric significance.

Findings

Sufficient conditions for Lyubeznik number vanishing

A formula for Lyubeznik numbers in graded $F$-pure rings

Lyubeznik numbers detect properties of $F$-split varieties

Abstract

In this article, we study certain local cohomology modules over $F$-pure rings. We give sufficient conditions for the vanishing of some Lyubeznik numbers, derive a formula for computing these invariants when the $F$-pure ring is standard graded and, by its means, we provide some new examples of Lyubeznik tables. We study associated primes of certain Ext-modules, showing that they are all compatible ideals. Finally, we focus on properties that Lyubeznik numbers detect over a globally $F$-split projective variety.