The defect of the F-pure threshold

TL;DR

This paper introduces the defect of the F-pure threshold as a new invariant for schemes, exploring its properties, semi-continuity, and behavior under various geometric operations.

Contribution

It defines the defect of the F-pure threshold for schemes and studies its properties, including semi-continuity and behavior under extensions and blow-ups.

Findings

The defect of the F-pure threshold is upper semi-continuous on schemes.

The defect satisfies Bertini-type theorems.

Behavior under flat extensions and blow-ups is characterized.

Abstract

Introduced by Takagi and Watanabe, the F-pure threshold is an invariant defined in terms of the Frobenius homomorphism. While it finds applications in various settings, it is primarily used as a local invariant. The purpose of this note is to start filling this gap by opening the study of its behavior on a scheme. To this end, we define the defect of the F-pure threshold of a local ring $(R,\mathfrak{m})$ by setting ${\rm dfpt}(R)=\dim (R) - {\rm fpt}(\mathfrak{m})$. It turns out that this invariant defines an upper semi-continuous function on a scheme and satisfies Bertini-type theorems. We also study the behavior of the defect of the F-pure threshold under flat extensions and after blowing up the maximal ideal of a local ring.