The defect of the F-pure threshold
Alessandro De Stefani, Luis Núñez-Betancourt, Ilya Smirnov
TL;DR
The work develops a global theory for the defect of the F-pure threshold via ${\rm dfpt}(R)=\dim(R)-{\rm fpt}(R)$, extending local F-purity invariants to schemes and proving upper semicontinuity and Bertini-type results. It introduces a differential-operator framework with $D^{(n,p^e)}_S$ and the Theta$_e(I)$-invariant, establishing a key formula ${\ell\ell}_R(R/I_e(R)) + \Theta_e(I) = \dim(S)(p^e-1)+1$ that links ${\rm dfpt}$ to a global convergent sequence. The paper then develops global Fedder-type criteria, proves semi-continuity and Bertini theorems for dfpt, analyzes behavior under flat extensions and blow-ups, and connects the theory to hypersurfaces, associated graded rings, and reductions to characteristic zero. The results yield structural insights into singularities, provide tools for measuring and controlling singularities across families, and situate F-pure thresholds within a broader birational-geometric and deformation-theoretic context. Overall, the work deepens the interaction between Frobenius-based invariants and global geometry, with implications for both algebraic and birational aspects of singularity theory.
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.