Diagonal F-thresholds for determinants and Pfaffians

TL;DR

This paper calculates the diagonal F-thresholds for determinantal and Pfaffian hypersurfaces, revealing minimal thresholds in generic cases and employing advanced cohomology techniques in characteristic p.

Contribution

It provides explicit computations of diagonal F-thresholds for determinantal and Pfaffian hypersurfaces, extending understanding in positive characteristic algebraic geometry.

Findings

Diagonal F-thresholds for generic matrices are minimal, equal to the negative a-invariant.

Computed thresholds for symmetric and skew-symmetric cases, with symmetric case requiring additional polynomiality results.

Established a cohomology vanishing theorem for line bundles on flag varieties in characteristic p.

Abstract

We compute the diagonal F-thresholds of determinantal hypersurfaces arising from a generic matrix and from a generic symmetric matrix, as well as of the Pfaffian hypersurface arising from a generic skew-symmetric matrix of even size. The main ingredient is a cohomology vanishing theorem for certain line bundles on flag varieties in characteristic $p$. In the cases of the generic matrix and the generic skew-symmetric matrix, we show that the diagonal F-threshold attains its minimal possible value, namely the negative of the a-invariant. The symmetric case is more subtle and relies in addition on a polynomiality result for representations afforded by cohomology, building on work of the second author with VandeBogert.