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Diagonal F-thresholds for determinants and Pfaffians

Barbara Betti, Claudiu Raicu, Francesco Romeo, Jyoti Singh

TL;DR

The paper determines diagonal $F$-thresholds for hypersurfaces defined by determinants and Pfaffians in positive characteristic. By developing cohomology vanishing on flag varieties and leveraging polynomial functors and modular representation theory, it yields explicit values for generic determinantal rings, generic symmetric determinants, and Pfaffians, with the symmetric case requiring a refined polynomiality argument and characteristic-dependent lower bounds. The results connect the diagonal $F$-threshold to the $a$-invariant and establish sharp equalities in several cases, including the Pfaffian via a degeneration to determinants. This work advances the understanding of Frobenius-power containment for determinantal and Pfaffian singularities and provides tools applicable to broader families of determinantal hypersurfaces in positive characteristic.

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.

Diagonal F-thresholds for determinants and Pfaffians

TL;DR

The paper determines diagonal -thresholds for hypersurfaces defined by determinants and Pfaffians in positive characteristic. By developing cohomology vanishing on flag varieties and leveraging polynomial functors and modular representation theory, it yields explicit values for generic determinantal rings, generic symmetric determinants, and Pfaffians, with the symmetric case requiring a refined polynomiality argument and characteristic-dependent lower bounds. The results connect the diagonal -threshold to the -invariant and establish sharp equalities in several cases, including the Pfaffian via a degeneration to determinants. This work advances the understanding of Frobenius-power containment for determinantal and Pfaffian singularities and provides tools applicable to broader families of determinantal hypersurfaces in positive characteristic.

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 . 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.
Paper Structure (15 sections, 9 theorems, 108 equations)

This paper contains 15 sections, 9 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hypersurface rings and duality bru-her
  4. Representation theory basics greenjantzen
  5. Polynomial functors greenfri-sus
  6. Flag varieties and flag bundles BCRV
  7. Cohomology vanishing and polynomiality
  8. Cohomology vanishing on flag bundles
  9. Cohomology as a polynomial representation
  10. Generic determinantal rings
  11. Generic symmetric determinant
  12. The upper bound
  13. The lower bound in characteristic $p=2$
  14. The lower bound via cohomology and representation theory
  15. Generic Pfaffian

Key Result

Lemma 2.1

Suppose that $\mu$ is $p$-restricted, and $\lambda=\lambda^0+p\lambda^1+\cdots+p^s\lambda^s$, where each $\lambda^i$ is $p$-restricted. If $L_{\lambda}$ occurs in a tensor product $L_{p^s\mu}\otimes L_{\nu}$ for some partition $\nu$ then $|\lambda^s|\geq |\mu|$.

Theorems & Definitions (20)

  • Example 1.1
  • Lemma 2.1
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • proof : Proof of Theorem \ref{['thm:relative-coh-vanishing-Flag']}
  • proof : Proof of \ref{['eq:vanishing-Polyfun']}
  • Theorem 3.3
  • Theorem 3.4
  • ...and 10 more