Diagonal F-thresholds for determinants and Pfaffians

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.

💡 Research Summary

In this paper the authors determine the diagonal F‑threshold c(R) for three families of hypersurface rings defined over an algebraically closed field k of characteristic p > 0. The hypersurfaces are given by (1) the determinant of a generic n × n matrix, (2) the determinant of a generic symmetric n × n matrix, and (3) the Pfaffian of a generic skew‑symmetric matrix of even size. The diagonal F‑threshold is defined as
c(R) = lim_{q→∞} v_R(q)/q, where v_R(q) = max{ d | 𝔪^d ⊆ 𝔪^{