$F$-pure threshold for the symmetric determinantal ring

TL;DR

This paper computes the $F$-pure threshold of the symmetric determinantal ring at its maximal ideal, providing a characteristic-independent value that also determines the log canonical threshold in characteristic zero.

Contribution

It establishes a characteristic-independent value for the $F$-pure threshold of symmetric determinantal rings, linking positive characteristic and characteristic zero invariants.

Findings

The $F$-pure threshold is explicitly calculated for symmetric determinantal rings.

The value is independent of the characteristic of the base field.

The result also determines the log canonical threshold in characteristic zero.

Abstract

We give a value for the $F$-pure threshold at the maximal homogeneous ideal $\mathfrak{m}$ of the symmetric determinantal ring over a field of prime characteristic. The answer is characteristic independent, so we immediately get the log canonical threshold in characteristic zero as well.