On lower bounds for the F-pure threshold of equigenerated ideals

TL;DR

This paper investigates the lower bounds of the F-pure threshold for equigenerated ideals in polynomial rings over fields of positive characteristic, classifying cases of equality and providing new bounds based on ideal invariants.

Contribution

It classifies ideals where the Takagi-Watanabe bound is tight and introduces a new lower bound for the F-pure threshold related to the height of a specific ideal.

Findings

Classified ideals attaining the Takagi-Watanabe bound.

Established a new lower bound for the F-pure threshold.

Connected the F-pure threshold to the height of the tau-invariant.

Abstract

Let $k$ be a field of positive characteristic and $R = k[x_0,\dots, x_n]$. We consider ideals $I\subseteq R$ generated by homogeneous polynomials of degree $d$. Takagi and Watanabe proved that $\mathrm{fpt}(I)\geq \mathrm{height}(I)/d$; we classify ideals $I$ for which equality is attained. Inspired by a result of de Fernex, Ein, and Musta\c{t}\u{a}, we give a lower bound on $\mathrm{fpt}(I)$ in terms of the height of $\tau(I^{\mathrm{fpt}(I)})$.