Lower Bounds on the F-pure Threshold and Extremal Singularities

TL;DR

This paper establishes lower bounds for the F-pure threshold of homogeneous polynomials, characterizes extremal cases, classifies associated singularities, and explores their geometric extremality in projective hypersurfaces.

Contribution

It provides a lower bound for the F-pure threshold of homogeneous polynomials and classifies extremal singularities up to coordinate changes.

Findings

F-pure threshold at least 1/(d-1) for homogeneous polynomials of degree d

Exact threshold characterization when f is in the q-th Frobenius power of the maximal ideal

Classification of extremal singularities with at most one isolated singularity

Abstract

We prove that if $f$ is a reduced homogenous polynomial of degree $d$, then its $F$-pure threshold at the unique homogeneous maximal ideal is at least $\frac{1}{d-1}$. We show, furthermore, that its $F$-pure threshold equals $\frac{1}{d-1}$ if and only if $f\in \mathfrak m^{[q]}$ and $d=q+1$, where $q$ is a power of $p$. Up to linear changes of coordinates (over a fixed algebraically closed field), we classify such "extremal singularities," and show that there is at most one with isolated singularity. Finally, we indicate several ways in which the projective hypersurfaces defined by such forms are "extremal," for example, in terms of the configurations of lines they can contain.