Homogeneous ideals with minimal singularity thresholds

TL;DR

This paper extends a lower bound on singularity thresholds from holomorphic function germs to arbitrary ideals in regular local rings, and classifies homogeneous ideals in polynomial rings that attain this bound.

Contribution

It generalizes the lower bound on the log canonical threshold to a broader algebraic setting and classifies homogeneous ideals achieving equality, confirming a conjecture.

Findings

Established a lower bound for F-thresholds in regular rings.

Classified homogeneous ideals in polynomial rings that attain the bound.

Resolved a conjecture of Biv ext{a}-Ausina in the graded case.

Abstract

Let $(\mathcal{O}_n, \mathfrak{m})$ denote the ring of germs of holomorphic functions $\mathbb{C}^n\to \mathbb{C}$, and let $I\subseteq \mathcal{O}_n$ be an $\mathfrak{m}$-primary ideal. Demailly and Pham showed that $\mathrm{lct}(I) \geq \frac{1}{e_1(I)} + \dots + \frac{e_{n-1}(I)}{e_n(I)}$, where $e_j(I)$ is the mixed multiplicity $e(I,\dots, I, \mathfrak{m},\dots, \mathfrak{m})$, with $I$ repeated $j$ times and $\mathfrak{m}$ repeated $n-j$ times. We generalize the lower bound to the case of an arbitrary ideal of an excellent regular local (or standard-graded) ring of equal characteristic, with $\mathrm{lct}(I)$ replaced by the $F$-threshold $c^{\mathfrak{m}}(I)$ in positive characteristic. Our main result is a classification of homogeneous ideals in polynomial rings for which the lower bound is attained, resolving a conjecture of Bivi\`a-Ausina in the graded case.