Homogeneous ideals with minimal singularity thresholds

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 Bivià-Ausina in the graded case.

Figures (5)

• Figure 1: Computation of $\mu$ for \ref{['ex:monomial-threshold']}
• Figure 2: $\Gamma(\mathfrak a_\bullet)$, together with $\mu\mathbf{1}$ and $H$.
• Figure 3: $\Gamma(\mathfrak a'_\bullet)$ in terms of $H$.
• Figure 4: $\Gamma(\mathfrak c)$, with $\mu\mathbf{1}$ computing $c(\mathfrak c)$ in red.
• Figure 5: $\Gamma(\mathfrak a)$, with $\mu\mathbf{1}$ computing $c(\mathfrak a)$ in red.

Theorems & Definitions (148)

• Theorem A: \ref{['thm:bound']}
• Theorem B: \ref{['thm:min-char']}
• Definition 2.2: Log Resolution
• Definition 2.4
• Definition 2.5: takagi_f-pure_2004
• Definition 2.6
• Proposition 2.7
• proof
• Proposition 2.8: hara_generalization_2003, Theorem 6.8
• Proposition 2.10: Properties of the singularity threshold