Density functions for epsilon multiplicity and families of ideals

TL;DR

The paper develops a comprehensive framework of density functions for algebraic invariants associated to filtrations, focusing on the epsilon multiplicity in graded settings. It constructs f_{A,{I^n}}, f_{A,{~I^n}}, and f_{ε(I)} by combining vector partition theory with volume theory of divisors, proving existence, continuity, differentiability, and invariance under integral closure. The results yield a piecewise-polynomial (or piecewise-polynomial-with-correctors) description of length functions, diagonal subalgebra multiplicities, and mixed multiplicities, and they connect to the geometry of the blowup X and the divisors H and E. In dimensions two and three, the density functions simplify to explicit rational or polynomial forms, yielding rational epsilon values and continuous families of rescaled multiplicities for diagonal subalgebras. Overall, the work provides concrete computational tools and structural insights into how saturated and ordinary powers of ideals encode asymptotic invariants via density functions.

Abstract

A density function for an algebraic invariant is a measurable function on $\mathbb{R}$ which measures the invariant on an $\mathbb{R}$-scale. This function carries a lot more information related to the invariant without seeking extra data. It has turned out to be a useful tool, which was introduced by the third author, to study the characteristic $p$ invariant, namely Hilbert-Kunz multiplicity of a homogeneous ${\bf m}$-primary ideal. Here we construct density functions $f_{A,\{I_n\}}$ for a Noetherian filtration $\{I_n\}_{n\in\mathbb{N}}$ of homogeneous ideals and $f_{A,\{\widetilde{I^n}\}}$ for a filtration given by the saturated powers of a homogeneous ideal $I$ in a standard graded domain $A$. As a consequence, we get a density function $f_{\varepsilon(I)}$ for the epsilon multiplicity $\varepsilon(I)$ of a homogeneous ideal $I$ in $A$. We further show that the function $f_{A,\{I_n\}}$ is continuous everywhere except possibly at one point, and $f_{A,\{\widetilde{I^n}\}}$ is a continuous function everywhere and is continuously differentiable except possibly at one point. As a corollary the epsilon density function $f_{\varepsilon(I)}$ is a compactly supported continuous function on $\mathbb{R}$ except at one point, such that $\int_{\mathbb{R}_{\geq 0}} f_{\varepsilon(I)} = \varepsilon(I)$. All the three functions $f_{A,\{I^n\}}$, $f_{A,\{\widetilde{I^n}\}}$ and $f_{\varepsilon(I)}$ remain invariant under passage to the integral closure of $I$. As a corollary of this theory, we observe that the `rescaled' Hilbert-Samuel multiplicities of the diagonal subalgebras form a continuous family.

Figures (2)

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Theorems & Definitions (93)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Remark 1.5
• Theorem 1.6
• Definition 2.1
• Theorem 2.3
• Lemma 2.5
• proof