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Asymptotic $\mathrm{v}$-number of graded families of ideals and the Newton-Okounkov region

Mousumi Mandal, Partha Phukan

Abstract

In this paper, we prove that for Noetherian graded families $\mathcal{I} = \{I_k\}_{k \ge 0}$ of homogeneous ideals, $\lim\limits_{k \to \infty} \frac{\mathrm{v}(I_k)}{k}$ exists, %equals $\lim\limits_{k \to \infty} \frac{α(I_k)}{k}$, and is given by $\frac{α(I_r)}{r}$ for some $r \ge 1$, where $α(I)$ denotes the initial degree. Extending these results to integral closures, we show that \( \lim\limits_{k\to\infty}\frac{\mathrm{v}(\overline{I_k})}{k} = \lim\limits_{k\to\infty}\frac{α(\overline{I_k})}{k}=\lim\limits_{k\to\infty}\frac{\mathrm{v}(I_k)}{k}=\lim\limits_{k\to\infty}\frac{α(I_k)}{k} \). For monomial ideals, we provide a combinatorial interpretation of these limits via Newton--Okounkov regions $Δ(\mathcal{I})$. %demonstrating that they equal $λ(Δ(\mathcal{I}))$, the minimum coordinate sum among vertices of $Δ(\mathcal{I})$. This connection is further generalized to arbitrary homogeneous ideals using good valuations. We also establish that both $\operatorname{reg}(I_k)$ and $\mathrm{v}(I_k)$ are eventually quasi-linear functions of $k$ for any Noetherian graded family. %Under suitable conditions, we prove the strict inequality $\mathrm{v}(I_k) < \operatorname{reg}(I_k)$. For stable monomial ideal $I$ we show that $\mathrm{v}(I) < \operatorname{reg}(I)$. Finally, for zero-dimensional homogeneous ideal $I$ in a polynomial ring $S$, we prove that $\mathrm{v}(I) < e(S/I)$, where $e(S/I)$ denote the multiplicity.

Asymptotic $\mathrm{v}$-number of graded families of ideals and the Newton-Okounkov region

Abstract

In this paper, we prove that for Noetherian graded families of homogeneous ideals, exists, %equals , and is given by for some , where denotes the initial degree. Extending these results to integral closures, we show that \( \lim\limits_{k\to\infty}\frac{\mathrm{v}(\overline{I_k})}{k} = \lim\limits_{k\to\infty}\frac{α(\overline{I_k})}{k}=\lim\limits_{k\to\infty}\frac{\mathrm{v}(I_k)}{k}=\lim\limits_{k\to\infty}\frac{α(I_k)}{k} \). For monomial ideals, we provide a combinatorial interpretation of these limits via Newton--Okounkov regions . %demonstrating that they equal , the minimum coordinate sum among vertices of . This connection is further generalized to arbitrary homogeneous ideals using good valuations. We also establish that both and are eventually quasi-linear functions of for any Noetherian graded family. %Under suitable conditions, we prove the strict inequality . For stable monomial ideal we show that . Finally, for zero-dimensional homogeneous ideal in a polynomial ring , we prove that , where denote the multiplicity.
Paper Structure (4 sections, 24 theorems, 73 equations)

This paper contains 4 sections, 24 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Asymptotic Behaviour of v-Number of a Noetherian graded family of homogeneous ideals
  4. A comparison of v-number with regularity and Multiplicity

Key Result

Theorem 1.2

Let $\mathcal{I} = \{I_k\}_{k \geq 0}$ be a Noetherian graded family of homogeneous ideals in a Noetherian $\mathbb{N}$-graded domain $R$. Then exists, and there exists $r \geq 1$ such that

Theorems & Definitions (59)

  • Definition 1.1
  • Theorem 1.2: Theorem \ref{['Proposition 1.4']}
  • Theorem 1.3: Theorem \ref{['mainresultofhomogeneousideals']}
  • Theorem 1.4: Theorem \ref{['analogue']}
  • Theorem 1.5: Theorem \ref{['NORforvaluations']}
  • Theorem 1.6: Theorem \ref{['reg/vnumberisquasilinear']}
  • Corollary 1.7: Corollary \ref{['v(I)<reg(I)forstablemonomialideals']}
  • Proposition 1.8: Proposition \ref{['vNumbervsHilbertMultiplicity']}
  • Definition 2.1
  • Definition 2.2
  • ...and 49 more