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 $\mathcal{I} = \{I_k\}_{k \ge 0}$ of homogeneous ideals, $\lim\limits_{k \to \infty} \frac{\mathrm{v}(I_k)}{k}$ exists, %equals $\lim\limits_{k \to \infty} \frac{\alpha(I_k)}{k}$, and is given by $\frac{\alpha(I_r)}{r}$ for some $r \ge 1$, where $\alpha(I)$ denotes the initial degree. Extending these results to integral closures, we show that \( \lim\limits_{k\to\infty}\frac{\mathrm{v}(\overline{I_k})}{k} = \lim\limits_{k\to\infty}\frac{\alpha(\overline{I_k})}{k}=\lim\limits_{k\to\infty}\frac{\mathrm{v}(I_k)}{k}=\lim\limits_{k\to\infty}\frac{\alpha(I_k)}{k} \). For monomial ideals, we provide a combinatorial interpretation of these limits via Newton--Okounkov regions $\Delta(\mathcal{I})$. %demonstrating that they equal $\lambda(\Delta(\mathcal{I}))$, the minimum coordinate sum among vertices of $\Delta(\mathcal{I})$. 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.