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

In this paper, we prove that for Noetherian graded families $\mathcal{I}=\{I_k{\}}_{k\ge 0}$ of homogeneous ideals, $\underset{k\to \mathrm{\infty }}{\lim }\frac{\mathrm{v}(I_k)}{k}$ exists, %equals $\underset{k\to \mathrm{\infty }}{\lim }\frac{\alpha (I_k)}{k}$, 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 $\underset{k\to \mathrm{\infty }}{\lim }\frac{\mathrm{v}(\overline{I_k})}{k}=\underset{k\to \mathrm{\infty }}{\lim }\frac{\alpha (\overline{I_k})}{k}=\underset{k\to \mathrm{\infty }}{\lim }\frac{\mathrm{v}(I_k)}{k}=\underset{k\to \mathrm{\infty }}{\lim }\frac{\alpha (I_k)}{k}$. For monomial ideals, we provide a combinatorial interpretation of these limits via Newton–Okounkov regions $\mathrm{\Delta }(\mathcal{I})$. %demonstrating that they equal $\lambda (\mathrm{\Delta }(\mathcal{I}))$, the minimum coordinate sum among vertices of $\mathrm{\Delta }(\mathcal{I})$. This connection is further generalized to arbitrary homogeneous ideals using good valuations. We also establish that both $\mathrm{reg}(I_k)$ 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 $\mathrm{v}(I_k)<\mathrm{reg}(I_k)$. For stable monomial ideal $I$ we show that $\mathrm{v}(I)<\mathrm{reg}(I)$. Finally, for zero-dimensional homogeneous ideal $I$ in a polynomial ring $S$, we prove that $\mathrm{v}(I)<e(S\mathrm{/}I)$, where $e(S\mathrm{/}I)$ denote the multiplicity.