The slope of the v-function and the Waldschmidt constant

In this paper, we study the asymptotic behavior of the v-number of a Noetherian graded filtration $I={\left\{I_{\left[k\right]}\right\}}_{k\ge 0}$ of a Noetherian $N$-graded domain R. Recently, it was shown that $v\left(I_{\left[k\right]}\right)$ is periodically linear in k for $k\gg 0$. We show that all these linear functions have the same slope, i.e. $\underset{k\to \infty }{\lim }\frac{v\left(I_{\left[k\right]}\right)}{k}$ exists, which is equal to $\underset{k\to \infty }{\lim }\frac{\alpha \left(I_{\left[k\right]}\right)}{k}$, where $\alpha \left(I\right)$ denotes the minimum degree of a non-zero element in I. In particular, for any Noetherian symbolic filtration $I={\left\{I^{\left(k\right)}\right\}}_{k\ge 0}$ of R, it follows that $\underset{k\to \infty }{\lim }\frac{v\left(I^{\left(k\right)}\right)}{k}=\hat{\alpha }\left(I\right)$, the Waldschmidt constant of I. Next, for a non-equigenerated square-free monomial ideal I, we prove that $v\left(I^{\left(k\right)}\right)\le \mathrm{reg}(R/I^{\left(k\right)})$ for $k\gg 0$. Also, for an ideal I having the symbolic strong persistence property, we give a linear upper bound on $v\left(I^{\left(k\right)}\right)$. As an application, we derive some criteria for a square-free monomial ideal I to satisfy $v\left(I^{\left(k\right)}\right)\le \mathrm{reg}(R/I^{\left(k\right)})$ for all $k\ge 1$, and provide several examples in support. In addition, for any simple graph G, we establish that $v\left(J{\left(G\right)}^{\left(k\right)}\right)\le \mathrm{reg}(R/J{\left(G\right)}^{\left(k\right)})$ for all $k\ge 1$, and $v\left(J{\left(G\right)}^{\left(k\right)}\right)=\mathrm{reg}(R/J{\left(G\right)}^{\left(k\right)})=\alpha \left(J{\left(G\right)}^{\left(k\right)}\right)-1$ for all $k\ge 1$ if and only if G is a Cohen-Macaulay very-well covered graph, where $J\left(G\right)$ is the cover ideal of G.