The slope of v-function and Waldschmidt Constant

In this paper, we study asymptotic behaviour of the v-numbers of a Noetherian filtration I= [k]≥ 0 of ideals in a Noetherian graded domain R. Recently, it is shown that v(I[k]) is periodically linear in k for $k>>0$. We show that limk → ∞v(I[k])k exists and limk → ∞v(I[k])k= limk → ∞α(I[k])k. That is, all these linear functions have same slope, which is equal to limk → ∞α(I[k])k. In particular, for Noetherian symbolic filtration, we have limk → ∞v(I⁽ᵏ⁾)k=α̂(I), the Waldschmidt constant of I. Also, for several classes of square-free monomial ideals, we show that v(I⁽ᵏ⁾)≤ reg(R/I⁽ᵏ⁾) for all k≥ 1. As a special case, for any simple graph G, we show that v(J(G)⁽ᵏ⁾) ≤ reg(R/J(G)⁽ᵏ⁾) for all k ≥ 1 and v(J(G)⁽ᵏ⁾) = reg(R/J(G)⁽ᵏ⁾)=α(J(G)⁽ᵏ⁾)-1 for all k≥ 1 if and only if G is a Cohen-Macaulay very-well covered graph, where $J(G)$ denotes the cover ideal of G.