The slope of the v-function and the Waldschmidt constant

In this talk, we discuss the asymptotic behaviour of the v-number of a Noetherian graded filtration I = {I_k} of a Noetherian N-graded domain R. Previous work by Ficcara and Sgroi showed that the v-number of the ideals of filtration, denoted as v(I_k) exhibits periodic linearity for sufficiently large values of k. Building on this foundation, we describe that the limit of the ratio v(I_k )/k as k approaches infinity not only exists but is also equal to the limit of the ratio of the minimum degree of non-zero elements of the ideal to k. This implies that all these linear functions associated with the v-numbers share the same slope. In particular, for a Noetherian symbolic filtration, the slope of these linear functions corresponds to the Waldschmidt constant of the ideal. In the last, we discuss the comparison of v-number and regularity of symbolic power of certain classes of monomial ideals.