Asymptotic behaviour and stability index of v-numbers of graded ideals

Recently, Ficarra and Sgroi initiated the study of v-numbers of powers of graded ideals. They proved that for a graded ideal I in a polynomial ring S, v (Iᵏ) is a linear function in k for k>>0. Later, Ficarra conjectured that if I is a monomial ideal with linear powers, then v (Iᵏ) = (I) k-1 for all k 1, where (I) denotes the initial degree of I. In this paper, we generalize this conjecture for graded ideals. We prove this conjecture for several classes of graded ideals: principal ideals, ideals I with depth (S/I) =0, cover ideals of graphs, t-path ideals, monomial ideals generated in degree 2, edge ideals of weighted oriented graphs. We reduce the conjecture for several classes of graded ideals (including square-free monomial ideals) by showing it is enough to prove the conjecture for k=1 only. We define the stability index of the v-number for graded ideals and investigate the stability index for edge ideals of graphs.