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

Abstract

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 $S$, $\mathrm{v}(I^k)$ is a linear function in $k$ for $k>>0$. Later, Ficarra conjectured that if $I$ is a monomial ideal with linear powers, then $\mathrm{v}(I^k)=\alpha(I)k-1$ for all $k\geq 1$, where $\alpha(I)$ 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 $\mathrm{depth}(S/I)=0$, 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 $\mathrm{v}$-number for graded ideals and investigate the stability index for edge ideals of graphs.