v-numbers of symbolic-ordinary discrepancy modules

Abstract

In this paper, we study the $\mathrm{v}$-number of the symbolic-ordinary discrepancy modules $M_t(I)=I^{(t)}/I^t$ over standard graded Noetherian rings. We show that this coincides with the $\mathrm{v}$-number of $I^t$ at embedded primes. Moreover, under mild assumptions on $I$, we show that the $\mathrm{v}$-number of $N_t(I)=I^{(t)}/I^{(t+1)}$ is the same as the $\mathrm{v}$-number of $I^{(t+1)}$. Furthermore, we provide combinatorial formulas for these $\mathrm{v}$-numbers when $t=2,3$ and $I$ is the edge ideal of a graph. Consequently, we classify all graphs for which $\mathrm{v}(I^2)=\mathrm{v}(I^{(2)})$.

arXiv comment14 Pages, Updated version