v-numbers of filtrations of homogenous ideals

We extend the notion of v-number and local v-number of a homogeneous ideal for Noetherian filtrations of homogeneous ideals in a Noetherian N-graded domain. We consider integral closure filtrations, generalized symbolic power filtrations and filtrations of valuation ideals. We show that v-number and local v-number of these filtrations are quasi-linear type and limn→∞v_ p(Iₙ)/n exists. We explicitly compute v-numbers of symbolic powers of cover ideals of cycles and show that they are bounded by the Castelnuovo-Mumford regularity of those ideals.