Comparing v-numbers of symbolic and ordinary powers of squarefree monomial ideals

Abstract

Let $I$ be a squarefree monomial ideal, and let $d$ denote the maximum degree of a minimal generator of $I$. We prove that \[ v(I^t)\le v(I^{(t)})+(t-1)(d-1) \] for all $t\ge1$, where $I^{(t)}$ denotes the $t$-th symbolic power of $I$. In particular, this bound does not depend on the number of variables in the ambient polynomial ring. On the other hand, for every fixed exponent $t\ge2$, the difference \[ v(I^{(t)})-v(I^t) \] can be arbitrarily large. Finally, we determine the $v$-numbers of the ordinary and symbolic powers of edge ideals of paths and cycles.