Asymptotic invariants of ideals with Noetherian symbolic Rees algebra and applications to cover ideals

Abstract

Let I be an ideal whose symbolic Rees algebra is Noetherian. For m≥1, the m-th symbolic defect, sdefect(I,m), of I is defined to be the minimal number of generators of the module I(m)Im. We prove that sdefect(I,m) is eventually quasi-polynomial as a function in m. We compute the symbolic defect explicitly for certain monomial ideals arising from graphs, termed cover ideals. We go on to give a formula for the Waldschmidt constant, an asymptotic invariant measuring the growth of the degrees of generators of symbolic powers, for ideals whose symbolic Rees algebra is Noetherian.