Comparing powers of edge ideals
Abstract
Given a nontrivial homogeneous ideal I⊆k[x1,x2,…,xd]I⊆k[x1,x2,…,xd], a problem of great recent interest has been the comparison of the rrth ordinary power of II and the mmth symbolic power I(m)I(m). This comparison has been undertaken directly via an exploration of which exponents mm and rr guarantee the subset containment I(m)⊆IrI(m)⊆Ir and asymptotically via a computation of the resurgence ρ(I)ρ(I), a number for which any m/r>ρ(I)m/r>ρ(I) guarantees I(m)⊆IrI(m)⊆Ir. Recently, a third quantity, the symbolic defect, was introduced; as It⊆I(t)It⊆I(t), the symbolic defect is the minimal number of generators required to add to ItIt in order to get I(t)I(t). We consider these various means of comparison when II is the edge ideal of certain graphs by describing an ideal JJ for which I(t)=It+JI(t)=It+J. When II is the edge ideal of an odd cycle, our description of the structure of I(t)I(t) yields solutions to both the direct and asymptotic containment questions, as well as a partial computation of the sequence of symbolic defects.
Communicated by T. H. Ha