Given a nontrivial homogeneous ideal I⊆k[x1,x2,…,xd], a
problem of great recent interest has been the comparison of the rth ordinary
power of I and the mth symbolic power I(m).
This comparison has been undertaken directly via an exploration of which
exponents m and r guarantee the subset containment I(m)⊆Ir
and asymptotically via a computation of the resurgence ρ(I), a number for
which any m/r>ρ(I) guarantees I(m)⊆Ir.
Recently, a third quantity, the symbolic defect, was introduced; as
It⊆I(t), the symbolic defect is the minimal number of generators
required to add to It in order to get I(t).
We consider these various means of comparison when I is the edge ideal of
certain graphs by describing an ideal J for which I(t)=It+J.
When I is the edge ideal of an odd cycle, our description of the structure
of I(t) yields solutions to both the direct and asymptotic containment
questions, as well as a partial computation of the sequence of symbolic
defects.Comment: Version 2: Revised based on referee suggestions. Lemma 5.12 was added
to clarify the proof of Theorem 5.13. To appear in the Journal of Algebra and
its Applications. Version 1: 20 pages. This project was supported by Dordt
College's undergraduate research program in summer 201