Comparing powers of edge ideals.

Janssen, Mike; Kamp, Thomas; Vander Woude, Jason

Given a nontrivial homogeneous ideal I ⊆ k [ x 1 , x 2 , ... , x d ] , a problem of great recent interest has been the comparison of the r th ordinary power of I and the m th 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) ⊆ I r and asymptotically via a computation of the resurgence ρ (I) , a number for which any m / r > ρ (I) guarantees I (m) ⊆ I r . Recently, a third quantity, the symbolic defect, was introduced; as I t ⊆ I (t) , the symbolic defect is the minimal number of generators required to add to I t 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) = I t 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.

EDGES (Geometry)

Journal of Algebra & Its Applications, 2019, Vol 18, Issue 10, pN.PAG

0219-4988

Academic Journal

10.1142/S0219498819501846