The symbolic defect sequence of edge ideals

Abstract

Symbolic powers of homogeneous ideals are challenging to study, even for square-free monomial ideals. In particular, the containments between symbolic and ordinary powers of ideals have been widely studied for decades. The symbolic defect provides a measure of the difference between symbolic and ordinary powers of ideals, introduced by Galetto, Geramita, Shin, and Van Tuyl in 2019. In this thesis, we investigate the symbolic defect sequence of edge ideals of nite simple graphs. We classify exactly which terms of this sequence are non-zero and which terms are equal to one. Further, we present formulae for both the rst and second non-zero terms. We describe a complete formula for the symbolic defect sequence of edge ideals of odd cycles, as well as a partial formula for the symbolic defect sequence of edge ideals of unicyclic graphs. We provide both lower and upper bounds on the terms that cannot be computed with this partial formula.