The symbolic defect sequence of edge ideals
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Date
2022-08-23
Authors
Reimer, Tessa
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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.
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Keywords
Commutative algebra, Edge ideals, Symbolic powers, Symbolic defect