On the Fluctuations of Associated Primes and Depths of Powers of Monomial Ideals

In this paper, we investigate the fluctuation phenomenon in the sets of associated primes and depths of powers of monomial ideals. We provide affirmative answers to two open questions posed in [8] concerning the existence of monomial ideals in K[x,y,z] whose associated primes exhibit prescribed patterns of appearance, disappearance, and reappearance of the maximal ideal across specified ranges of powers. For each construction, we also derive the corresponding fluctuation in depths. Furthermore, we construct a square-free monomial ideal that is nearly normally torsion-free but fails to satisfy the persistence property, thereby answering a question posed in [20] in the negative. This example arises as the edge ideal of the ideal minimally non-packing clutter Q_10, revealing an intriguing connection between combinatorial optimization and the algebraic properties of monomial ideals.