A Note on the $k$-Buchsbaum Property of Symbolic Powers of Stanley-Reisner Ideals

Abstract

Let $I$ be the Stanley-Reisner ideal of pure simplicial complex $\Delta$ of dimension one. We shall give a formula for $S/I^{(r)}$ to be a $k$-Buchsbaum ring for each $r>0$, where $I^{(r)}$ is the $r$-th symbolic power of $I$. The main result is an improvement of the previous result in [MN] on the $k$-Buchsbaumness of $S/I^{(r)}$.