Powers of Alexander Duals of Stanley-Reisner Ideals of Chessboard Complexes

Abstract

Let $\Delta_{m,n}$ be the chessboard complex, and let $J_{m,n}$ be the Stanley-Reisner ideal of its Alexander dual. We study the symbolic and ordinary powers of $J_{m,n}$, focusing on their homological and algebraic invariants. First, we characterize when $J_{m,n}^{(q)}$ has linear quotients. For the ordinary powers, we give sufficient conditions on $m$ and $n$ for $J_{m,n}^q$ to have linear quotients. We determine the maximal degree of a minimal generator and the Castelnuovo-Mumford regularity of $J_{m,n}^{(q)}$, obtaining explicit formulas for all $q\ge2$ and showing that both invariants have the same asymptotic slope. We also compute the initial degrees and Waldschmidt constant, proving that $\alpha(J_{m,n}^{(q)})$ is a degree-one quasi-polynomial of period two and $\widehat{\alpha}(J_{m,n})=mn/2$. Explicit formulas for the $\mathrm{v}$-numbers of symbolic and ordinary powers are obtained. Finally, for every $m\ge3$, we show that $J_{m,2m-1}$ is not weakly polymatroidal, disproving a conjecture of Lu and Wang in this setting.