On the Hilbert depth of the quotient ring of the edge ideal of a complete bipartite graph

Abstract

Let $n\geq m$ be two positive integers, $S_{n,m}=K[x_1,\ldots,x_n,y_1,\ldots,y_m]$ and $I_{n,m}=(x_iy_j\;:\;1\leq i\leq n,1\leq j\leq m)\subset S_{n,m}$ the edge ideal of a complete bipartite graph. Denote $h(n,m)=\operatorname{hdepth}(S_{n,m}/I_{n,m})$. We prove that $h(n,m)\geq \left\lceil \frac{n}{2} \right\rceil$ and the equality holds if $m$ belong to a certain interval centered in $\left\lceil \frac{n} {2} \right\rceil$. Also, we find some tight bounds for $h(n,n)$ and we prove several inequalities between $h(n,m)$ and $h(n,m')$.