Remarks on the Hilbert depth of squarefree monomial ideals

Let $K$ be a infinite field, $S=K[x_1,\dots ,x_n]$ and $0\subset I⊊J\subset S$ two squarefree monomial ideals. In a previous paper we proved a new formula for the Hilbert depth of $J\mathrm{/}I$. In this paper, we illustrate how one can use the Stanley-Reisner correspondence between (relative) simplicial complexes and (quotients of) squarefree monomial ideals, in order to reobtain some basic properties of the Hilbert depth. More precisely, we show that $\mathrm{depth}(J\mathrm{/}I)\le \mathrm{hdepth}(J\mathrm{/}I)\le \dim (J\mathrm{/}I)$. Also, we show that $\mathrm{hdepth}(I)\ge \mathrm{hdepth}(S\mathrm{/}I)+1$, if $S\mathrm{/}I$ is Cohen-Macaulay.