On the second powers of Stanley-Reisner ideals

In this paper, we study several properties of the second power $I_{\Delta}^2$ of a Stanley-Reisner ideal $I_{\Delta}$ of any dimension. As the main result, we prove that $S/I_{\Delta}$ is Gorenstein whenever $S/I_{\Delta}^2$ is Cohen-Macaulay over any field $K$. Moreover, we give a criterion for the second symbolic power of $I_{\Delta}$ to satisfy $(S_2)$ and to coincide with the ordinary power, respectively. Finally, we provide new examples of Stanley-Reisner ideals whose second powers are Cohen-Macaulay.