Depth of some square free monomial ideals

Abstract

Let $I\supsetneq J$ be two square free monomial ideals of a polynomial algebra over a field generated in degree $\geq 1$, resp. $\geq 2$ . Almost always when $I$ contains precisely one variable, the other generators having degrees $\geq 2$, if the Stanley depth of $I/J$ is $\leq 2$ then the usual depth of $I/J$ is $\leq 2$ too, that is the Stanley Conjecture holds in these cases.