# On the depths of symbolic powers of the Stanley-Reisner ideal of a bow-tie complex

Consider the polynomial ring $$S=k[x_1,...,x_5]$$.
Consider the Stanley-Reisner ideal $$I$$ (i.e. the face ideal) of the simplicial complex which is a bow-tie $$\Delta:=\left$$.
So $$I=\left$$.
Let $$I^{(m)}$$ denote the $$m$$-th symbolic power of $$I$$ .

Is it true that depth $$S/I^{(m)}\ge 2, \forall m\ge 1$$ ?

Let $$R[x]$$ be a polynomial ring over a ring R. If $$J$$ is an $$R$$-ideal, then $$R[x]/JR[x] \cong R/J [x]$$.
Notice that the ideal $$I$$ has a primary decomposition $$I = (x_1,x_2) \cap (x_4,x_5),$$ and the generators of $$I$$ do not involve the variable $$x_3$$.
Now, to show that depth $$S/I^{(m)} \ge 2$$, as $$x_3$$ is a nonzerodivior (use the quoted statement above), if suffices to show that depth $$S'/J^{(m)} \ge 1$$ in $$S' = k[x_1,x_2,x_4,x_5]$$ with $$J = (x_1x_4,x_1x_5,x_2x_4,x_2x_5)S'$$. As the associated primes of $$J^{(m)}$$ are $$(x_1,x_2)S'$$ and $$(x_4,x_5)S'$$, an element such as $$x_1 + x_4$$ is a nonzerodivisor on $$S'/J^{(m)}$$.