Let $J$ be an ideal in a Noetherian local ring $(R,m)$. It is well known that for any prime ideal $p\in Spec(R)$, $l(J\_p)\leq l(J)$, where $l(J)$ is the analytic spread of $J$. 


**Q) Are there examples of ideals $J$ such that $l(J\_p)\leq l(J)-1$ for all $p\supset J$ such that $ht p=ht J+1$ and $J^n\neq J^{(n)}$ for some $n$ (where $J^{(n)}$ is the nth symbolic power)?** 



Analytic spread of localization of an ideal
Let $(R, \mathfrak{m}): = \mathbb{C}[[x,y,z]]$ a formal power series. Let $J = (x^2, xy, xz) = \mathfrak{m}(x)$. We can check that $\ell(J) = 3$, $J^{(n)} = (x^n)$. This ideal satisfies the requirements.


