Can someone please give me an example of a Noetherian normal local domain of dimension two such that there exists a prime ideal $P$ of height one having the property $P^{(n)}$ is not a principal ideal for any $n \geq 1$. Here $P^{(n)}$ is the [symbolic $n$-power](https://en.wikipedia.org/wiki/Symbolic_power_of_an_ideal).



Symbolic powers of a prime ideal of height one
Take $E$ an elliptic curve $zy^2 - x(x-z)(x-tz)$ say over $\mathbb{C}$ and choose a point $Q$ of infinite order (or so for instance the divisor $Q - O$ has infinite order in the divisor class group, here $O$ is the point at infinity).


It follows that in the graded ring of dimension $2$,
$$\mathbb{C}[x,y,z]/(zy^2 - x(x-z)(x-tz))$$
that the homogeneous ideal corresponding to $Q$, call it $P$, has the property that $P^{(n)}$ is never principal. Indeed, if $P^{(n)} = (f)$, then $(f)$ is homogeneous and the corresponding divisor $\mathrm{Div}\_E(f/z^{\deg f})$ is linearly equivalent to 0. This contradicts the infinite order of $P$.


One should point out that for any rational double point, the divisor class group is finite by a result of Lipman (if and only if under some hypotheses), so one has to leave the setting of rational singularities. The cone over an elliptic curve is the probably the simplest singularity that is not rational.


