# Q-factorial and rational singularities on surfaces

Let $X$ be a normal surface. Is any rational singularity $\mathbf{Q}$-factorial? I've seen this somewhere for surfaces over fields, but what about the general case of an integral 2-dimensional excellent normal scheme?

In this generality it might not hold so what if we assume that $X$ is fibered ( = flat projective) over a Dedekind scheme? When can we hope for such a result to hold. Probably there are some problems depending on the characteristic.

I know that every surface fibered over $\mathrm{Spec} \mathbf{Z}$ is $\mathbf{Q}$-factorial. Are all its singularities rational?

I know that one has to be careful with the base scheme. Probably if the base scheme is a smooth projective curve over a field things might not work so well, but maybe if the base is $\mathrm{Spec} \mathbf{Z}$ things might become better.

• Just to clarify, the converse here means: "if the class group is finite locally ...", which is stronger than being $Q$-factorial. Mar 27, 2012 at 6:08
About the converse: one does need all the assumptions Karl mentioned in his answer. There are $2$-dim. complete local rings which are UFD but does not have rational singularity. One such example (due to Salmon) is $k(u)[[x,y,z]]/(x^2+y^3+uz^6)$ which is factorial for any field $k$.
Removing the hensenlian condition is also a problem: $R=k[x,y,z]_{(x,y,z)}/(x^r+y^s+z^t)$ where $r,s,t$ are pairwise prime, is factorial over any field $k$!