TITLE: Where can I find the divisor class groups of du Val singularities?
QUESTION [9 upvotes]: The du Val singularities are the simplest type of surface singularities.  Each type of du Val singularity has a divisor class group.  Specifically, let $X$ be a surface with an isolated singularity at $P$; then the (analytic or étale) local ring at P depends only on the type of the singularity, and has a divisor class group.
The most familiar example is the quadric cone (A1 singularity), found in many algebraic geometry textbooks.  A line $L$ passing though the vertex of the cone is not locally principal, but $2L$ is, and we find that the divisor class group has order $2$.  (Note: in general an A1 singularity will be étale locally, but not Zariski locally, isomorphic to the vertex of the cone.  As far as I can see, there's no reason in general to expect the generator of the divisor class group to come from a divisor on the ambient surface; we may well have to pass to an étale (or analytic) neighbourhood.)
In a beautiful article, Lipman (Pub. Math. IHES 1969) studied these and computed the (finite) divisor class group of each du Val singularity.  However, as far as I can see, he does not give explicit generators like we have in the example of the quadric cone.
So:

Is there in the literature an explicit description (i.e. with explicit generators)
  of the divisor class groups of the du Val singularities?

REPLY [4 votes]: I do not know a reference, but here is my guess. I will use the notion of the wikipedia article. The order is: type, class group, the generator ideals. ($i^2=-1$, and I assume char. 0 for simplicity)
$A_n$, $\mathbb Z/(n+1)$, $(w+ix, y)$. 
$D_n$ ($n$ even), $\mathbb Z/(2)\oplus \mathbb Z/(2)$, $(w,y), (w, x+iy^{(n-2)/2})$. 
$D_n$ ($n$ odd), $\mathbb Z/(4)$, $(w,y)$. 
$E_6$, $\mathbb Z/(3)$, $(x, w+iy^2)$.
$E_7$, $\mathbb Z/(2)$, $(w,x)$. 
$E_8$, $0$.