Where can I find the divisor class groups of du Val singularities?

Question

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?

Answer 1

You can mimic the quadric cone construction (if I did not make any mistakes in my computation). An $A_{2k-1}$ singularity is the vertex of the cone $S$ given by $x^2+y^2+z^{2k}=0$ in the weighted projective space $P(k,k,1,1)$ (note that the ambient wps is smooth at the vertex of $S$). Any point $p$ on the curve $C$ given by $x^2+y^2+z^{2k}=0$ in $P(k,k,1)$ yields a rational curve on $S$, which is not a principal divisor. Is this enough for your purposes?

You can do something similar for arbitrary $A_{2k}$, $D_m$ and $E_n$ singularities.

Answer 2

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), $\mathbb Z/(4)$, $(w,y)$.

$E_6$, $\mathbb Z/(3)$, $(x, w+iy^2)$.

$E_7$, $\mathbb Z/(2)$, $(w,x)$.

$E_8$, $0$.