Class group of hypersurfaces of finite representation type

Question

Let $k$ be an algebraically closed field of characteristic different from $2,3$ and $5$, and let $R=k[[x,y,x_2,\dots,x_d]]/(f)$, where $f\in(x,y,x_2,\dots,x_d)^2$, $f\neq0$. By results of Buchweitz-Greuel-Schreyer and Knörrer, $R$ has finite Cohen-Macaulay representation type if and only if $R\cong k[[x,y,x_2,\dots,x_d]]/(g+x_2^2+\cdots+x_d^2)$, where $g\in k[x,y]$ is one of the following polynomials (one-dimensional ADE singularities):

• ($A_n$) $x^2+y^{n+1}$, $n\geq1$
• ($D_n$) $x^2y+y^{n-1}$, $n\geq4$
• ($E_6$) $x^3+y^4$
• ($E_7$) $x^3+xy^3$
• ($E_8$) $x^3+y^5$

I am interested in the class group $\mathrm{Cl(R)}$ of these rings, for any $d\geq1$. Has anyone already computed it?

For $d=2$, i.e., two-dimensional ADE singularities, the class group is known and can be computed using the Auslander-Reiten quiver of $R$ (see e.g. [Yos90, $\S13$]). However, I could not find similar (even partial) computations for $d>2$ in the literature. Could you point me to some references in this direction?

Thank you in advance!

References:

[Yos90] Yoshino, Yuji, Cohen-Macaulay modules over Cohen-Macaulay rings. London Mathematical Society Lecture Note Series, 146. Cambridge University Press, Cambridge, 1990.

Answer 1

When $d\geq 3$, these are isolated hypersurface singularities of dimension at least $4$, so are UFD by the Grothendieck's local Lefschetz Theorem.

When $d=2$ and the field has characteristic $0$, the class group is $\mathbb Z^{r-1}$ where $r$ is the number of branches of $g$. See 2.2.6 of Kollár's paper "Flip, flops, minimal models, etc". That result was for the Picard group of the punctured spectrum of the affine hypersurface, but it should agree with the class group of the completed local rings (you might need to use old results by Danilov here).

I do not know if anyone has worked out the case of positive char. But at least we know that the class group is torsion-free, see the references in this MO question and answer.

A good reference for local Picard group (which for a normal local point of dimension $2$ or higher is just the class group at that point) is Kollár's paper "Maps between local Picard groups", where you can also find reference to the Grothendieck-Lefchetz theorem mentioned above.