Divisor class group of an A-type surface singularity ($\mathrm{Cl}(k[x,y,z]/(xy-z^n))$)

For $n\ge 2$, the normal affine surfacehas divisor class group $\mathbb{Z}/n\mathbb{Z}$. Inverting $x$ gives $A_n[x^{-1}]\cong k[x,x^{-1},z]$, a unique factorization domain. The only prime Weil divisor over $x=0$ is $D=V(x,z)$, so the Nagata theorem for divisor class groups makes $[D]$ a generator.

At the generic point of $D$, $y$ is a unit and $x=z^n/y$. The local discrete valuation ring has uniformizer $z$, giving $\mathrm{div}(x)=nD$. This proves that the order divides $n$; exactness requires a further step. If $mD=\mathrm{div}(q)$, then $q$ is a unit on $X_n\setminus D$. The units of $k[x,x^{-1},z]$ are precisely $c{x}^r$, so $mD=rnD$. Thus every relation is a multiple of $nD$, proving exact order $n$.