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 $\mathrm{Cl}(X_n)\cong \mathbb{Z}/n\mathbb{Z}$. The class of the prime divisor $D=V(x,z)$ generates: localization at $x$ is a unique factorization domain, so the divisor-class localization sequence leaves only $D$, while $\mathrm{div}(x)=nD$ gives its exact order.