I've been spending some time looking at the surface $X=\mathcal{Z}(z^3-y(y^2-x^2)(x-1))$ in $\mathbb{A}^3$ (over an algebraically closed field of characteristic different from 3). The surface has four singularities, three of which are quite simple.

The singularity at the origin, however, seems to be a different story. I guess it's not rational, because the intersection of the blownup surface with the exceptional surface produces what seems to be an elliptic curve.

Let's let $T=A(X)$, the affine coordinate ring of $X$, and let $\mathfrak{m}=(x,y,z)\in \mathrm{MaxSpec}(T)$. I've been trying to figure out how to compute things such as $\mathrm{Pic}(T),\mathrm{Cl}(T_\mathfrak{m}),\hat{T}_\mathfrak{m}$.

I'm not really sure what I have at my disposal. I know that $\mathrm{Cl}(T)$ maps onto $\mathrm{Cl}(T_\mathfrak{m})$, and I know what the class group of $T$ is ($\mathrm{Cl}(T)\cong (\mathbb{Z}/3\mathbb{Z})^{(3)}\oplus \mathbb{Z}^{(2)}$ generated by $\mathfrak{p}_i=(y,z),(y-x,z),(y+x,z)$, each of order 3, and $\mathfrak{q}_i=(x,y+z),(x,y+\omega z)$, each of infinite order, where $\omega$ is a primitive third root of unity in $k$). But all of these divisors on the surface go through the origin, so I'm not sure that's much help.

Any ideas? Or even suggestions on where to look for something that could be helpful?

elliptic Gorenstein singularity. This paper by Prokhorov: springerlink.com/content/j56g17k172344p02 might help you. – Francesco Polizzi Nov 10 '11 at 13:26