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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?

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For a singular scheme, you should be using Cartier divisors instead of Weil divisors. See Hartshorne, for instance, for a treatment of these. – Will Sawin Nov 8 '11 at 5:06
Yes...Cartier divisors for the Picard group. I'd still like to unravel a little more about the singularity at (0,0,0). It seems to be very different from the others, and any time I search for references, I can only ever seem to fins stuff about rational singularities. – topspin1617 Nov 10 '11 at 5:00
The singularity in $(0,0,0)$ seems to be an elliptic Gorenstein singularity. This paper by Prokhorov: might help you. – Francesco Polizzi Nov 10 '11 at 13:26
Ah... at least I have a proper name now, thanks haha. Thanks for the reference... I don't know if anyone here knows much about these types of singularities, but are they really as... complicated as looking at that paper suggests? Just at a first look, there is so much terminology that is completely unfamiliar. – topspin1617 Nov 12 '11 at 0:18
So, I've spent some more time working on it, and I haven't really been able to accomplish all that much. I can show that there are certain elements NOT in $\mathrm{Pic}(T)$; for example, any torsion element of $\mathrm{Cl}(T)$. What techniques are there to use to study this surface? In Hartshorne, it seems the entire chapter is devoted to non-singular surfaces. Now, this one DOES only have 4 singularities (and none at infinity), so maybe those singularities can be somehow worked around. Even though I do want to look at the singularity at the origin as well, I'm still stumped by it... – topspin1617 Nov 20 '11 at 0:35

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