# What is the Picard group of z3=y(y2−x2)(x−1)z^3=y(y^2-x^2)(x-1)?

I’m actually doing much more with this affine surface than just looking for the Picard group. I have already proved many things about this surface, and have many more things to look at it, but the Picard group continues to elude me.

One of the biggest problems seems to be that I’m not really sure what tools I have at my disposal to attempt such a problem. This surface has 4 singularities, one of which (the origin) is particularly nasty (the exceptional fiber over the origin when blowing-up is an elliptic curve).

Let $X=\mathcal{Z}(z^3-y(y^2-x^2)(x-1))$ be the surface. I know that the divisor class group of the surface is $\mathrm{Cl}(X)\cong (\mathbb{Z}/3\mathbb{Z})^{3}\oplus \mathbb{Z}^{2}$, and that the Picard group is (isomorphic to) a subgroup of this. If we let $p_i,i=1,2,3,4$ be the singular points of $X$, then there is an exact sequence

$0\rightarrow\mathrm{Pic}(X)\rightarrow\mathrm{Cl}(X)\rightarrow\bigoplus\mathrm{Cl}(\mathcal{\hat{O}}_{X,p_i})$,

though the hat (for completion) only really matters on the singularity at the origin. I have shown that the three generators for the torsion part of $\mathrm{Cl}(X)$ map to linearly independent elements of this last direct sum, so nothing in the torsion subgroup can be in the kernel of that map, which by exactness equals $\mathrm{Pic}(X)$.

This is where I get stuck. I don’t really know what else I can do; most of the things I can find in the literature seems to be only for nonsingular surfaces, or surfaces where the singularities are more simple than the mess at $(0,0,0)$.

I’d like to thank in advance anyone who takes some time to help me out.