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Let $X\subset\mathbb{P}^3$ be a normal quartic surface with divisor class group $Cl(X)\cong\mathbb{Z}[H]$ generated by the hyperplane section.

What can we say about the singularities of $X$?

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Here are a few comments-too long to be in a comment.

First, it is clear that a singular point can only have multiplicity at most 4 and if it is of multiplicity 4, then it is a cone over a smooth quartic plane curve, and then your hypothesis will be violated.

If the multiplicity is 3, again, by projectiing, you see that it is birational to the plane and I think your hypothesis will be violated, though I haven't thought through this.

If multiplicity is 2, then I think it must be a rational double point and there is only one which is a UFD upto completion, namely the $E_8$ singularity.

As I said, these are just thoughts, not fully analyzed.

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  • $\begingroup$ @gbp No, you can not have a triple point. Consider the projection from the triple point, which gives a birational map to the plane. If $\Gamma$ is its graph, then, using the hypothesis, you can show that the projection to the plane from $\Gamma$ is an isomorphism, by showing that the map must be quasi-finite (since any curve contracted must be a line in the quartic, and there are no lines in the quartic) and then appealing to Zariski's Main Theorem . Then you have a birational morphism from the plane to the quartic, which is absurd. I do not know tacnodes which are UFDs, but not sure. $\endgroup$
    – Mohan
    Commented Mar 3, 2017 at 17:11

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