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 $X$?



Divisor class group of quartic surfaces
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.


