Suppose that $X=Spec(A)$ is an affine variety over an algebraically closed field $k$ which is normal and such that $Cl(X)=0$. I am interested in hypersurfaces of $X$ which again satisfy this condition.
Q: Do there always exist hypersurfaces $X'\subset X$ such that $X'$ is again normal and $Cl(X')=0$? How 'many' such hypersurfaces do exist?
Since $A$ being a UFD is equivalent to the conditions above an algebraic formulation is as follows.
Q: Are there prime elements $p\in A$ such that $A/p$ is again a UFD? How 'many' such elements exist?
I am also interested in the same questions if the condition $Cl(X)=0$ is weakened to the condition that $Cl(X)$ is torsion.
I am aware that there is some work in this direction if $A$ is a polynomial ring. At least if $X$ is smooth and $dim(X)\geq 4$ we could also use the Lefschetz theorem.