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.


The answer is NO in small dimension. Take a general surface $\bar{X}$ of degree $d\gg 0$ in $\mathbb{P}^3$, and take for $X$ the complement of a general hyperplane section. Then $X$ is smooth, $\mathrm{Pic}(X)=\mathrm{Cl}(X)=0$, but $\bar{X}$ contains no rational curve, so any normal (= smooth) curve in $X$ has a large Picard group.


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