Let $X$ be a projective Calabi-Yau threefold with a single ordinary double point at $x \in X$, and smooth elsewhere. Is $X$ necessarily factorial?


I suspect that the answer is "yes", for the following reason. Some neighbourhood of $x$ is analytically isomorphic to the 'conifold' geometry
$$
xy - wz = 0 ~~\mathrm{in}~~\mathbb{C}^4~.
$$
This is not factorial; $x = w = 0$ is a non-Cartier divisor, and blowing up along this gives a small projective resolution of the conifold. However, analytic isomorphisms do not preserve the local class group, and $\mathcal{O}\_{X,x}$ may nevertheless be a UFD. I suspect that this is necessarily true, since if not, we could blow up along the non-Cartier divisor, but one-nodal projective Calabi-Yau threefolds do not admit projective resolutions. I would like to know whether this argument can be made rigorous, or replaced by a simpler one.




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Edit: It may be important to assume that $X$ is smoothable, and this is the case which matters to me.



Factoriality of one-nodal Calabi-Yau threefolds
Let $X$ be a Calabi-Yau three-fold with only ordinary double points. One can always find a small resolution $Y\rightarrow X$ where $Y$ is a (not necessarily Kaehler) complex manifold. Let $C\_1,\ldots,C\_n$ be the exceptional curves. Friedman proved in his paper "Simultaneous Resolution of Threefold Double Points" that $X$ has an infinitesimal smoothing if and only if there is a linear dependence relation $\sum\_i a\_i[C\_i]=0$ in $H\_2(Y,{\mathbb R})$ with all $a\_i$ non-zero. Furthermore, by unobstructedness of the deformation theory of Calabi-Yau varieties with isolated ordinary double points (proved by Ran and Tian), having an infinitesimal smoothing is equivalent to having a smoothing.


In your case, $X$ just has a single ODP, so there is one exceptional curve $C$.
So there is a smoothing if and only if $[C]=0$ in $H\_2(Y,{\mathbb R})$. This is in turn
equivalent to $X$ being factorial (there is no algebraic small resolution).


So what you want is true: if $X$ is smoothable, it is factorial. However, there are examples of non-smoothable $X$ with one ODP, and these are not factorial.


