Let $X \subset \mathbf{P}^4$ be a complex threefold hypersurface with isolated singularities.

We denote as usual by $\textrm{Cl}(X)$ the group of Weil divisors modulo linear equivalence and by $\textrm{Pic}(X)$ the group of Cartier divisors modulo linear equivalence. We also write $G(X):=\textrm{Cl}(X)/\textrm{Pic}(X)$.

Recall that

$\bullet$ $X$ is called factorial if every Weil divisor is a Cartier divisor; by Lefschetz theorem, this is equivalent to say that $\textrm{Pic}(X)=\textrm{Cl}(X)=\mathbf{Z}$, generated by the hyperplane section;

$\bullet$ $X$ is called $\mathbf{Q}$-factorial if every Weil divisor has a multiple which is a Cartier divisor; this is equivalent to say that $G(X)$ is a torsion group.

Of course if $X$ factorial then $X$ is $\mathbf{Q}$-factorial, because if $X$ is factorial then $G(X)=0$. I'm interested in the other implication, so my first question is

Question 1. Assume that $X \subset \mathbf{P}^4$ is a $\mathbf{Q}$-factorial threefold with isolated singularities. Does this imply that $X$ is factorial? If not, what is a counterexample?

It is known that the answer to Question 1 is yes when $X$ is nodal, i.e. contains only ordinary double points. The way I see this is the following. There is an exact sequence (I think it is called Jaffe's exact sequence) $$0 \to \textrm{Pic}(X) \to \textrm{Cl}(X) \to \bigoplus_{p \in \textrm{Sing}(X)} \textrm{Cl}(\mathscr{O}_{X, p})$$ and the last group injects into $\bigoplus_{p \in \textrm{Sing}(X)} \textrm{Cl}(\widehat{\mathscr{O}}_{X, p})$. On the other hand, if $p$ is a node then $\textrm{Cl}(\widehat{\mathscr{O}}_{X, p}) \cong \mathbf{Z}$, so we have an inclusion $$G(X) \hookrightarrow \bigoplus \mathbf{Z}.$$ This implies that if $G(X)$ is a torsion group then necessarily $G(X)=0$, so factoriality and $\mathbf{Q}$-factoriality are equivalent conditions in this case. These considerations led me to the following

Question 2. Let $X \subset \mathbf{P}^4$ be a threefold hypersurface with isolated singularities and let $p \in \textrm{Sing}(X)$. Is it true that $\textrm{Cl}(\mathscr{O}_{X, p})$ is either torsion-free or zero? Or, still better, is it true that $\textrm{Cl}(\widehat{\mathscr{O}}_{X, p})$ is either torsion-free or zero?

An affirmative answer to Question 2 would imply an affirmative answer to Question 1, by the same argument used in the nodal case.

I'm particulary interested in the case where all the singularities of $X$ are $ordinary$, i.e. the corresponding tangent cone is a cone over a smooth surface in $\mathbf{P}^3$.

Any answer or reference to the existing literature will be greatly appreciated. Thank you!

EDIT. As pointed out by Artie in his comment, Question 1 has a positive answer when $X$ is a Gorenstein threefold with terminal singularities (here the assumption $X \subset \mathbb{P}^4$ is not necessary).

  • 2
    $\begingroup$ Another data point: the answer to Question 1 is positive whenever X has only terminal singularities. A reference is Cutkosky, "Elementary contractions of Gorenstein threefolds." $\endgroup$ – user5117 Jan 29 '13 at 14:38
  • $\begingroup$ @Artie: Thank you for the reference! Terminal 3-folds singularities are isolated cDV, hence rational. I'm espacially looking for results about ordinary singularities, but what you are pointing out is surely worth knowing. $\endgroup$ – Francesco Polizzi Jan 29 '13 at 14:59

The answer to Question 2 is true. If $(R,m)$ is a local complete intersection of equicharacterstic $0$ and of dimension $3$ then the Picard group of $Y = \text{Spec} R-\{m\}$ is torsion-free. This group agrees with the class group of $R$ when $R$ has isolated singularity. The proof in this case essentially follows from the proof of the Grothendieck-Lefschetz theorem (SGA somewhere), which states that if $R$ is a complete intersection and $\dim R\geq 4$ then $Y$ has trivial Picard group.

For reference and some extensions see the papers by Badescu and Robbiano quoted here.

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  • $\begingroup$ Dear Hailong, thank you very much for your nice answer! It was the best thing I was hoping for :-) Since I started to work on these questions quite recently, and so I'm always worried of making some trivial mistake, may I ask you if the argument I sketched giving the implication (Question 2 true => Question 1 true) seems to you correct? Best, Francesco $\endgroup$ – Francesco Polizzi Jan 31 '13 at 8:58
  • $\begingroup$ Dear Francesco, it looks fine to me. By the way, I forgot a more recent reference by Hartshorne-Polini, front.math.ucdavis.edu/1301.3222. $\endgroup$ – Hailong Dao Jan 31 '13 at 9:37
  • $\begingroup$ Dear Hailong, thank you again. I find your answer and the included references very satisfactory, so I will accept it. Best, Francesco $\endgroup$ – Francesco Polizzi Jan 31 '13 at 12:15
  • $\begingroup$ There is also a recent preprint by Kollár, Grothendieck-Lefschetz type theorems for the local Picard group (arxiv.org/pdf/1211.0317v2.pdf). It does not address this question directly, but obtains some nice results on the local Picard group. $\endgroup$ – Sándor Kovács Jan 31 '13 at 21:22

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