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. $\endgroup$ 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.

  • $\begingroup$ 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 $\endgroup$ 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. $\endgroup$ 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 $\endgroup$ 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. $\endgroup$ Jan 31 '13 at 21:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.