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Let R be a local, normal domain of dimension 2. Suppose that R contains a finite field. I am interested in knowing when the class group of R is torsion. In characteristic 0, this is known to be related to R being a rational singularity.

Lipman showed that if X is a desingularization of Spec(R), then one has an exact sequence:

$0 \to Pic^{0}(X) \to Cl(R) \to H $

Here $Pic^{0}(X)$ is the numerically trivial part of the Picard group of $X$, and $H$ is a finite group. Thus the second one is torsion if and only if the first one is. I do not have much understanding of the first group, unfortunately.

Does anyone know an answer or reference to this? Does anyone know an example in positive characteristic such that $Cl(R)$ is not torsion? Thanks a lot.

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I was going to point you to this nice paper I read recently about Frobenius and torsion in the class group ... then I saw who the poster was! –  Graham Leuschke Nov 29 '09 at 23:20
Hi Graham! Yes, unfortunately we need dimension 3! I think a general answer is probably not easy, but there are many smart people here. –  Hailong Dao Nov 30 '09 at 2:07
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2 Answers

up vote 22 down vote accepted

As requested in the comments, here's an example of a local, normal $2$-dimensional domain R in positive characteristic such that $\mathrm{Cl}(R)$ is not torsion: choose an elliptic curve $E \subset \mathbf{P}^2$ over a field $k$ such that $E(k)$ is not torsion, and take R to be the local ring at the origin of the affine cone on $E$ (i.e., $R = k[x,y,z]/(f)_{(x,y,z)}$ where $f$ is a homoegenous cubic defining $E$). This can be done over $k = \overline{\mathbf{F}_p(t)}$.

Proof: The normality follows from the fact that R is a hypersurface singularity (hence even Gorenstein) and isolated and $2$-dimensional (hence regular in codim 1). Blowing up at the origin defines a map $f:X \to \mathrm{Spec}(R)$. One can then show the following: $X$ is smooth, and $X$ can be identified with the Zariski localisation along the zero section of the total space of the line bundle $L = \mathcal{O}_{\mathbf{P^2}}(-1)|_E$ (these are general facts about cones). By Lipman's theorem, it suffices to show that $\mathrm{Pic}^0(X)$ contains non-torsion elements. As $X$ is fibered over $E$ with a section, the pullback $\mathrm{Pic}^0(E) \to \mathrm{Pic}^0(X)$ is a direct summand. As $\mathrm{Pic}^0(E) \simeq E(k)$ has non-torsion elements by assumption, so does $\mathrm{Pic}^0(X)$.

Also, an additional comment: In general, Lipman's theorem tells you that $\mathrm{Cl}(R)$ is torsion if and only if $\mathrm{Pic}^0(X)$ is torsion. Now $\mathrm{Pic}(X) \simeq \lim_n \mathrm{Pic}(X_n)$ where $X_n$ is the $n$-th order thickening of the exceptional fibre $E$. Because we are blowing up a point, the sheaf of ideals $I$ defining $E$ is ample on $E$. The kernel and cokernel of $\mathrm{Pic}(X_n) \to \mathrm{Pic}(X_{n-1})$ are identified with $H^1(E,I|_E^{\otimes n+1})$ and $H^2(E,I|_E^{\otimes n+1})$. As $I|_E$ is ample, it follows that the system "$\lim_n \mathrm{Pic}(X_n)$" is eventually stable. Thus, $\mathrm{Pic}(X) \simeq \mathrm{Pic}(X_n)$ for $n$ sufficiently big. As $X_n$ is a proper variety, it follows that if we are working over a finite field (resp. an algebraic closure of a finite field), then $\mathrm{Pic}^0(X)$ is finite (resp. ind-finite).

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Excellent, thanks! Is there a reference about your fact on elliptic curves over big fields? –  Hailong Dao Jan 17 '10 at 5:07
Also, why $X_n$ proper implies $Pic^0$ finite? –  Hailong Dao Jan 17 '10 at 5:13
Take any old elliptic curve E over F_q, and view it as an elliptic curve E' = E \times_{F_q} k over its function field k = k(E). The generic point in E'(k) will be non-torsion (as all the torsion is defined over F_q-bar). –  Bhargav Jan 17 '10 at 11:39
The connected component $\mathrm{Pic}^0(Y)$ of the Picard group $\mathrm{Pic}(Y)$ is representable by a group scheme of finite type when Y is a proper variety, so its F_q-points form a finite set. (For spaces like the X_n appearing above, the finiteness follows from that of $\mathrm{Pic}^0(X_0)$ using the description of the kernel and cokernel I mentioned in the answer). –  Bhargav Jan 17 '10 at 11:43
Beautiful! I wish I can upvote more. Please upvote this answer, this is really what MO is about. –  Hailong Dao Jan 17 '10 at 15:52
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I recently found some references: Theorem 4.5 of this paper and Theorem 4 + next Corollary of this paper which says:

If $(R,m,k)$ is a complete normal local domain of dimension $2$ such that $k$ is the algebraic closure of some finite field, then $Cl(R)$ is torsion!

It remains open what happens in other situations.

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You mean only the case $k=\mathbb F_q$ remains? This case follows from the case $\overline{F}_p$ at once, I think. But what if $k$ is algebraically closed and not equal to $\mathbb F_p$? –  VA. Jan 17 '10 at 0:02
Why would the $F_p$ follow? The class group may get smaller if one enlarge the field. But my comment was not very clear, thanks for pointing it out –  Hailong Dao Jan 17 '10 at 0:29
The statement will not be true for big fields k. Take the (completed) affine cone Y on an elliptic curve E in P^2 over k. Blowing up at the origin resolution X -> Y. On the other hand, X is the total space of O(-1) over E (suitably completed). In particular, Pic^0(X) contains Pic^0(E) =~ E(k) as a subgroup, and this can certainly have non-torsion elements if k is not algebraic over the prime field. –  Bhargav Jan 17 '10 at 1:38
@bhargav: would you write up your answer? I want to upvote it. –  Hailong Dao Jan 17 '10 at 2:03
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