# Do divisors of degree g with this property exist in general

I have the following question. It's a long shot, but worth the try.

Let X be a compact connected Riemann surface of genus $g\geq 2$. Does there exist an effective divisor $D$ on $X$ of degree $g$ such that, if $D=\sum_{i=1}^g D_i$ is the prime decomposition of $D$, the image of the line bundle $\mathcal{O}_X(D_i-D_j)$ is torsion in the Picard group of $X$ for all $i,j = 1,\ldots,g$?

(Note that the $D_i$ are not necessarily distinct.)

The answer is yes for modular curves by Manin-Drinfeld. Namely, take $D$ to be a degree $g$ divisor supported on the cusps. The answer is also yes for Fermat curves.

The answer would be no if there would exist a Jacobian variety without torsion. Fortunately, Jacobians have a lot of torsion.

If it helps assume that $X$ can be defined over some number field (as an algebraic curve).

If the answer to my question is not completely trivial, there should be some general theory about such divisors. If yes, does there exist a good reference?

• What is to stop $D = gP$ for some point $P\in X$? Also, is $\mathcal O_X(D_i-D_j)$ supposed to be torsion for every $i,j$? – Jack Huizenga Dec 14 '11 at 17:38
• You're right. I wanted to exclude this example. And yes, I want $\mathcal{O}_X(D_i-D_j)$ to be a torsion element in the Picard group for all $i,j=1,\ldots,g$. Moreover, the $D_i$ are points (with multiplicity 1). – Ariyan Javanpeykar Dec 14 '11 at 18:18
• It is not true that the answer is always yes for modular curves : for example, there could be only one cusp. In this direction Matthew Baker has proved that if $N>479$ is prime then there is no torsion packet at all on the modular curve $X_0^+(N) = X_0(N)/W_N$ (see his article Torsion points on modular curves). – François Brunault Dec 14 '11 at 22:42

Are the $D_i$ supposed to be points? This is not made clear in the question. If the $D_i$ are points and you exclude Jack's example where they are all the same, then the answer is no for the general curve. That is because, for the general curve (of genus at least three) $P-Q$ is not torsion for any pair of distinct points of the curve. I don't have a reference for this but it's not very hard.
• To answer your question: yes the $D_i$ are points. Sorry for being unclear about thas. Say we fix $g$. Do I understand correctly that the answer to my question is no for all but finitely many curves of genus $g$? – Ariyan Javanpeykar Dec 14 '11 at 18:22
• I understand. But I don't see how to prove this easily. Fixing a point Q on X, the claim is that if X is general of genus at least three, we have that the image of C in the Jacobian via the map $P\mapsto P-Q$ contains no torsion elements...What kind of argument should I look for? – Ariyan Javanpeykar Dec 14 '11 at 19:39
• For any given $n$, the set of curves that has distinct points $P,Q$ with $P-Q$ of order $n$ is closed in moduli, that's more or less clear. Now you have to find, for each $n$, a curve not in this closed subset and conclude that this closed subset is not the whole moduli space. Off the top of my head, I don't remember all the details. – Felipe Voloch Dec 14 '11 at 19:59