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?



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


