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?