Let C be an integral projective curve over $\mathbb{C}$ and Jac(C) be its jacobian. Let $\overline{Jac(C)}$ be the compactified jacobian of C (the moduli space of rank 1 torsion free sheaves of degree 0 on C). Jac(C) acts on $\overline{Jac(C)}$ by tensor product. The question is :

What are the fixed points of this action ?

$\overline{Jac(C)}$ contains Jac(C) and obviously there is no fixed point on Jac(C) because the restriction to the action to it is a action by translation. So the fixed points are on $\overline{Jac(C)}$ minus Jac(C).

If C is rational (of geometric genus 0) and if $f: \mathbb{P}^{1} \longrightarrow C$ is the normalisation, there is a natural point in $\overline{Jac(C)}$ : F = $f_{*} \mathcal{O}_{\mathbb{P}^{1}}$. A more restricted question is :

Is F always a fixed point of the action ? Is it the once ?

(it is true for a nodal or cuspidal C but surely these examples are too simple ...)