Torsors of the Jacobians of the universal Fermat curves

Let m≥3 be an integer. We show that every torsor of the Jacobian of the universal family of degree-m Fermat curve is necessarily a connected component of the Picard scheme. We show that the Jacobian of the generic degree-m Fermat curve has uncountably many non-isomorphic torsors. We give some results towards the Franchetta type problem for torsors of the Jacobian of the universal family of genus-g curves over Mg.