Brauer class over the Picard scheme of curves

We study the Brauer class rising from the obstruction to the existence of a tautological line bundle on Picard scheme of curves. If we consider the universal totally degenerate curve with a fixed dual graph, then, using symmetries of the graph, we give bounds on the period and index of the Brauer classes. As a result, we provide some division algebra of prime degree, serving as candidates for the cyclicity problem.