Tautological relations via the Abel-Jacobi map

Given a curve with marked points and integer weights on the markings, we consider the corresponding divisor. This defines an Abel-Jacobi map from the moduli space of marked curves to the universal Jacobian. This map extends to the space of curves of compact type. Pulling back relations from the universal Jacobian for various values of the weights gives a plethora of tautological relations on the space of curves of compact type. These relations include those discovered by Faber, Getzler, Belorousski-Pandharipande, and Tavakol.