A global description of the fine Simpson moduli space of 1-dimensional sheaves supported on plane quartics

A global description of the fine Simpson moduli spaces of 1-dimensional sheaves supported on plane quartics is given: we describe the gluing of the Brill-Noether loci described by Drézet and Maican and show that the Simpson moduli space M=M_{4m±1}(P_2) is a blow-down of a blow-up of a projective bundle over a smooth moduli space of Kronecker modules. An easy computation of the Poincaré polynomial of M is presented.