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 Dr\'ezet and Maican and show that the Simpson moduli space $M=M_{4m\pm 1}(\mathbb P_2)$ 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 Poincar\'e polynomial of $M$ is presented.