A parametrization 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, provide a common parameter space for these loci, and show that the Simpson moduli space M = M4m ± 1(ℙ2) is a blow-down of a blow-up of a projective bundle over a smooth moduli space of Kronecker modules. Two different proofs of this statement are given.
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