Abstract. We investigate the geometry of the Simpson moduli space M P (P 3 ) of stable sheaves with Hilbert polynomial P (m) = 3m + 1. It consists of two smooth, rational components M 0 and M 1 of dimensions 12 and 13 intersecting each other transversally along an 11-dimensional, smooth, rational subvariety. The component M 0 is isomorphic to the closure of the space of twisted cubics in the Hilbert scheme Hilb P (P 3 ) and M 1 is isomorphic to the incidence variety of the relative Hilbert scheme of cubic curves contained in planes. In order to obtain the result and to classify the sheaves, we characterize M P (P 3 ) as geometric quotient of a certain matrix parameter space by a non-reductive group. We also compute the Betti numbers of the Chow groups of the moduli space.