We investigate the geometry of the Simpson moduli space M P ([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]) of stable sheaves with Hilbert polynomial P ( m ) = 3 m + 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 ([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]) 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 MP ([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /]) as geometric quotient of a certain matrix parameter space by a nonreductive group. We also compute the Betti numbers of the Chow groups of the moduli space.