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 xubics 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 nonreductive group. We also compute the Betti numbers of the Chow groups of the moduli space.
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