On the moduli scheme of stable sheaves supported on cubic space curves

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) = 3m + 1. It consists of two smooth, rational components M₀ and M₁ of dimensions 12 and 13 intersecting each other transversally along an 11-dimensional, smooth, rational subvariety. The component M₀ 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 M₁ 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.