  This paper studies the components of the moduli space of rank 1, torsion-free
sheaves, or compactified Jacobian, of a non-Gorenstein curve. We exhibit a
generically reduced component of dimension equal to the arithmetic genus and
prove that this is the only non-smoothable component when the curve has a
unique singularity that is of finite representation type. Analogous results are
proven for the Hilbert scheme of points and the Quot scheme parameterizing
quotients of the dualizing sheaf.
