  We prove that the moduli space of stable maps of degree 2 to the moduli space
of rank 2 stable bundles of fixed determinant O(-x) over a smooth projective
curve of genus g>2 has two irre- ducible components which intersect
transversely. One of them is Kir- wan's partial desingularization of the moduli
space of rank 2 semistable bundles with determinant isomorphic to O(y-x) for
some y in X. The other component is the partial desingularization of
PHom(sl(2)^*;W)//PGL(2) for a vector bundle W of rank g over the Jacobian of X.
We also show that the Hilbert scheme H, the Chow scheme C of conics in N and
the moduli space of stable maps of degree 2 are related by explicit
contractions.
