  Hausel introduced a commutative algebra -- the multiplicity algebra --
associated to a fixed point of the C^*-action on the Higgs bundle moduli space.
Here we describe this algebra for a fixed point consisting of a very stable
rank 2 vector bundle and zero Higgs field for a curve of low genus.
Geometrically, the relations in the algebra are described by a family of
quadrics and we focus on the discriminant of this family, providing a new
viewpoint on the moduli space of stable bundles. The discriminant in our
examples demonstrates that as the bundle varies, we obtain a continuous
variation in the isomorphism class of the algebra.
