An example of an Sl₂-Hilbert scheme with multiplicities

We determine the invariant Hilbert scheme of the zero fibre of the moment map of an action of Sl_2 on (C^2)^6 as one of the first examples of invariant Hilbert schemes with multiplicities. While doing this, we present a general procedure how to realise the calculation of invariant Hilbert schemes, which have been introduced by Alexeev and Brion. We also consider questions of smoothness and connectedness and thereby show that our Hilbert scheme gives a resolution of singularities of the symplectic reduction of the action.