An example of an SL2-Hilbert scheme with multiplicities

In AB05], Alexeev and Brion have introduced the notion of invariant Hilbert schemes. We determine the invariant Hilbert scheme of the zero fibre of the moment map of an action of SL₂ on ( \mathbbC² ) ^?6 {\left( {{\mathbb{C}^2}} \right)^{ \oplus 6}} as one of the first examples of invariant Hilbert schemes with multiplicities. While doing this, we present a general procedure for realizing these calculations. 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.