An example of an Sl_2-Hilbert scheme with multiplicities

TL;DR

This paper computes a specific invariant Hilbert scheme related to an Sl_2 action on six copies of C^2, illustrating a general calculation method and showing it resolves singularities in symplectic reduction.

Contribution

It presents the first example of an invariant Hilbert scheme with multiplicities and introduces a general procedure for their calculation.

Findings

Determined the invariant Hilbert scheme for a specific Sl_2 action.

Showed the Hilbert scheme provides a resolution of singularities.

Discussed smoothness and connectedness properties.

Abstract

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.