Invariant Hilbert scheme of the Cox realization of the nilpotent cone in $\mathfrak{sl}(n)$

Abstract

We pose a question asking if the main component of the invariant Hilbert scheme of the Cox realization of a given singularity is a resolution of singularities. As a concrete example, we consider the case where the singularity is the nilpotent cone in $\mathfrak{sl}(n)$, and we see that in this case the answer to the question is positive.