Invariant Hilbert scheme resolution of Popov's $SL(2)$-varieties I: the toric case

TL;DR

This paper demonstrates that all 3D affine normal quasihomogeneous SL(2)-varieties that are toric can be resolved of singularities using invariant Hilbert schemes, focusing on the toric case.

Contribution

It provides the first detailed analysis of invariant Hilbert scheme resolutions for toric SL(2)-varieties, extending the understanding of singularity resolutions in this context.

Findings

Every 3D affine normal quasihomogeneous SL(2)-variety admits an equivariant resolution via invariant Hilbert schemes.

The paper specifically addresses the toric case of these varieties.

Non-toric cases are to be explored in future work.

Abstract

We show that every 3-dimensional affine normal quasihomogeneous $SL(2)$-variety has an equivariant resolution of singularities given by an invariant Hilbert scheme. This article treats the case where such $SL(2)$-variety is toric. The non-toric case is considered in the forthcoming article [Kub18].