Gorenstein Threefold Singularities with Small Resolutions via Invariant Theory for Weyl Groups

TL;DR

This paper classifies certain Gorenstein threefold singularities with small resolutions using invariant theory related to Weyl groups, identifying six families distinguished by Kollár's length invariant.

Contribution

It provides a classification of simple flops on smooth threefolds and explicitly describes generators of invariant rings for Weyl group actions.

Findings

Six families of Gorenstein threefold singularities with small resolutions.

Explicit generators of invariant rings for Weyl group actions.

Application of invariant theory to classify and construct small resolutions.

Abstract

We classify simple flops on smooth threefolds, or equivalently, Gorenstein threefold singularities with irreducible small resolution. There are only six families of such singularities, distinguished by Koll{\'a}r's {\em length} invariant. The method is to apply invariant theory to Pinkham's construction of small resolutions. As a by-product, generators of the ring of invariants are given for the standard action of the Weyl group of each of the irreducible root systems.