Threefold extremal curve germs with one non-Gorenstein point

TL;DR

This paper reviews classifications of threefold extremal curve germs with terminal singularities, focusing on cases where the central fiber is irreducible and contains a single non-Gorenstein point, advancing understanding of their structure.

Contribution

It provides a comprehensive review of extremal curve germs with one non-Gorenstein point and irreducible central fiber, summarizing recent classification results.

Findings

Classification of extremal curve germs with one non-Gorenstein point

Description of the structure of such germs

Summary of recent classification results

Abstract

An extremal curve germ is the analytic germ of a threefold with terminal singularities along a reduced complete curve admitting a contraction whose fibers have dimension at most one. The aim of the present paper is to review the results concerning those contractions whose central fiber is irreducible and contains only one non-Gorenstein point.