Toward the classification of threefold extremal contractions with one-dimensional fibers

TL;DR

This paper advances the classification of threefold extremal contractions with one-dimensional fibers, focusing on cases where the central curve is reducible, contributing to the understanding of their structure and singularities.

Contribution

It provides a preliminary classification of extremal curve germs with reducible central curves, expanding the understanding of threefold contractions with terminal singularities.

Findings

Classification of extremal curve germs with reducible central curves

Analysis of the structure of threefold contractions with terminal singularities

Insights into the nature of one-dimensional fibers in extremal contractions

Abstract

An extremal curve germ is a germ of a threefold $X$ with terminal singularities along a connected reduced complete curve~$C$ such that there exists a $K_X$-negative contraction $f : X \to Z$ with~$C$ being a fiber. We give a rough classification of extremal curve germs with reducible central curve~$C$.