Threefold extremal contractions of types (IC) and (IIB)

TL;DR

This paper classifies threefold extremal contractions of types (IC) and (IIB) with terminal singularities along a curve, focusing on the structure of the germ and the properties of a general member containing the curve.

Contribution

It completes the classification of threefold extremal contractions of types (IC) and (IIB) with terminal singularities along a curve, expanding understanding of their structure.

Findings

Classification of (IC) and (IIB) types completed

Characterization of general members containing the curve

Insights into the structure of extremal contractions

Abstract

Let $(X,C)$ be a germ of a threefold $X$ with terminal singularities along an irreducible reduced complete curve $C$ with a contraction $f: (X,C)\to (Z,o)$ such that $C=f^{-1}(o)_{red}$ and $-K_X$ is ample. Assume that $(X,C)$ contains a point of type (IC) or (IIB). We complete the classification of such germs in terms of a general member $H\in |\mathcal O_X|$ containing $C$.