Threefold extremal contractions of type IA

TL;DR

This paper classifies certain threefold extremal contractions of type IA with specific singularity and intersection properties, providing a detailed description based on general members of the anticanonical linear system.

Contribution

It offers a complete classification of threefold extremal contractions of type IA under specified conditions, expanding understanding of their structure and singularities.

Findings

Classification of all such germs in terms of a general member containing C

Identification of Du Val singularities of type A at a specific point

Description of the structure of contractions with terminal singularities

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 a general member $F\in |-K_X|$ meets $C$ only at one point $P$ and furthermore $(F,P)$ is Du Val of type A if index$(X,P)=4$. We classify all such germs in terms of a general member $H\in |O_X|$ containing $C$.