Threefold extremal contractions of type (IA)

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 assume that $(F,P)$ is Du Val of type A if index $(X,P)=4$. We classify all such germs in terms of a general member $H\in \left|{\mathcal{O}}_X\right|$ containing $C$.