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

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~$C$.