General elephants for threefold extremal contractions with one-dimensional fibers: exceptional case

Abstract

Let $(X,C)$ be a germ of a threefold $X$ with terminal singularities along a connected reduced complete curve $C$ with a contraction $f:(X,C)\to (Z,o)$ such that $C=f^{-1}(o{)}_{\mathrm{red}}$ and $-K_X$ is $f$-ample. Assume that each irreducible component of $C$ contains at most one point of index $>2$. We prove that a general member $D\in |-K_X|$ is a normal surface with Du Val singularities.

22 pages, one sentence in the introduction is corrected and Appendix B is added