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

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Abstract

Let $(X, C)$ be a germ of a threefold $X$ with terminal singularities along a connected reduced complete curve $C$ with a contraction $f \colon (X, C) \to (Z, o)$ 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. Bibliography: 16 titles.

About the authors

Shigefumi Mori

Kyoto University Institute for Advanced Study; Research Institute for Mathematical Sciences, Kyoto University; Chubu University

Email: mori@kurims.kyoto-u.ac.jp

Yuri Gennadievich Prokhorov

Steklov Mathematical Institute of Russian Academy of Sciences

Email: prokhoro@mi-ras.ru
Doctor of physico-mathematical sciences, Professor

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