Geodesic
General elephants for threefold extremal contractions with one-dimensional fibres: exceptional case
Sbornik. Mathematics, Tome 212 (2021) no. 3, pp. 351-373 Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: terminal singularity, extremal curve germ, flip, divisorial contraction, $\mathbb{Q}$-conic bundle. 
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S. Mori; Yu. G. Prokhorov. General elephants for threefold extremal contractions with one-dimensional fibres: exceptional case. Sbornik. Mathematics, Tome 212 (2021) no. 3, pp. 351-373. http://geodesic.mathdoc.fr/item/SM_2021_212_3_a6/

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