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

Abstract Let be a germ of threefold $X$?> with terminal singularities along connected reduced complete curve $C$?> contraction $f \colon (X, C) \to (Z, o)$?> such that $C = f^{-1} (o)_{\mathrm{red}}$?> and $-K_X$?> is $f$?> -ample. Assume each irreducible component contains at most one point index ${>2}$?> . We prove general member $D\in |{-}K_X|$?> normal surface Du Val singularities. Bibliography: 16 titles.