Geodesic
Threefold extremal contractions of type (IIA).~I
Izvestiya. Mathematics , Tome 80 (2016) no. 5, pp. 884-909

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Let $(X,C)$ be a germ of a threefold $X$ with terminal singularities along an irreducible 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 ample. Assume that $(X,C)$ contains a point of type $(\mathrm{IIA})$ and that a general member $H\in|\mathscr O_X|$ containing $C$ is normal. We classify such germs in terms of $H$.
Keywords: extremal contraction, threefold, extremal curve germ, terminal singularity, sheaf. 
@article{IM2_2016_80_5_a4,
     author = {S. Mori and Yu. G. Prokhorov},
     title = {Threefold extremal contractions of type {(IIA).~I}},
     journal = {Izvestiya. Mathematics },
     pages = {884--909},
     publisher = {mathdoc},
     volume = {80},
     number = {5},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2016_80_5_a4/}
}
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S. Mori; Yu. G. Prokhorov. Threefold extremal contractions of type (IIA).~I. Izvestiya. Mathematics , Tome 80 (2016) no. 5, pp. 884-909. http://geodesic.mathdoc.fr/item/IM2_2016_80_5_a4/