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Terminal 3-fold divisorial contractions of a surface to a curve. I. (English) Zbl 1058.14028
Extremal divisorial contractions and flips (flops) are fundamental birational operations essential to the minimal model program of projective algebraic varieties with at most terminal singularities in dimension 3 over the complex number field. Flips in dimension 3 are completely classified by J. Kollár and S. Mori [J. Am. Math. Soc. 5, 533–703 (1992; Zbl 0773.14004)]. In the divisorial contraction case, Y. Kawamata [in: Higher dimensional complex varieties. Proc. int. conf. Trento 1994, 241–246 (1996; Zbl 0894.14019)], M. Kawakita [Invent. Math. 145, 105–119 (2001; Zbl 1091.14007)] and T. Luo [Am. J. Math. 120, 441–451 (1998; Zbl 0919.14021)] investigated divisor to point contractions and had classifications in various special cases.
The paper under review studies divisor to curve contractions with the assumption that terminal singularities along the curve can be at most of index 1. All possible contractions are classified when the general hypersurface containing the curve has singularities of type $$A_1$$, $$A_2$$, $$A_3$$, and $$D_{2n}$$.

##### MSC:
 14E30 Minimal model program (Mori theory, extremal rays) 14J30 $$3$$-folds
##### Keywords:
flips; minimal model program
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