Terminal 3-fold Divisorial Contractions of a Surface to a Curve I

Let X be a smooth curve on a 3-fold which has only index 1 terminal singularities along . In this paper we investigate the existence of extremal terminal divisorial contractions E Y X, contracting an irreducible surface E to . We consider cases with respect to the singularities of the general hypersurface section S of X through . We completely classify the cases when S is A _i , i 3, and D _2n for any n.