2 9 N ov 2 00 1 BLOW - UPS OF THREE - DIMENSIONAL TERMINAL SINGULARITIES : cA CASE

Divisors with minimal discrepancy over cA points are classified. The problem of birational classification of algebraic varieties is highly interconnected with the problem of description of singularities on them. One of the most important class of three-dimensional singularities is terminal singularities, which arise within minimal models programm. Despite the analytical classification of the singularities [2],[3],[4],[5], this description does not help one to fully understand many birationl properties of them. In particular, the problem of description of resolution of such singularities and the problem of classification of morphisms of terminal varieties are still up-to-date. Divisorial contractions to cyclic quotient singularities were described by Y.Kawamata [6], S.Mori [7] and S. Cutkosky [8] classified contractions from terminal Gorenstein threefolds. T.Luo [9] set out contractions when the index is not increase. Recently M.Kawakita [10], [11], [12] has gave a description of contractions to a smooth and cA points. In the paper we describe divisors with minimal discrepancy in Mori's category over cA points after M.Hayakawa [15], when he did the same for the cases of non Gorenstein terminal singularities. We will deal with varieties over C. The basic results and notions are contained in [1], [16], [15]. 1.1. A singularity has cA type if there is an embedding j : X ≃ {xy + f (z, u) = 0} ֒→ C 4 x,y,z,u (see [17]). This embedding we will call standard.