Divisorial Contractions to Some Gorenstein Singularities

Divisorial contractions to singularities, defined by equations xy + zn + un = 0 n ≥ 3 and xy + z + u = 0 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 [4],[17],[15],[13], 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 [14] and S. Cutkosky [3] classified contractions from terminal Gorenstein threefolds. T.Luo [12] set out contractions when the index is not increase. A.Corti [1] with M.Mella [2] described divisorial contractions to xy + z + u = 0 points, where n = 2, 3. Recently M.Kawakita [8], [9], [10] has gave a description of contractions to a smooth and cA points. In this paper we classify divisorial contractions from a terminal 3-folds to a germ of a point defined by the equation xy + z + u = 0, where n ≥ 3 and to a germ of a singularity defined by the equation xy + z + u = 0 using quite different method then the one introduced in [10]. Our method allow us to deal with all terminal Gorenstein singularities an with non Gorenstein of a type cA/m. The author would like to thank Professor V.A. Iskovskikh and Professor Yu.G. Prokhorov for their vulnerable discussions and encouragement. The author was partially supported by grants RFBR-99-0101132, RFBR-96-15-96146 and INTAS-OPEN-97-2072. 1. Preliminary results We will deal with varieties over C. The basic results and notions are contained in [7], [18].