On classification of non-Gorenstein Q-Fano 3-folds of Fano index 1

摘要

Definition1.1 A d-dimensional normal complex projective variety X is called a Q-Fano d-fold if it has only terminal singularities and the anti-canonical Weil divisor —K_X is ample (cf. [KMM]). The index of singular point p is defined to be the smallest positive integer i_p such that i_pK_X is a Cartier divisor near p. A singular point of singularity index one is called Gorenstein singularity. Singularity index I(X) of X is defined to be the smallest positive integer such that IK_X is a Cartier divisor. Hence there is a positive integer r and a Cartier divisor H such that —IK_X～rH. Taking the largest number of such r, we call r/I the Fano index of X.