Mathematical Notes, Vol. 68, No. 5, 2000 Sparseness of Exceptional Quotient Singularities Yu. G. Prokhorov Key words: exceptional quotient singularity, log-canonical singularity, effective divisor, finite group, log- terminal singularity, imprimitive subgroup. 1. Introduction The present note is a continuation of [1] and is devoted to the study of exceptional quotient singularities. Throughout the note, the ground field is assumed to be the field C . We make use of the main notions from the log-program of minimal models (see [2]). Definition 1.1 (see [3, 5.6]). Let (X  P ) be a normal singularity and let D = ∑ d i D i be an effective Q -divisor on X such that the pair (X,D) has log-canonical singularities only. A pair (X,D) is said to be exceptional if there is at most one divisor E of the function field K(X) with discrepancy a(E,D)= −1. A log-canonical singularity (X  P ) is said to be exceptional if a pair (X,D) is exceptional whenever it is log-canonical. We consider, from the viewpoint of exceptionality, a special class of singularities, namely, quotient sin- gularities by finite groups. These singularities are log-terminal [2, 20.3.3]. By the well-known Chephard– Todd–Chevalley theorem (see, e.g., [4, 4.2.5]), when studying quotient singularities, we can restrict our- selves (without loss of generality) to the groups G ⊂ GL n (C) that contain no reflections (i.e., elements g such that rk(g − id) = 1). The main result of the present note is the following theorem. Theorem 1.1. Choose an ε> 0 . The set of finite reflection-free subgroups G ⊂ GL n (C) such that C n /G is an exceptional ε -log-terminal singularity is finite up to conjugations. Corollary 1.1. The set of finite reflection-free subgroups G ⊂ GL n (C) such that C n /G is an excep- tional canonical singularity is finite up to conjugations. A similar result for hypersurface singularities was obtained in [5]. 2. Exceptional groups are primitive In this section we prove Theorem 1.1. The assertion is a consequence of Proposition 2.1 and of the classical Jordan theorem (see Theorem 2.1). Notation. Write V := C n . Let G ⊂ GL n (C) be a finite subgroup without reflections and let π : V → X = V/G be the factorization morphism. Let σ :  V → V be a blow-up of the point 0, and let E := σ −1 (0) be an exceptional divisor. Let us consider the commutative diagram  V ϕ −−−−→  X σ    f    V π −−−−→ X , where  X := X/G and set S := ϕ(E). Translated from Matematicheskie Zametki, Vol. 68, No. 5, pp. 786–789, November, 2000. Original article submitted March 20, 2000. 664 0001–4346/2000/6856–0664$25.00 c 2000 Kluwer Academic/Plenum Publishers