On Resolutions of Cyclic Quotient Singularities

where n, pt, 1 ^ i ^ m, are integers satisfying 0 ̂ ^ < n and e£* = exp(27iv/ — Ipi/ri). Then the quotient space X=C /G has the natural structure of a normal affine algebraic variety such that the quotient map n: C-+X is a morphism of algebraic varieties [10] [13]. We call this X a cyclic quotient singularity and denote it often by Nntplt_tpm according to the particular expression of the generator g as above. Now the main purpose of this paper is to show the existence of certain natural ways of resolution of these cyclic quotient singularities, which have some good properties. (For the precise statement, see Theorem 1.) Such resolutions were first constructed by Hirzebruch in [3] when m=2 and then, by Ueno [12], when m = 3, p±=l and p2=Pi in (1). On the other hand, the author recently learned that Mumford has found the equivariant resolutions of toroidal singularities, which contain cyclic quotient singularities as special cases [6]. However, our method is different from his and is connected more closely with the above expression of g. So it may be of some interest to compare the resolutions obtained here with those in [6]. In § 1 we prove Theorem 1 and then, in § 2 we apply this theorem to obtain resolutions of the general isolated quotient singularities and the isolated singularity with C* action in the case where the dimension