On a series of Gorenstein cyclic quotient singularities admitting a unique projective crepant resolution

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In this paper we prove that the Gorenstein cyclic quotient singularities of type 1l (1,..., 1,l-(r-1)) with l≥ r≥ 2, have a uniquetorus-equivariant projective, crepant, partial resolution, which is ``full'' iff either l≡ 0 mod % (r-1) or l≡ 1 mod (r-1). As it turns out, if one of these two conditions is fulfilled, then the exceptional locus of the full desingularization consists of l/r-1 prime divisors, l/r-1 - 1 of which are isomorphic to the total spaces of PC¹-bundles over PC%ʳ⁻². Moreover, it is shown that intersection numbers are computable explicitly and that the resolution morphism can be viewed as a composite of successive (normalized) blow-ups. Obviously, the monoparametrized singularity-series of the above type contains (as its ``first member'') the well-known Gorenstein singularity defined by the origin of the affine cone which lies over the r-tuple Veronese embedding of PCʳ⁻¹.