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

Let G be a finite subgroup of SL(r,C). In dimensions r = 2 and r = 3, McKay correspondence provides a natural bijection between the set of irreducible representations of G and a cohomology-ring basis of the overlying space of a projective, crepant desingularization of C/G. For r = 2 this desingularization is unique and is known to be determined by the Hilbert scheme of the Gorbits. Similar statements (including a method of distinguishing just one among all possible smooth minimal models of C/G), are very probably true for all G’s ⊂ SL(3,C) too, and recent Hilbert-scheme-techniques due to Ito, Nakamura and Reid, are expected to lead to a new fascinating uniform theory. For dimensions r ≥ 4, however, to apply analogous techniques one needs extra modifications. In addition, minimal models of C/G are smooth only under special circumstances. C/ (involution), for instance, cannot have any smooth minimal model. On the other hand, all abelian quotient spaces which are c.i.’s can always be fully resolved by torus-equivariant, crepant, projective morphisms. Hence, from the very beginning, the question whether a given Gorenstein quotient space C/G, r ≥ 4, admits special desingularizations of this kind, seems to be absolutely crucial. In the present paper, after a brief introduction to the existence-problem of such desingularizations (for abelian G’s) from the point of view of toric geometry, we prove that the Gorenstein cyclic quotient singularities of type 1 l (1, . . . , 1, l− (r − 1)) with l ≥ r ≥ 2, have a unique torus-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 PC-bundles over P r−2 C . 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 Pr−1 C .