On crepant resolutions of 2-parameter series of Gorenstein cyclic quotient singularities

Abstract

An immediate generalization of the classical McKay correspondence for Gorenstein quotient spaces C r /G in dimensions r ≥ 4 would primarily demand the existence of projective, crepant, full desingularizations. Since this is not always possible, it is natural to ask about special classes of such quotient spaces which would satisfy the above property. In this paper we give explicit necessary and sufficient conditions under which 2-parameter series of Gorenstein cyclic quotient singularities have torus-equivariant resolutions of this specific sort in all dimensions.• The most important aspects of the three-dimensional generalization of McKay's bijections for C 3 /G's, G ⊂ SL(3, C), were only recently clarified by the paper [29] of Ito and Reid; considering a canonical grading on the Tate-twist of the acting G's by the so-called "ages", they proved that for any projective crepant resolution f : X → X = C 3 /G, there are one-to-one correspondences between the elements of G of age 1 and the exceptional prime divisors of f , and between them and the members of a basis of H 2 X, Q , respectively. On the other hand, the existence of crepant f 's was proved by Markushevich, Ito and Roan: Theorem 1.1 (Existence-Theorem in Dimension 3). The underlying spaces of all 3-dimensional Gorenstein quotient singularities possess crepant resolutions.

Kähler Moduli Space for a D-Brane at Orbifold Singularities

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