All Abelian Quotient C.I.-Singularities Admit Projective Crepant Resolutions in All Dimensions

Abstract

For Gorenstein quotient spaces C d ÂG, a direct generalization of the classical McKay correspondence in dimensions d 4 would primarily demand the existence of projective, crepant desingularizations. Since this turned out to be not always possible, Reid asked about special classes of such quotient spaces that would satisfy the above property. We prove that the underlying spaces of all Gorenstein abelian quotient singularities, which are embeddable as complete intersections of hypersurfaces in an affine space, have torus-equivariant projective crepant resolutions in all dimensions. We use techniques from toric and discrete geometry. 1998Academic Press

Smooth toric G-Hilbert schemes via G-graphs

“…The example we provide in this note is not covered by previous results. The groups G n are not symplectic for n 3, the toric varieties we deal with are not local complete intersections (so one cannot apply [3,4]) and for n 4, [1] does not apply because the exceptional locus is too big.…”

On the existence of crepant resolutions of Gorenstein abelian quotient singularities in dimensions ≥4

“…In particular, it includes more background material than the average research paper has. The new results are essentially in sections 3, 8, 9 and 10, together with some parts of §4 and of Appendices A and D. (In §5, §6 , and §7 we summarize results from [27,29] and from the unpublished manuscript [28].) More precisely, the paper is organized as follows: In §2 we recall fundamental notions from toric geometry and introduce our notation.…”

“…[29]). All abelian quotient c.i.-singularities admit projective, crepant resolutions in all dimensions.An extensive technical part of its proof is devoted to the rendering of the original (purely algebraic) group classification of Watanabe[124] into graph-theoretic terms and to a subsequent convenient description of the corresponding junior simplices.As it turns out, an AQS is a c.i.-singularity if and only if the junior simplex s G is (what we call) a Watanabe simplex w.r.t.…”

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

“…In this section we first briefly recall some general theorems concerning the projective, crepant resolutions of Gorenstein abelian quotient singularities in terms of appropriate lattice triangulations of the junior simplex. (For detailed expositions we refer to [7], [8], [9]). After that we formulate our main theorems.…”

“…• On the other hand, as it was proved in [8] by making use of Watanabe's classification of all abelian quotient singularities (C r /G, [0]), G ⊂ SL(r, C), (up to analytic isomorphism) whose underlying spaces are embeddable as complete intersections ("c.i. 's") of hypersurfaces into an affine complex space, and methods of toric and discrete geometry, Hence, the expected answer(s) to the main question will be surely of special nature, depending crucially on the generators of the acting groups or at least on the properties of the ring C [x 1 , .…”