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

1997/04/10 by Dimitrios I. Dais, Dais, Dimitrios I., Martin Henk +3
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Rings, Modules, and Algebras #alg-geom #math.AG

paper · pdf · doi:10.48550/arxiv.alg-geom/9704007

revised version of MPI-preprint 97/4, 35 pages, 13 figures, latex2e-file (preprint.tex), macro packages and eps-files

arxiv created 1997/04/10 · openalex publication_date 1997/04/10 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For Gorenstein quotient spaces C^d/G, 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 which 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.

Citations

• Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties

Related

• All toric l.c.i.-singularities admit projective crepant resolutions
• On the existence of crepant resolutions of Gorenstein Abelian quotient singularities in dimensions ≥ 4
• Crepant Property of Fujiki-Oka Resolutions for Gorenstein Abelian Quotient Singularities
• On a series of Gorenstein cyclic quotient singularities admitting a unique projective crepant resolution
• Crepant Resolutions of Gorenstein Toric Singularities and Upper Bound Theorem