234 research outputs found
Gorenstein simplices and the associated finite abelian groups
It is known that a lattice simplex of dimension corresponds a finite
abelian subgroup of . Conversely, given a finite
abelian subgroup of such that the sum of all
entries of each element is an integer, we can obtain a lattice simplex of
dimension . In this paper, we discuss a characterization of Gorenstein
simplices in terms of the associated finite abelian groups. In particular, we
present complete characterizations of Gorenstein simplices whose normalized
volume equals and , where and are prime numbers with . Moreover, we compute the volume of the dual simplices of Gorenstein
simplices.Comment: 18 pages, to appear in European Journal of Combinatoric
All Abelian Quotient C.I.-Singularities Admit Projective Crepant Resolutions in All Dimensions
For Gorenstein quotient spaces , a direct generalization of the
classical McKay correspondence in dimensions 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.Comment: revised version of MPI-preprint 97/4, 35 pages, 13 figures,
latex2e-file (preprint.tex), macro packages and eps-file
Volume and lattice points of reflexive simplices
We prove sharp upper bounds on the volume and the number of lattice points on
edges of higher-dimensional reflexive simplices. These convex-geometric results
are derived from new number-theoretic bounds on the denominators of unit
fractions summing up to one. The main algebro-geometric application is a sharp
upper bound on the anticanonical degree of higher-dimensional Q-factorial
Gorenstein toric Fano varieties with Picard number one, where we completely
characterize the case of equality.Comment: AMS-LaTeX, 19 pages; paper reorganized, introduction added,
bibliography updated; typos correcte
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