Equivariant embeddings of Tori

Kempf, George R.; Knudsen, Finn F.; Mumford, David; Saint-Donat, B.

doi:10.1007/bfb0070319

Cited by 39 publications

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“…Remark 0.8. We follow the convention in [KKMS73], where one requires a lifting function to be "convex down" on each cell, namely f (λx + µy) ≥ λf (x) + µf (y). Also, all our lifting functions take rational values on the lattices in the cells.…”

Section: The Restriction Of F To Umentioning

confidence: 99%

“…Since the integral generator of every edge in Σ ′ X maps to the generator of the image edge in Σ B 1 , and since B is nonsingular, all the fibers of f 1 are reduced [ℵ-K97]. By the construction of [KKMS73], f 1 is semistable in codimension 1. Since ∆ ′ X is simplicial, Y has at worst quotient singularities.…”

Section: The Restriction Of F To Umentioning

confidence: 99%

“…We will need some terminology in order to state the weaker version of semistable reduction we actually address here. We will follow [KKMS73] for the basic definitions. 1 Definition 0.4.…”

Section: Introductionmentioning

confidence: 99%

“…If U B ⊂ B is a toroidal embedding, then we may write B \ U B as a union of divisors D 1 ∪· · ·∪ D k . More precisely, recall that B \ U B can be decomposed into strata of varying dimensions (see [KKMS73] or [GM88]). In particular, let us define U…”

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confidence: 99%

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Extending Triangulations and Semistable Reduction

Abramovich

Rojas

2002

Foundations of Computational Mathematics

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Section: The Restriction Of F To Umentioning

confidence: 99%

Section: The Restriction Of F To Umentioning

confidence: 99%

“…We will need some terminology in order to state the weaker version of semistable reduction we actually address here. We will follow [KKMS73] for the basic definitions. 1 Definition 0.4.…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Extending Triangulations and Semistable Reduction

Abramovich

Rojas

2002

Foundations of Computational Mathematics

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“…In a sense, the correspondence RA-+ PA is a logarithmic transformation. How it can be applied has been thoroughly studied by Hochster [8], Kempf et al[9],Danilov [S], and others. Any time algebraic-geometric properties about singularities or blowups can be defined in local analytic coordinates by monomials, one may have a chance to apply the combinatorial-geometric technique of exponent semigroups.…”

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confidence: 99%

Convex Bodies and Algebraic Geometry

Ewald¹

1985

Annals of the New York Academy of Sciences

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During the last decade a new area of research has developed relating two subjects which until now had very little in common: convexity and algebraic geometry. An initial success was Stanley's solution of the so-called upper bound conjecture for combinatorial spheres ([16]; for polytopes solved by McMullen, published in [14]): Among all combinatorial spheres with v vertices (faces are convex polytopes), the convex hulls of v points on a moment curve {(t, t2, ..., t")lt E R} (called cyclic polytopes) possess maximal numbers of faces in all dimensions. This result has been pursued further by Kind and Kleinschmidt [lo]. Another result is the solution of McMullen's conjecture about characterizing those vectors f (P) = ( fo(P), . . . , fd-l(P)) for which f'P) is the number ofj-faces of a polytope P. The necessity of McMullen's condition has been shown by Stanley [17] and the sufficiency by Billera and Lee [2]. A foundation for Stanley's, Billera's, and Lee's work was laid by Hochster [S].This paper by Hochster was also one of the starting points for a development which is to be outlined in what follows. We emphasize two main achievements: First, a characterization of Milnor's number of critical points of complex algebraic functions by numbers assigned to convex polytopes (Kouchnirenko [ll]); second, characterizations of mixed volumes of convex bodies as the intersection index of certain varieties, and an alternative proof of the Alexandrov-Fenchel inequality (Bernstein [l], Burago and Zalgaller [4], Teissier [19]). In order to make the exposition comprehensible to nonspecialists, we restrict the discussion to elementary explanations.

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

Dais¹,

Henk²,

Ziegler³

1998

Advances in Mathematics

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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

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Equivariant embeddings of Tori

Cited by 39 publications

References 0 publications

Extending Triangulations and Semistable Reduction

Extending Triangulations and Semistable Reduction

Convex Bodies and Algebraic Geometry

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

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