The existence of generating families for the cohomology ring of a graph

Guillemin, Victor; Zara, Catalin

doi:10.1016/s0001-8708(02)00057-9

Cited by 31 publications

(34 citation statements)

References 7 publications

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“…To have a structure which determines the equivariant cohomology or Chow groups, we should label each multiedge with the topological type of the corresponding family. Guillemin and Zara [2001;2002;2003] have explored the combinatorial properties of moment graphs.…”

Section: T -Invariant Curvesmentioning

confidence: 99%

The equivariant Chow rings of quot schemes

Braden¹,

Chen²,

Sottile³

2008

Pacific J. Math.

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We give a presentation for the (integral) torus-equivariant Chow ring of the quot scheme, a smooth compactification of the space of rational curves of degree d in the Grassmannian. For this presentation, we refine Evain's extension of the method of Goresky, Kottwitz, and MacPherson to express the torus-equivariant Chow ring in terms of the torus-fixed points and explicit relations coming from the geometry of families of torus-invariant curves. As part of this calculation, we give a complete description of the torusinvariant curves on the quot scheme and show that each family is a product of projective spaces.

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Section: T -Invariant Curvesmentioning

confidence: 99%

The equivariant Chow rings of quot schemes

Braden¹,

Chen²,

Sottile³

2008

Pacific J. Math.

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“…§6.3, Remark 6.11). In recent years, combinatorially defined algebras of the type we are considering have been studied by Guillemin and Zara [GZ99,GZ01,GZ03] following the work of GoreskyKottwitz-MacPherson on equivariant cohomology [GKM98], but their focus is different, since they are interested in cases where the algebra in question is actually free over the underlying polynomial ring.…”

Section: Introductionmentioning

confidence: 99%

Relation spaces of hyperplane arrangements and modules defined by graphs of fiber zonotopes

Finis

Lapid

2014

Isr. J. Math.

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Abstract. We study the exactness of certain combinatorially defined complexes which generalize the Orlik-Solomon algebra of a geometric lattice. The main results pertain to complex reflection arrangements and their restrictions. In particular, we consider the corresponding relation complexes and give a simple proof of the n-formality of these hyperplane arrangements. As an application, we are able to bound the Castelnouvo-Mumford regularity of certain modules over polynomial rings associated to Coxeter arrangements (real reflection arrangements) and their restrictions. The modules in question are defined using the relation complex of the Coxeter arrangement and fiber polytopes of the dual Coxeter zonotope. They generalize the algebra of piecewise polynomial functions on the original arrangement.

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“…In a series of papers [4][5][6], Guillemin and Zara formulated the notion of a 1-skeleton as described above, and showed that many topological techniques and theorems regarding GKM manifolds have combinatorial analogues on 1-skeleta. In particular, Guillemin and Zara described combinatorial analogues for symplectic blow-up and symplectic reduction, and developed a combinatorial analogue of Morse theory for 1-skeleta.…”

Section: Introductionmentioning

confidence: 99%

Generalized 1-Skeleta and a Lifting Result

McDaniel

2013

Discrete Comput Geom

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Abstract. In [5] Guillemin and Zara described some beautiful constructions enabling them to use Morse theory on a certain class 1-skeleta that includes all 1-skeleta of simple polytopes. In this paper we extend some of the notions and constructions from [5] to a larger class of 1-skeleta that includes all 1-skeleta of projected simple polytopes. As an application of these ideas we prove a lifting result for 1-skeleta, which yields a characterization of 1-skeleta of projected simple polytopes.

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The existence of generating families for the cohomology ring of a graph

Cited by 31 publications

References 7 publications

The equivariant Chow rings of quot schemes

The equivariant Chow rings of quot schemes

Relation spaces of hyperplane arrangements and modules defined by graphs of fiber zonotopes

Generalized 1-Skeleta and a Lifting Result

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