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- Graham C. Denham
- Electr. J. Comb.
- 2003

Let a1, a2, . . . , an be distinct, positive integers with (a1, a2, . . . , an) = 1, and let k be an arbitrary field. Let H(a1, . . . , an; z) denote the Hilbert series of the graded algebra k[ta1 , ta2 , . . . , tan ]. We show that, when n = 3, this rational function has a simple expression in terms of a1, a2, a3; in particular, the numerator has at mostâ€¦ (More)

- Graham C. Denham
- 1999

We examine a bilinear form associated with a real arrangement of hyperplanes introduced in [Schechtman and Varchenko 1991]. Our main objective is to show that the linear algebraic properties of this bilinear form are related to the combinatorics and topology of the hyperplane arrangement. We will survey results and state a number of open problems whichâ€¦ (More)

In a recent paper, Dimca and NÃ©methi pose the problem of finding a homogeneous polynomial f such that the homology of the complement of the hypersurface defined by f is torsion-free, but the homology of the Milnor fiber of f has torsion. We prove that this is indeed possible, and show by construction that, for each prime p , there is a polynomial withâ€¦ (More)

We describe dualities and complexes of logarithmic forms and differentials for central affine and corresponding projective arrangements. We generalize the Borelâ€“Serre formula from vector bundles to sheaves on P with locally free resolutions of length one. Combining these results we present a generalization of a formula due to MustaÅ£Äƒ and Schenck, relatingâ€¦ (More)

- Graham C. Denham, Phil Hanlon, Olga Taussky-Todd
- 1997

Let A = {H1, . . . , H`} be an arrangement of hyperplanes in R and let r(A) = {R1, . . . , Rm} denote the set of regions in the complement of the union of A. Let L(A) denote the collection of intersections of hyperplanes in A including the empty intersection which we take to be R. We order the elements of L(A) by reverse inclusion thus making it into aâ€¦ (More)

Let A be a graded-commutative, connected k-algebra generated in degree 1. The homotopy Lie algebra gA is defined to be the Lie algebra of primitives of the Yoneda algebra, ExtA(k, k). Under certain homological assumptions on A and its quadratic closure, we express gA as a semi-direct product of the well-understood holonomy Lie algebra hA with a certainâ€¦ (More)

- Doron J. Cohen, Graham C. Denham, Michael Falk, A. VARCHENKO
- 2009

If Î¦Î» is a master function corresponding to a hyperplane arrangement A and a collection of weights Î», we investigate the relationship between the critical set of Î¦Î», the variety defined by the vanishing of the one-form Ï‰Î» = d log Î¦Î», and the resonance of Î». For arrangements satisfying certain conditions, we show that if Î» is resonant in dimension p, thenâ€¦ (More)

The characteristic varieties of a space are the jump loci for homology of rank 1 local systems. The way in which the geometry of these varieties may vary with the characteristic of the ground field is reflected in the homology of finite cyclic covers. We exploit this phenomenon to detect torsion in the homology of Milnor fibers of projective hypersurfaces.â€¦ (More)

In this paper, we recover the characteristic polynomial of an arrangement of hyperplanes by computing the rational equivalence class of the variety defined by the logarithmic ideal of the arrangement. The logarithmic ideal was introduced in [CDFV] in a study of the critical points of the master function. The above result is used to understand the asymptoticâ€¦ (More)

- Graham C. Denham
- 2007

We give a short, case-free and combinatorial proof of de Concini and Procesiâ€™s formula from [1] for the volume of the simplicial cone spanned by the simple roots of any finite root system. The argument presented here also extends their formula to include the non-crystallographic root systems.