Graham C. Denham

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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 most(More)
We describe dualities and complexes of logarithmic forms and differentials for central affine and corresponding projective arrangements. We generalize the Borel–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 Mustaţă and Schenck, relating(More)
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 a(More)
If Φλ is a master function corresponding to a hyperplane arrangement A and a collection of weights λ, we investigate the relationship between the critical set of Φλ, the variety defined by the vanishing of the one-form ωλ = d log Φλ, and the resonance of λ. For arrangements satisfying certain conditions, we show that if λ is resonant in dimension p, then(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.(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 asymptotic(More)