540 research outputs found

    Freeness and multirestriction of hyperplane arrangements

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    Generalizing a result of Yoshinaga in dimension 3, we show that a central hyperplane arrangement in 4-space is free exactly if its restriction with multiplicities to a fixed hyperplane of the arrangement is free and its reduced characteristic polynomial equals the characteristic polynomial of this restriction. We show that the same statement holds true in any dimension when imposing certain tameness hypotheses.Comment: 8 page

    Cohen-Macaulayness and computation of Newton graded toric rings

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    Let HH be a positive semigroup in Zd\mathbb{Z}^d generated by AA, and let K[H]K[H] be the associated semigroup ring over a field KK. We investigate heredity of the Cohen-Macaulay property from K[H]K[H] to both its AA-Newton graded ring and to its face rings. We show by example that neither one inherits in general the Cohen-Macaulay property. On the positive side we show that for every HH there exist generating sets AA for which the Newton graduation preserves Cohen-Macaulayness. This gives an elementary proof for an important vanishing result on AA-hypergeometric Euler-Koszul homology. As a tool for our investigations we develop an algorithm to compute algorithmically the Newton filtration on a toric ring.Comment: 20 pages, 4 figure

    Irregularity of hypergeometric systems via slopes along coordinate subspaces

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    We study the irregularity sheaves attached to the AA-hypergeometric DD-module MA(β)M_A(\beta) introduced by Gel'fand et al., where AZd×nA\in\mathbb{Z}^{d\times n} is pointed of full rank and βCd\beta\in\mathbb{C}^d. More precisely, we investigate the slopes of this module along coordinate subspaces. In the process we describe the associated graded ring to a positive semigroup ring for a filtration defined by an arbitrary weight vector LL on torus equivariant generators. To this end we introduce the (A,L)(A,L)-umbrella, a simplicial complex determined by AA and LL, and identify its facets with the components of the associated graded ring. We then establish a correspondence between the full (A,L)(A,L)-umbrella and the components of the LL-characteristic variety of MA(β)M_A(\beta). We compute in combinatorial terms the multiplicities of these components in the LL-characteristic cycle of the associated Euler-Koszul complex, identifying them with certain intersection multiplicities. We deduce from this that slopes of MA(β)M_A(\beta) are combinatorial, independent of β\beta, and in one-to-one correspondence with jumps of the (A,L)(A,L)-umbrella. This confirms a conjecture of Sturmfels and gives a converse of a theorem of Hotta: MA(β)M_A(\beta) is regular if and only if AA defines a projective variety.Comment: 44 pages, 3 figures, choose PS or PDF to see figures, new Lemma 2.8 fills gap in previous version of Lemma 2.12, error in previous version of Theorem 3.2 repaired by considering L-holonomic modules in Sections 3.2 and 4.

    Quasihomogeneity of curves and the Jacobian endomorphism ring

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    We give a quasihomogeneity criterion for Gorenstein curves. For complete intersections, it is related to the first step of Vasconcelos' normalization algorithm. In the process, we give a simplified proof of the Kunz-Ruppert criterion.Comment: 9 page

    On the formal structure of logarithmic vector fields

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    In this article, we prove that a free divisor in a three dimensional complex manifold must be Euler homogeneous in a strong sense if the cohomology of its complement is the hypercohomology of its logarithmic differential forms. F.J. Calderon-Moreno et al. conjectured this implication in all dimensions and proved it in dimension two. We prove a theorem which describes in all dimensions a special minimal system of generators for the module of formal logarithmic vector fields. This formal structure theorem is closely related to the formal decomposition of a vector field by Kyoji Saito and is used in the proof of the above result. Another consequence of the formal structure theorem is that the truncated Lie algebras of logarithmic vector fields up to dimension three are solvable. We give an example that this may fail in higher dimensions.Comment: 13 page
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