ABSTRACT. We propose an approach to study logarithmic sheaves n (− log D ) associated with hyperplane arrangements on the projective space n , based on projective duality, direct image functors and vector bundles methods. We focus on freeness of line arrangements having a point with high multiplicity.
In a recent paper, Mezzetti, Miró‐Roig and Ottaviani [Mezzetti et al., ‘Laplace equations and the weak Lefschetz property’, Canad. J. Math. 65 (2013) 634–654] highlight the link between rational varieties satisfying a Laplace equation and artinian ideals failing the weak Lefschetz property. Continuing their work, we extend this link to the more general situation of artinian ideals failing the strong Lefschetz property. We characterize the failure of the SLP (which includes WLP) by the existence of special singular hypersurfaces (cones for WLP). This characterization allows us to solve three problems posed in J. C. Migliore and U. Nagel [‘A tour of the weak and strong Lefschetz properties’, Preprint, 2011, arXiv:1109.5718, September 2011. J Commutative Algebra, to appear] and to give new examples of ideals failing the SLP. Finally, line arrangements are related to artinian ideals and the unstability of the associated derivation bundle is linked to the failure of the SLP. Moreover, we reformulate the so‐called Terao's conjecture for free line arrangements in terms of artinian ideals failing the SLP.
Soit S n,k la famille des fibrés de Steiner S sur P n définis par une suite exacte (k > 0)Nous montrons le résultat suivant : Soient S ∈ S n,k et H 1 , · · · , H n+k+2 des hyperplans distincts tels que h 0 (S ∨ H i ) = 0. Alors il existe une courbe rationnelle normale C n ⊂ P ∨ n telle que H i ∈ C n pour i = 1, ..., n + k + 2 et S E n+k−1 (C n ), où E n+k−1 (C n ) est le fibré de Schwarzenberger sur P n appartenantà S n,k associéà la courbe C n ⊂ P ∨ n . On en déduit qu'un fibré de Steiner S ∈ S n,k , s'il n'est pas un fibré de Schwarzenberger, possède au plus (n + k + 1) hyperplans instables; ceci prouve dans tous les cas un résultat de Dolgachev et Kapranov ([DK], thm. 7.2) concernant les fibrés logarithmiques.Abstract. Let S n,k denote the family of Steiner's bundle S on P n defined by the exact sequence (k > 0)We show the following result : Let S ∈ S n,k and H 1 , · · · , H n+k+2 distincts hyperplanes such that h 0 (S ∨ H i ) = 0. Then it exists a rational normal curve C n ⊂ P ∨ n such that H i ∈ C n for i = 1, ..., n+k +2 and S E n+k−1 (C n ), where E n+k−1 (C n ) is the Schwarzenberger's bundle on P n which belongs to S n,k associated to C n ⊂ P ∨ n It implies that a Steiner's bundle S ∈ S n,k , if it isn't a Schwarzenberger's bundle, possesses no more than (n + k + 1) unstable hyperplanes; this proves in any case a result of Dolgachev and Kapranov ([DK], thm 7.2) about logarithmic bundles. 508 J. Vallès
Many papers are devoted to study logarithmic sheaves associated to reduced divisors, in particular logarithmic bundles associated to plane curves since forty years in differential and algebraic topology or geometry. An interesting family of these curves are the so-called free ones for which the associated logarithmic sheaf is the direct sum of two line bundles. When the curve is a finite set of distinct lines (i.e. a line arrangement), Terao conjectured thirty years ago that its freeness depends only on its combinatorics. A lot of efforts were done to prove it but at this time it is only proved up to 12 lines. If one wants to find a counter example to this conjecture a new family of curves arises naturally: the nearly free curves introduced by Dimca and Sticlaru. We prove here that the logarithmic bundle associated to a nearly free curve possesses a minimal non zero section that vanishes on one single point P , called jumping point, and that characterizes the bundle. Then we give a precise description of the behaviour of P . In particular we show, based on detailed examples, that the position of P relatively to its corresponding nearly free arrangement of lines may or may not be a combinatorial invariant, depending on the chosen combinatorics.
Abstract. A plane curve D ⊂ P 2 (k) where k is a field of characteristic zero is free if its associated sheaf TD of vector fields tangent to D is a free O P 2 (k) -module (see [7] or [5] for a definition in a more general context). Relatively few free curves are known. Here we prove that a divisor D consisting of a union of curves of a pencil of plane projective curves with the same degree and with a smooth base locus is a free divisor if and only if D contains all the singular members of the pencil and its Jacobian ideal is locally a complete intersection.
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