Pencil type line arrangements of low degree: classification and monodromy

Dimca, Alexandru; Ibadula, Denis; Măcinic, Daniela Anca

doi:10.2422/2036-2145.201306_003

Cited by 6 publications

(20 citation statements)

References 21 publications

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“…Despite its apparent simplicity, this case already poses quite a challenge. It was previously attacked in a number of papers, including [14,18,37], but only partial answers were obtained as a result. Our approach, though, provides a complete answer in this setting.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy

Papadima

Suciu

2017

Proc. London Math. Soc.

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A central question in arrangement theory is to determine whether the characteristic polynomial Δq of the algebraic monodromy acting on the homology group Hqfalse(F(A),double-struckCfalse) of the Milnor fiber of a complex hyperplane arrangement scriptA is determined by the intersection lattice L(A). Under simple combinatorial conditions, we show that the multiplicities of the factors of Δ1 corresponding to certain eigenvalues of order a power of a prime p are equal to the Aomoto–Betti numbers βpfalse(scriptAfalse), which in turn are extracted from L(A). When scriptA defines an arrangement of projective lines with only double and triple points, this leads to a combinatorial formula for the algebraic monodromy. To obtain these results, we relate nets on the underlying matroid of scriptA to resonance varieties in positive characteristic. Using modular invariants of nets, we find a new realizability obstruction (over double-struckC) for matroids, and estimate the number of essential components in the first complex resonance variety of scriptA. Our approach also reveals a rather unexpected connection of modular resonance with the geometry of prefixSL2false(double-struckCfalse)‐representation varieties, which are governed by the Maurer–Cartan equation.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy

Papadima

Suciu

2017

Proc. London Math. Soc.

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“…But it is unknown whether the first Betti number of an arrangement in 3-space is a function of the lattice alone. By [89], this is so for collections of up to 14 lines with up to 5-fold intersections in the projective plane. See also [134] for the origins of the approach.…”

Section: Milnor Fibersmentioning

confidence: 96%

The Jacobian module, the Milnor fiber, and the D-module generated by $$f^s$$ f s

Walther

2016

Invent. math.

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Abstract. For a germ f on a complex manifold X, we introduce a complex derived from the Liouville form acting on logarithmic differential forms, and give an exactness criterion. We use this Liouville complex to connect properties of the D-module generated by f s to homological data of the Jacobian ideal; specifically we show that for a large class of germs the annihilator of f s is generated by derivations. Through local cohomology, we connect the cohomology of the Milnor fiber to the Jacobian module via logarithmic differentials.In particular, we consider (not necessarily reduced) hyperplane arrangements: we prove a conjecture of Terao on the annihilator of 1/f ; we confirm in many cases a corresponding conjecture on the annihilator of f s but we disprove it in general; we show that the Bernstein-Sato polynomial of an arrangement is not determined by its intersection lattice; we prove that arrangements for which the annihilator of f s is generated by derivations fulfill the Strong Monodromy Conjecture, and that this includes as very special cases all arrangements of Coxeter and of crystallographic type, and all multi-arrangements in dimension 3.

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“…First, we have with (iv) that 2z(1 + vx) = x(−2vx + 5v + 3). (7) If 1 + vx = 0, then x = 5v+3 2v with the previous equality. Since x = − 1 v we get v = −1, which contradicts our assumption.…”

Section: Assume Now (Umentioning

confidence: 99%

On two pencils of cubic curves

Bailet

2016

Preprint

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We give two examples of curve arrangements C : f = 0 ⊂ P 2 C of pencil type which are very close to line arrangements, though the action of the monodromy h 1 on the cohomology H 1 (F, C) of the Milnor Fiber F := f −1 (1) ⊂ C 3 has eigenvalues of order 5 and 6, showing that surprising situations can occur for larger classes of curve arrangements than for line arrangements. Our computations rely on the algorithm given by A. Dimca and G. Sticlaru in [8] which detects the non trivial monodromy eigenspaces of free curves.

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Pencil type line arrangements of low degree: classification and monodromy

Cited by 6 publications

References 21 publications

The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy

The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy

The Jacobian module, the Milnor fiber, and the D-module generated by $$f^s$$ f s

On two pencils of cubic curves

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