Computing the monodromy and pole order filtration on Milnor fiber cohomology of plane curves

Dimca, Alexandru; Sticlaru, Gabriel

doi:10.1016/j.jsc.2018.06.015

Cited by 11 publications

(21 citation statements)

References 31 publications

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“…One has the following result, the second part of which answers positively a conjecture made in Remark 2.9 (i) in [29]. (1) If V has only isolated singularities, then the Hodge filtration F p and the pole order filtration P p coincide on H n−1 (F, C).…”

Section: Gauss-manin Complexes Koszul Complexes and Milnor Fiber Cosupporting

confidence: 53%

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Computing Milnor fiber monodromy for some projective hypersurfaces

Dimca¹,

Sticlaru²

2020

A Panorama of Singularities

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We describe an algorithm computing the monodromy and the pole order filtration on the top Milnor fiber cohomology of hypersurfaces in P n whose pole order spectral sequence degenerates at the second page. In the case of hyperplane arrangements and free, locally quasi-homogeneous hypersurfaces, and assuming a key conjecture, this algorithm is much faster than for a hypersurface as above. Our conjecture is supported by the results due to L. Narvéz Macarro and M. Saito on the roots of Bernstein-Sato polynomials of such hypersurfaces, by all the examples computed so far, and by one partial result. For hyperplane arrangements coming from reflection groups, a surprising symmetry of their pole order spectra on top cohomology is displayed in our examples. We also improve our previous results in the case of plane curves.2010 Mathematics Subject Classification. Primary 32S40; Secondary 32S22, 32S55.

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Section: Gauss-manin Complexes Koszul Complexes and Milnor Fiber Cosupporting

confidence: 53%

“…In the case n = 2, this approach was already described in [27,29] in the case of a reduced plane curve C : f = 0. However, even in this case, we bring here valuable new information, see Proposition 2.2 and Corollary 2.4.…”

Section: Introductionmentioning

confidence: 99%

Computing Milnor fiber monodromy for some projective hypersurfaces

Dimca¹,

Sticlaru²

2020

A Panorama of Singularities

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“…First we state in down-to-earth terms some of our results in [14]. For more details on this spectral sequence approach to the computation of the Milnor fiber monodromy we refer to [10,12,13,23,24].…”

Section: Conic Pencils With One Point Base Locusmentioning

confidence: 99%

Free and Nearly Free Curves vs. Rational Cuspidal Plane Curves

Dimca

Sticlaru

2018

Publ. Res. Inst. Math. Sci.

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We construct some infinite series of free and nearly free curves using pencils of conics with a base locus of cardinality at most two. These curves have an interesting topology, e.g. a high degree Alexander polynomial that can be explicitly determined, a Milnor fiber homotopy equivalent to a bouquet of circles, or an irreducible translated component in the characteristic variety of their complement. Monodromy eigenspaces in the first cohomology group of the corresponding Milnor fibers are also described in terms of explicit differential forms.

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“…The equation (3.2) tells us that this is equivalent to dim H 1 (F, C) λ = 0. Using [16, Proposition 2.2], see also [15,Remark 4.4], it follows that…”

Section: Some Facts About Free and Nearly Free Curvesmentioning

confidence: 99%

“…Here E 1,0 2 (f ) k and E 1,0 2 (f ) d−k denote some terms of the second page of spectral sequences used to compute the monodromy action on the Milnor fiber F , see [5,12,15,26] for details. Note also that the weaker result in [12, Theorem 1.2] is enough for this proof.…”

Section: The Proofsmentioning

confidence: 99%