Milnor fibers of real line arrangements

Yoshinaga, Masahiko

doi:10.5427/jsing.2013.7l

Cited by 19 publications

(48 citation statements)

References 13 publications

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“…a point in T =1 in the notation from section 3) of the rank one local system L, then H 1 (M(A), L) = 0 if some very mild extra condition holds. This result was known for line arrangements defined over the real numbers without this mild extra condition, see [18,19]. A similar result is also known for certain monodromy eigenspaces of the Milnor fiber of a Date: October 12, 2018October 12, .…”

Section: Introductionsupporting

confidence: 54%

A vanishing result for the first twisted cohomology of affine varieties and applications to line arrangements

Bailet¹,

Dimca²,

Yoshinaga³

2018

manuscripta math.

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Section: Introductionsupporting

confidence: 54%

A vanishing result for the first twisted cohomology of affine varieties and applications to line arrangements

Bailet¹,

Dimca²,

Yoshinaga³

2018

manuscripta math.

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“…The many examples we discuss in this paper show a strikingly similar pattern, whereby the only interesting primes, as far as the algebraic monodromy of the Milnor fibration goes, are p = 2 and p = 3. Furthermore, all rank-3 simplicial arrangements examined by Yoshinaga in [57] satisfy e 3 (A) = 0 or 1, and e d (A) = 0, otherwise. Finally, we do not know of any arrangement A of rank at least 3 for which β p (A) = 0 if p > 3.…”

Section: Resultsmentioning

confidence: 99%

“…Consequently, ρ 3 belongs to the component T = f * (H 1 (S, C * )) of V 1 (A m ). Hence, by formula (57), e 3 (A m ) 1.…”

Section: More Examplesmentioning

confidence: 94%

“…Finally, Figure shows a simplicial arrangement of 12 lines in

{CP}^{2}

supporting a reduced

(3, 4)

‐multinet which is not a 3‐net. For more examples, we refer to .…”

Section: Matroids and Multinetsmentioning

confidence: 99%

“…Taking ζ α = e 2πi/k in the above, we see that ρ k = f * (ρ) belongs to the subtorus T = f * (H 1 (S, C * )). Since T lies inside V k−2 (A), formula (57) shows that e k (A) = depth(ρ k ) k − 2.…”

Section: Milnor Fibration and Multinetsmentioning

confidence: 99%

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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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Configurations of points and topology of real line arrangements

Guerville-Ballé

Viu-Sos

2018

Math. Ann.

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A central question in the study of line arrangements in the complex projective plane CP 2 is the following: when does the combinatorial data of the arrangement determine its topological properties? In the present work, we introduce a topological invariant of complexified real line arrangements, called the chamber weight. This invariant is based on the weight counting over the points of the associated dual configuration, located in particular chambers of the real projective plane RP 2 .Using this dual setting, we construct several examples of complexified real line arrangements with the same combinatorial data and different embeddings in CP 2 (i.e. Zariski pairs) which are distinguished by this invariant. In particular, we obtain new Zariski pairs of 13, 15 and 17 lines defined over Q and containing only double and triple points. For each one of our examples, we derive some degenerations containing points of multiplicity 2, 3 and 5, which are also Zariski pairs.We compute explicitly the moduli space of the combinatorics of one of these examples, and prove that it has exactly two connected components. We also obtain three geometric characterizations of these components: the existence of two smooth conics, one tangent to six lines and the other containing six triple points, as well as the collinearity of three specific triple points.

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Milnor fibers of real line arrangements

Cited by 19 publications

References 13 publications

A vanishing result for the first twisted cohomology of affine varieties and applications to line arrangements

A vanishing result for the first twisted cohomology of affine varieties and applications to line arrangements

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

Configurations of points and topology of real line arrangements

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