The Non-vanishing Cohomology of Orlik-Solomon Algebras

Kawahara, Yukihito

doi:10.3836/tjm/1184963658

Cited by 7 publications

(21 citation statements)

References 16 publications

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“…The work of Kawahara from [, § 3] shows that nets on matroids abound. In particular, this work shows that, for any

k ⩾ 3

, there is a simple matroid

scriptM

supporting a nontrivial

k

‐net.…”

Section: Matroids 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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“…The work of Kawahara from [, § 3] shows that nets on matroids abound. In particular, this work shows that, for any

k ⩾ 3

, there is a simple matroid

scriptM

supporting a nontrivial

k

‐net.…”

Section: Matroids and Multinetsmentioning

confidence: 99%

“…Example Particularly simple is the following construction, due to Kawahara : given any Latin square, there is a matroid with a 3‐net realizing it, such that each submatroid obtained by restricting to the parts of the 3‐net is a uniform matroid. In turn, some of these matroids may be realized by line arrangements in

{CP}^{2}

.…”

Section: Matroids and Multinetsmentioning

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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“…Needless to say, the same is true for the complement of 9 lines in Pappus's theorem ( [Fa]). In the degree four case, there are two different arrangements of 12 lines in the Kirkman Theorem and the Steiner Theorem ( [Ka3]). Those arrangements are 3-nets, whose combinatorial structures are matroids associated to Latin squares ( [LY,Yu2,Ka3] [Fa,Ka3]).…”

Section: Affine Casementioning

confidence: 99%

“…The n = 2 case was found in [CS]. Note that the underlying matroid of A is a degeneration of the matroid associated to the Latin n-dimensional hypercube given by the addition table for (Z d ) n (see [Ka3]). …”

Section: Affine Casementioning

confidence: 99%

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Non-vanishing of the twisted cohomology on the complement of hypersurfaces

Kawahara

2008

Proc. Amer. Math. Soc.

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Abstract. Generically, the cohomology with coefficients in a local system of rank one on the complement in P n of the union of a finite number of hypersurfaces vanishes except in the highest dimension. We study the non-generic case, in which the cohomology in other dimensions does not vanish. When the hypersurfaces are hyperplanes, many examples of this kind are known. In this paper, we consider the case in which the hypersurfaces need not be hyperplanes. We prove that the hypersurfaces given by some particular linear systems have non-vanishing local system cohomologies.

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Multinets, resonance varieties, and pencils of plane curves

2007

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We show that a line arrangement in the complex projective plane supports a nontrivial resonance variety if and only if it is the underlying arrangement of a "multinet," a multi-arrangement with a partition into three or more equinumerous classes which have equal multiplicities at each inter-class intersection point, and satisfy a connectivity condition. We also prove that this combinatorial structure is equivalent to the existence of a pencil of plane curves, also satisfying a connectivity condition, whose singular fibers include at least three products of lines, which comprise the arrangement. We derive numerical conditions which impose restrictions on the number of classes, and the line and point multiplicities that can appear in multinets, and allow us to detect whether the associated pencils yield nonlinear fiberings of the complement.2000 Mathematics Subject Classification. 52C35,14H10,05B30.

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The Non-vanishing Cohomology of Orlik-Solomon Algebras

Cited by 7 publications

References 16 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

Non-vanishing of the twisted cohomology on the complement of hypersurfaces

Multinets, resonance varieties, and pencils of plane curves

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