On line arrangements with applications to 3-nets

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 realizability of 3‐nets by line arrangements in

{CP}^{2}

has been studied by several authors, including Yuzvinsky , Urzúa , and Dimca, Ibadula, and Măcinic .…”

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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“…Because of Kapranov's description of M 0,d+1 [11,12], this produces bijections between arrangements and curves in P d−2 (Corollary 4.2). For instance, arrangements of d lines in P 2 correspond to lines in P d−2 (as in [21]), arrangements of d conics in x 2 + y 2 + z 2 = 0 correspond to conics in P d−2 , etc. To Date: October 30, 2018.…”

mentioning

confidence: 99%

“…As in [21], we introduced ordered pairs (A, P ), where A is an arrangement of d lines, and P is a point in P 2 \ d i=1 L i . If (A, P ) and (A ′ , P ′ ) are two such pairs, we say that they are isomorphic if there exists an automorphism T of P 2 such that T (L i ) = L ′ i for every i, and T (P ) = P ′ .…”

mentioning

confidence: 99%

Arrangements of rational sections over curves and the varieties they define

Urzúa¹

2011

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

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We introduce arrangements of rational sections over curves. They generalize line arrangements on P 2 . Each arrangement of d sections defines a single curve in P d−2 through the Kapranov's construction of M 0,d+1 . We show a one-to-one correspondence between arrangements of d sections and irreducible curves in M 0,d+1 , giving also correspondences for two distinguished subclasses: transversal and simple crossing. Then, we associate to each arrangement A (and so to each irreducible curve in M 0,d+1 ) several families of nonsingular projective surfaces X of general type with Chern numbers asymptotically proportional to various log Chern numbers defined by A. For example, for the main families and over C, any such X is of positive index and π 1 (X) ≃ π 1 (A), where A is the normalization of A. In this way, any rational curve in M 0,d+1 produces simply connected surfaces with 2 < c 2 1 c 2 arbitrarily close to the Miyaoka-Yau bound 3? Hence, at least when A is a rational curve, it is important for us to know about sharp upper bounds forc 2 1 (A) c 2 (A) (also for A p∆ , see Remark 7.3). So far, we only know that this bound is strictly smaller than 3 (Corollaries 6.2 and 6.4). On the other hand, in positive characteristic, we use our method to producé etale simply connected nonsingular projective surfaces of general type which violate any sort of Miyaoka-Yau inequality for any given characteristic (Example 1.4, Remark 6.5, Example 7.2).2 1 these are the type of singularities for arrangements in [22]. 6 7 1. We have V 0 (P 1 , . . . , P d ) ∼ = M 0,d . 9

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“…It is clear that (Λ 1 , Λ 2 , Λ 3 ) can be seen as an abstract dual 3-net, embedded in PG(2, K) and labeled by Q. Important examples of quasigroup realizations were given by Yuzvinsky [22] when Q is an abelian group, by Stipins [20] when Q is a nonassociative loop of order 5, by Korchmáros, Nagy and Pace when Q is a finite dihedral group, and by Urzúa [21] when Q is the quaternion group of order 8. In fact, it turns out that these are essentially all projective realizations of finite groups.…”

Section: 5mentioning

confidence: 99%

Light dual multinets of order six in the projective plane

Bogya

Nagy

2019

Acta Math. Hungar.

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The aim of this paper is twofold: First we classify all abstract light dual multinets of order 6 which have a unique line of length at least two. Then we classify the weak projective embeddings of these objects in projective planes over fields of characteristic zero. For the latter we present a computational algebraic method for the study of weak projective embeddings of finite point-line incidence structures.

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On line arrangements with applications to 3-nets

Cited by 18 publications

References 14 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

Arrangements of rational sections over curves and the varieties they define

Light dual multinets of order six in the projective plane

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