On the connectivity of the realization spaces of line arrangements

Nazir, Shaheen; Yoshinaga, Masahiko

doi:10.2422/2036-2145.201009_003

Cited by 23 publications

(30 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…On the contrary direction, Garber, Teicher and Vishne [GTV03] proved that there is no Zariski pair of arrangement of upto 8 real lines which covered the result of Fan [Fan97] on arrangements of 6 lines. This result was recently generalized to arrangements of 8 complex lines by Nazir and Yoshinaga [NY10].…”

Section: Introductionmentioning

confidence: 80%

Classification of moduli spaces of arrangements of nine projective lines

Ye¹

2013

Pacific J. Math.

View full text Add to dashboard Cite

In the study of line arrangements, searching for minimal examples of line arrangements whose fundamental groups are not combinatorially invariant is a very interesting and hard problem. It was known that such a minimal arrangement must have at least 9 lines. In this paper, we extend the number to 10 by a new method. We classify arrangements of 9 projective lines according to the irreducibility of their moduli spaces and show that fundamental groups of complements of arrangements of 9 projective lines are combinatorially invariant. The idea and results have been used to classify arrangements of 10 projective lines.2000 Mathematics Subject Classification. 14N20, 32S22, 52C35.

show abstract

Section: Introductionmentioning

confidence: 80%

Classification of moduli spaces of arrangements of nine projective lines

Ye¹

2013

Pacific J. Math.

View full text Add to dashboard Cite

show abstract

“…The aim of this section is to prove that complex central hyperplane arrangements with up to 7 hyperplanes and same underlying matroid are isotopic, improving the results of [NY12] and [Ye13] to any rank. The central idea of our proof is to exploit the connectedness of the reduced realization space of the underlying matroid of these arrangements to apply Proposition 2.1.…”

Section: Applicationsmentioning

confidence: 99%

“…Several results in this direction appeared in the literature. In particular, Jiang and Yau [JY98], Nazir and Yoshinaga [NY12] and Amram, Teicher and Ye [ATY13] focused their attention on some specific classes of line arrangements in the complex projective plane. One-parameter families of isotopic arrangements have also been studied in [WY07], [WY08] and [YY09].…”

Section: Introductionmentioning

confidence: 99%

The diffeomorphism type of small hyperplane arrangements is combinatorially determined

Gallet¹,

Saini²

2016

Preprint

View full text Add to dashboard Cite

It is known that there exist hyperplane arrangements with same underlying matroid that admit non-homotopy equivalent complement manifolds. In this work we show that, in any rank, complex central hyperplane arrangements with up to 7 hyperplanes and same underlying matroid are isotopic. In particular, the diffeomorphism type of the complement manifold and the Milnor fiber and fibration of these arrangements are combinatorially determined, that is, they depend uniquely on the underlying matroid. To do this, we associate to every such matroid a topological space, that we call the reduced realization space; its connectedness -showed by means of symbolic computation -implies the desired result.

show abstract

“…We illustrate the relevance of this method by reconstructing of the all the classical arrangements which have a non-connected moduli space: the MacLane [16], the Nazir-Yoshinaga [18], the Falk-Sturmfels [9] and the Rybnikov [21] arrangements. As a final illustration of this method, we construct several arrangements of eleven lines which have moduli spaces form by four connected components.…”

Section: Introductionmentioning

confidence: 99%

On the non-connectivity of moduli spaces of arrangements: the splitting-polygon structure

Guerville-Ballé¹

2021

Preprint

View full text Add to dashboard Cite

Questions that seek to determine whether a hyperplane arrangement property, be it geometric, arithmetic or topological, is of a combinatorial nature (that is determined by the intersection lattice) are abundant in the literature. To tackle such questions and provide a negative answer, one of the most effective methods is to produce a counterexample. To this end, it is essential to know how to construct arrangements that are lattice-equivalent. The more different they are, the more efficient it will be.In this paper, we present a method to construct arrangements of complex projective lines that are lattice-equivalent but lie in distinct connected components of their moduli space. To illustrate the efficiency of the method, we apply it to reconstruct all the classical examples of arrangements with disconnected moduli spaces: MacLane, Falk-Sturmfels, Nasir-Yoshinaga and Rybnikov. Moreover, we employ this method to produce novel examples of arrangements of eleven lines whose moduli spaces are formed by four connected components.

show abstract

On the connectivity of the realization spaces of line arrangements

Abstract: We prove that, under certain combinatorial conditions, the realization spaces of line arrangements on the complex projective plane are connected. We also give several examples of arrangements with eight, nine and ten lines that have disconnected realization spaces.

Cited by 23 publications

References 12 publications

Classification of moduli spaces of arrangements of nine projective lines

Classification of moduli spaces of arrangements of nine projective lines

The diffeomorphism type of small hyperplane arrangements is combinatorially determined

On the non-connectivity of moduli spaces of arrangements: the splitting-polygon structure

Contact Info

Product

Resources

About