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On the diffeomorphic type of the complement to a line arrangement in a projective plane. (English) Zbl 1243.14047

In the paper under review, the incidence matrix associated to a line arrangement is defined. Two line arrangements are combinatorially equivalent if their incidence matrices are the same up to permutations in rows and columns. The parameter variety of a line arrangement \(\mathcal{L}\), viewed as a subvariety of \(({\hat{\mathbb{P}}}{}^2)^n\setminus \tilde{\Delta}\), consists of line arrangements combinatorially equivalent to \(\mathcal{L}\).
The authors discover three gluing operations on line arrangements such that the parameter variety of the derived line arrangement under the gluing operations is irreducible if the parameter varieties of the contributing arrangements are irreducible. As a corollary, the authors prove that the parameter variety of a line arrangement is irreducible if each line passes through at most two multiple points of multiplicities greater than \(2\). Another corollary is an alternative proof of the main result of [T. Jiang and S. S.-T. Yau, Compos. Math. 92, No. 2, 133–155 (1994; Zbl 0828.57018)].
Reviewer: Fei Ye (Pokfulam)

MSC:

14N20 Configurations and arrangements of linear subspaces
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
32S22 Relations with arrangements of hyperplanes

Citations:

Zbl 0828.57018
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References:

[1] Eliyahu M., Garber D., Teicher M., A conjugation-free geometric presentation of fundamental groups of arrangements, Manuscripta Math., 2010, 133(1-2), 247-271 http://dx.doi.org/10.1007/s00229-010-0380-2; · Zbl 1205.14034
[2] Fan K.-M., Direct product of free groups as the fundamental group of the complement of a union of lines, Michigan Math. J., 1997, 44(2), 283-291 http://dx.doi.org/10.1307/mmj/1029005704; · Zbl 0911.14007
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[4] Jiang T., Yau S.S.-T., Diffeomorphic types of the complements of arrangements of hyperplanes, Compositio Math., 1994, 92(2), 133-155; · Zbl 0828.57018
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[6] Wang S., Yau S.S.-T., Rigidity of differentiable structure for new class of line arrangements, Comm. Anal. Geom., 2005, 13(5), 1057-1075; · Zbl 1115.52010
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