On the connectivity of the realization spaces of line arrangements
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 4, p. 921-937

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.

Published online : 2018-06-21
Classification:  52C35
@article{ASNSP_2012_5_11_4_921_0,
     author = {Nazir, Shaheen and Yoshinaga, Masahiko},
     title = {On the connectivity of the realization spaces of line arrangements},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 11},
     number = {4},
     year = {2012},
     pages = {921-937},
     mrnumber = {3060685},
     zbl = {06142478},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2012_5_11_4_921_0}
}
Nazir, Shaheen; Yoshinaga, Masahiko. On the connectivity of the realization spaces of line arrangements. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 4, pp. 921-937. http://www.numdam.org/item/ASNSP_2012_5_11_4_921_0/

[1] E. Artal Bartolo, J. Carmona Ruber, J. I. Cogolludo-Agustín and M. Marco Buzunáriz, Topology and combinatorics of real line arrangements, Compos. Math. 141 (2005), 1578–1588. | MR 2188450 | Zbl 1085.32012

[2] E. Artal Bartolo, J. Carmona Ruber, J. I. Cogolludo Agustín and M. Á. Marco Buzunáriz, Invariants of combinatorial line arrangements and Rybnikov’s example, In: “Singularity Theory and its Applications”, Adv. Stud. Pure Math., 43, Math. Soc. Japan, Tokyo, 2006, 1–34. | MR 2313406 | Zbl 1135.32025

[3] D. C. Cohen and A. Suciu, The braid monodromy of plane algebraic curves and hyperplane arrangements, Comment. Math. Helv. 72 (1997), 285–315. | MR 1470093 | Zbl 0959.52018

[4] M. Eliyahu, D. Garber and M. Teicher, A conjugation-free geometric presentation of fundamental groups of arrangements, Manuscripta Math. 133 (2010), 247–271. | MR 2672548 | Zbl 1205.14034

[5] K. M. Fan, Position of singularities and fundamental group of the complement of a union of lines, Proc. Amer. Math. Soc. 124 (1996), 3299–3303. | MR 1343691 | Zbl 0860.14020

[6] K. M. Fan, Direct product of free groups as the fundamental group of the complement of a union of lines, Michigan Math. J. 44 (1997), 283–291. | MR 1460414 | Zbl 0911.14007

[7] D. Garber, M. Teicher and U. Vishne, π 1 -classification of real arrangements with up to eight lines, Topology 42 (2003), 265–289. | MR 1928653 | Zbl 1074.14050

[8] T. Jiang and S. S.-T. Yau, Diffeomorphic types of the complements of arrangements of hyperplanes, Compositio Math. 92 (1994), 133–155. | Numdam | MR 1283226 | Zbl 0828.57018

[9] S. Nazir and Z. Raza, Admissible local systems for a class of line arrangements, Proc. Amer. Math. Soc. 137 (2009), 1307–1313. | MR 2465653 | Zbl 1162.32016

[10] R. Randell, Lattice-isotopic arrangements are topologically isomorphic, Proc. Amer. Math. Soc. 107 (1989), 555–559. | MR 984812 | Zbl 0681.57016

[11] G. Rybnikov, On the fundamental group of the complement of a complex hyperplane arrangement, Preprint available at arXiv:math.AG/9805056 | MR 2848779 | Zbl 1271.14085

[12] I. R. Shafarevich, “Basic Algebraic Geometry. 2. Schemes and complex manifolds”, Second edition. Translated from the 1988 Russian edition by Miles Reid. Springer-Verlag, Berlin, 1994. xiv+269 pp. | MR 1328834 | Zbl 0362.14001

[13] S. Wang and S.S.-T. Yau, Rigidity of differentiable structure for new class of line arrangements, Comm. Anal. Geom. 13 (2005), 1057–1075. | MR 2216152 | Zbl 1115.52010