Mina TEICHER, Moshe COHEN, Hao Max SUN, Meirav AMRAM

The height of a permutation and applications to distance between real line arrangements

We present a new notion of a distance between two real line arrangements. We define the height of a permutation and use this idea in our main theorem, which gives us a lower bound on the distance between the pair. We apply these techniques to the seven special cases of real arrangements with ten lines found in previous work by the authors.

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[1] Amram M, Cohen M, Sun H, Teicher M, Ye F et al. Combinatorial symmetry of line arrangements and applications. Topology and its Applications 2015; 193: 226-247. doi: 10.1016/j.topol.2015.07.004
[2] Amram M, Cohen M, Teicher M, Ye F. Moduli spaces of ten-line arrangements with double and triple points. arXiv 2013. arxiv:1306.6105
[3] Amram M, Teicher M, Ye F. Moduli spaces of arrangements of 10 projective lines with quadruple points. Advances in Applied Mathematics 2013; 51 (3): 392-418. doi: 10.1016/j.aam.2013.05.002
[4] Artal Bartolo E, Cogolludo-Agustín J.I, Guerville-Ballé B, Marco-Buzunariz M. An arithmetic Zariski pair of line arrangements with non-isomorphic fundamental group. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 2017; 111 (2): 377-402. doi: 10.1007/s13398-016-0298-y
[5] Bartolo EA, Ruber JC, Cogolludo-Agustín JI, Buzunáriz MM. Topology and combinatorics of real line arrangements. Compositio Mathematica 2005; 141 (6): 1578-1588. doi: 10.1112/S0010437X05001405
[6] Guerville-Ballé B. Zariski pairs of line arrangements with twelve lines. Geometry & Topology 2016; 20: 537-553. doi: 10.2140/gt.2016.20.537
[7] Nazir S, Yoshinaga M. On the connectivity of the realization spaces of line arrangements. Annali Della Scuola Normale Superiore Di Pisa 2012; 11 (4): 921-937. doi: 10.2422/2036 − 2145.201009003
[8] Randell R. Lattice-isotopic arrangements are topologically isomorphic. Proceedings of the American Mathematical Society 1989; 107 (2): 555-559. doi: 10.1090/S0002-9939-1989-0984812-7
[9] Rybnikov G.L. On the fundamental group of the complement of a complex hyperplane arrangement. Functional Analysis and Its Applications 2011; 45 (2): 137-148. doi: 10.1007/s10688-011-0015-8
[10] Ye F. Classification of moduli spaces of arrangements of 9 projective lines. Pacific Journal of Mathematics 2013; 265 (1): 243-256. doi: 10.2140/pjm.2013.265.243