2021/10/31 by Benoît Guerville-Ballé, Guerville-Ballé, Benoît
Mathematics · #14H10 #14N10 #14N20 #51A45 #51M15 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #math.AG #math.CO #msc:14H10 #msc:14N10 #msc:14N20 #msc:51A45 #msc:51M15
paper · pdf · doi:10.48550/arxiv.2111.00399
13 pages, 4 figures
arxiv created 2021/10/31 · arxiv updated 2021/11/02
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.