Combinatorial symmetry of line arrangements and applications

Abstract

Abstract. We introduce an algorithm that exploits a combinatorial symmetry of an arrangement in order to produce a geometric reflection between two disconnected components of its moduli space. We apply this method to disqualify three real examples found in previous work by the authors from being Zariski pairs. Robustness is shown by its application to complex cases, as well.

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

“…We investigate d min (C a , C b ) of the seven real examples out of the eighteen potential Zariski pairs of ten lines mentioned above and summarize the results in Table . We note here that Cases 1, 6, and 7 from [1] all have distance d min (C a , C b ) = 2 , and we know these arrangements are not Zariski pairs. Beyond the scope of this paper, we question the relationship between these two properties, and we wonder whether this notion of distance could be used to measure the "strength" of a Zariski pair.…”

“…When t = 1 2 , the combinatorics of all C t 's are the same. Since the moduli space of this case has two irreducible components, any parametrization of 1.A and 1.B has at least one degenerate arrangement.…”

“…In this subsection we define a parametrization between two line arrangements and we present our main theorem (Theorem 3.9). The motivation here is the study of the automorphism group Aut(A) of the arrangement A that appears in [1]. Definition 3.5 Let C t for t ∈ I = [a, b] be a parametrization of A and B that is defined by the polynomial…”

“…Three of these seven were proven not to be Zariski pairs in [1] using the automorphism group Aut(A), the group of all lattice isomorphisms or symmetries of L(A). We consider these three cases anyway so as to compare their geometry to the remaining four real cases.…”

“…In Section 4 we consider the three pairs of real line arrangements that appear in [1]. These three pairs have a Z 2 symmetry.…”