Moduli spaces of arrangements of 10 projective lines with quadruple points

Abstract

We classify moduli spaces of arrangements of 10 lines with quadruple points. We show that moduli spaces of arrangements of 10 lines with quadruple points may consist of more than 2 disconnected components, namely 3 or 4 distinct points. We also present defining equations to those arrangements whose moduli spaces are still reducible after taking quotients of complex conjugations.2 sifying moduli spaces of arrangements of 10 lines. Section 3 shows that moduli spaces of arrangements with multiple points of high multiplicity are most likely irreducible. Section 4 and Section 5 deal with arrangements of 10 lines with a quadruple point. All possible arrangements of 10 lines with quadruple point whose moduli spaces are reducible, for instance, consisting of 2 points, 3 points, 4 points, or 2 one-dimensional components, can be found there.Acknowledgements: This work was partially supported by the Oswald Veblen Fund and by the Minerva Foundation of Germany. The authors thank M. Falk and the referee for their comments and suggestions that helped improve the clarity of the paper. PreliminariesLet A = {L 1 , L 2 , · · · , L n } be a line arrangement in CP 2 . We say a singularity of L 1 ∪ L 2 ∪ · · · ∪ L n is a multiple point of A, if it has multiplicity at least 3.Definition 2.1. A line arrangement A is said to be C ≤3 if all multiple points of A are on at most three lines, say L 1 , L 2 and L 3 . A line arrangement is called simple C ≤3 if it is C ≤3 and one of the following conditions holds:Theorem 3.1. Let A be an arrangement of 10 lines. If there is a multiple point of multiplicity ≥ 6, then the moduli space M A is irreducible.Proof. Assume that L 1 ∩L 2 ∩· · ·∩L 6 = ∅. It is easy to check that at least one of the six lines contains at most two multiple points. By Lemma 2.3 and classification of arrangements of 9 lines (see [Ye11] Proposition 3.3), we see that M A is irreducible.When the highest multiplicity is 5, we notice that there is a case that the moduli space is reducible. However, a close look (see Remark 3.3) shows that the fundamental groups are still isomorphic.Theorem 3.2. Let A be a non-reductive arrangement of 10 lines with a quintuple point and no multiple points of higher multiplicities. Then A contains a Falk-Sturmfels arrangement as a sub-arrangement.Proof. By Lemma 2.4 and Theorem 2.5, we have the following inequality n 3 +n 4 ≤ 140−44n 5 9 .Since n 5 ≥ 1, thus n 3 + n 4 ≤ 10. On the other hand, there must be at least 11 − n 5 multiple points so that each line will pass through at least 3 multiple points. Therefore, 11 − n 5 ≤ n 3 + n 4 . The two inequalities together tell us that n 5 = 1 and n 3 + n 4 = 10. Apply Lemma 2.4 again, and we see that n 4 ≤ 1.

Combinatorial symmetry of line arrangements and applications

“…In the second subsection we consider the three arrangements that contain a Z 2 subgroup out of the five complex conjugate arrangements that were produced in [ACTY13]. 5.1.…”

“…Although seven real cases of ten-line arrangements with disconnected moduli space were produced in [ATY13] and [ACTY13], we only apply our algorithm to those that have a Z 2 subgroup of the automorphism group. We start treating each of the cases {1}, {6} and {7} by producing such a subgroup.…”

“…The complex cases are considered in Section 5 with Subsection 5.1 on two examples from the literature, the MacLane arrangement and the Nazir-Yoshinaga arrangement, and Subsection 5.2 on examples from [ACTY13]. The geometric reflection given by complex conjugation can already be used to disqualify these as Zariski pairs, but our geometric reflection ϕ works, as well.…”

“…Following this methodology, a classification of the moduli spaces of arrangements of ten lines was completed by the authors in [ATY13] and [ACTY13]. By Theorem 5.3 and Corollary 5.5. in [ACTY13], this gives a list of eighteen potential Zariski pairs, as determined via disconnected moduli spaces, for all complex line arrangements of ten lines satisfying a reasonable assumption [ACTY13, Assumption 1.2].…”

“…By Theorem 5.3 and Corollary 5.5. in [ACTY13], this gives a list of eighteen potential Zariski pairs, as determined via disconnected moduli spaces, for all complex line arrangements of ten lines satisfying a reasonable assumption [ACTY13, Assumption 1.2].…”