[Paper Review] Moduli spaces of ten-line arrangements with double and triple points

Meirav Amram, Moshe Cohen+2 more|arXiv (Cornell University)|Jun 25, 2013

Advanced Combinatorial Mathematics3 references8 citations

TL;DR

This paper classifies the moduli spaces of ten-line arrangements in the complex projective plane with only double and triple points, using combinatorial and geometric methods. It identifies at most nine Zariski pairs—lattice-isomorphic arrangements with non-isomorphic fundamental groups—among such configurations, providing a complete combinatorial classification of 71 distinct arrangements, including nine non-realizable ones.

ABSTRACT

Two arrangements with the same combinatorial intersection lattice but whose complements have different fundamental groups are called a Zariski pair. This work finds that there are at most nine such pairs amongst all ten line arrangements whose intersection points are doubles or triples. This result is obtained by considering the moduli space of a given configuration table which describes the intersection lattice. A complete combinatorial classification is given of all arrangements of this type under a suitable assumption, producing a list of seventy-one described in a table, most of which do not explicitly appear in the literature. This list also includes other important counterexamples: nine combinatorial arrangements that are not geometrically realizable.

Motivation & Objective

• To classify all moduli spaces of ten-line arrangements in ℂP² with only double and triple points.
• To determine whether such arrangements can form Zariski pairs—lattice-isomorphic arrangements with different fundamental groups.
• To provide a complete combinatorial classification of all such arrangements under a suitable assumption, identifying both realizable and non-realizable configurations.
• To extend prior classifications of nine-line arrangements and explore the existence of Zariski pairs in ten-line configurations.
• To offer a systematic framework for detecting topological distinctions (via fundamental groups) through moduli space analysis and geometric realizability.

Proposed method

• Constructs the intersection lattice (combinatorial configuration table) for each ten-line arrangement with only double and triple points.
• Applies a geometric methodology based on moduli space analysis to determine realizability and topological invariants.
• Uses a practical algorithm to systematically analyze subarrangements (e.g., easel, central, generic) and reduce complex configurations.
• Employs algebraic constraints derived from line equations and intersection conditions (e.g., $ b - (1 + a - a^2) = 0 $) to classify realizations.
• Analyzes the fundamental group of the complement via lower central series and Alexander module invariants, following Rybnikov’s approach.
• Validates results through case-by-case analysis of subarrangements and parameterized line arrangements, identifying contradictions or consistent solutions.

Experimental results

Research questions

• RQ1 How many Zariski pairs exist among ten-line arrangements with only double and triple points?
• RQ2 Which combinatorial configurations of ten lines with only double and triple points are geometrically realizable?
• RQ3 Can the moduli space of a given configuration table detect differences in the fundamental group of the complement?
• RQ4 What is the complete combinatorial classification of ten-line arrangements with only double and triple points under the assumption of realizability?
• RQ5 Are there configurations that are combinatorially defined but not geometrically realizable, and how many such cases exist?

Key findings

• There are at most nine Zariski pairs among all ten-line arrangements with only double and triple points.
• A complete list of 71 distinct combinatorial configurations is provided, most of which are not previously documented in the literature.
• Nine of the 71 configurations are combinatorially defined but not geometrically realizable, serving as important counterexamples.
• The classification confirms the absence of Zariski pairs in arrangements of nine lines, extending prior results.
• The moduli space analysis successfully identifies topological distinctions via fundamental group invariants, even when the intersection lattice is identical.
• The method successfully reduces complex configurations using subarrangement decomposition (e.g., easel, central, generic), enabling systematic enumeration and validation.