Moduli spaces of ten-line arrangements with double and triple points (1306.6105v1)

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.

• The paper introduces a complete classification of ten-line arrangements with double and triple intersections, detailing 71 distinct configurations.
• It employs advanced computational methods and matroid theory to analyze the topology and geometric realizability of these arrangements.
• The study reveals nine potential Zariski pairs and non-geometric arrangements, paving the way for future research in combinatorial geometry.

Moduli Spaces of Ten-Line Arrangements with Double and Triple Points

This paper considers the intricate field of line arrangements, focusing specifically on ten-line arrangements characterized by double and triple points. Within algebraic geometry, line arrangements are a compelling subject due to their complex topological properties and combinatorial structures. The paper explores the exploration of moduli spaces, which serves as a parameter space representing all arrangements with certain shared properties. The authors undertake the arduous task of classifying these arrangements and delineating their moduli spaces, producing new results of significant interest to the community of combinatorial geometry.

Summary of Core Contributions

The authors address the phenomenon of Zariski pairs, which are pairs of arrangements possessing isomorphic combinatorial intersection lattices while having complements with non-isomorphic fundamental groups. They successfully classify ten-line arrangements under the assumption that the intersections are only double and triple points, albeit this classification process acknowledges departures from previously examined intersection phenomena influenced by lines with configurations of higher multiplicity points.

The authors present a complete list of seventy-one ten-line arrangements, a notable majority of which are not explicitly reported in previous studies. The paper identifies nine particularly intriguing combinatorial arrangements which are not geometrically realizable, posing significant implications in the paper of line arrangements as well as advancing understanding of matroid theory when considering configurations irreducible over certain fields.

Methodological Insight

By leveraging an interdisciplinary approach combining elements from classical algebraic geometry with advanced computational techniques, the authors advance existing knowledge on the topology of line arrangements. This includes the use of configuration tables and oriented matroids which facilitate an efficient enumerative approach to characterizing ten-line arrangements.

A substantial part of the paper is dedicated to analyzing different configurations of line arrangements and assessing their realizability. The authors develop algorithms based on matroid theory to ascertain the conditions under which configurations can be geometrically realized in the projective plane. Additionally, these algorithms help distinguish between those arrangements belonging to irreducible moduli spaces and those suggestive of Zariski pairs.

Results and Implications

The paper underscores two primary contributions—the combinatorial classification of ten-line arrangements and the identification of nine potential Zariski pairs alongside the nine non-geometric arrangements. These findings have profound implications for the paper of the topology of the complements of arrangements, with particular interest in the role of fundamental groups as combinatorial invariants.

The constructed catalog of arrangements holds the potential for future explorations into smaller Zariski pairs. Moreover, discussions speculating on new forms of line arrangements set the stage for greater examination under the frameworks both of algebraic geometry and matroid theory. These explorations can lead to further insights into the properties of excluded minors in matroid theory, particularly relevant to complex and real fields.

Speculation and Future Directions

By providing a fertile ground of new open questions, the authors facilitate an ongoing inquiry into the field. Delving into extended classes of line arrangements through rigorous empirical and theoretical techniques opens multiple avenues for future research. Continuing to focus on configurations undermined by their specific moduli spaces may yield exciting breakthroughs in our understanding of line arrangements and their associated topologies.

In conclusion, this paper marks a significant step forward in the field of combinatorial geometry, offering researchers an unprecedented classification of ten-line arrangements, and laying groundwork for future inquiries into the nature and consequences of Zariski pairs. The findings enrich the theoretical landscape of the moduli spaces and pose intriguing challenges that align with ongoing trends in analyzing geometric and combinatorial structures within mathematics.