Related papers: Connectedness and combinatorial interplay in the m…
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…
Two line arrangements in $\mathbb{CP}^2$ can have different topological properties even if they are combinatorially isomorphic. Results by Dan Cohen and Suciu and by Randell show that a reducible moduli space under complex conjugation is a…
We investigate several topological and combinatorial properties of line arrangements. We associate to a line arrangement a link obtained by intersecting the arrangement with some sphere. Several topics are discussed: (a) some link…
It is well known that knots are countable in ordinary knot theory. Recently, knots {\it with intersections} have raised a certain interest, and have been found to have physical applications. We point out that such knots --equivalence…
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…
*This paper is from 2018* In this paper, we try to classify moduli spaces of arrangements of $12$ lines with sextic points. We show that moduli spaces of arrangements of $12$ lines with sextic points can consist of more than two connected…
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…
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…
We prove that under certain combinatorial conditions, the realization spaces of line arrangements on the complex projective plane are connected. We also give several examples of arrangements with eight, nine and ten lines which have…
Compositionality is a key strategy for addressing combinatorial complexity and the curse of dimensionality. Recent work has shown that compositional solutions can be learned and offer substantial gains across a variety of domains, including…
The aim of this study is to construct and compose an instructional design in combinatorial learning, particularly in the concept of counting. A composed design is expected to optimize students' combinatorial-thinking skill. This research…
The boundary of the Milnor fiber associated with a complex line arrangement is a three dimensional plumbed manifold, and it is a combinatorial invariant. We prove the reverse implication, which was conjectured N\'emethi and Szil\'ard. That…
A ribbon is a first-order thickening of a non-singular curve. Motivated by a question of Eisenbud and Green, we show that a compactification of the moduli space of line bundles on a ribbon is given by the moduli space of semi-stable…
Using several numerical invariants, we study a partition of the space of line arrangements in the complex projective plane, given by the intersection lattice types. We offer also a new characterization of the free plane curves using the…
In this paper, we give a classification of the 3-dimensional associative algebras over the complex numbers, including a construction of the moduli space, using versal deformations to determine how the space is glued together.
The geometric, topological, and symplectic properties of moduli spaces (spaces of configurations modulo rotations and translations) of polygonal linkages have been studied by Kapovich, Millson, and Kamiyama, et. al. One can form a polygonal…
An effective way to design structured coherent wave interference patterns that builds on the theory of coherent lattices, is presented. The technique combines prime number factorization in the complex plane with moir\'e theory to provide a…
We show that a line arrangement in the complex projective plane supports a nontrivial resonance variety if and only if it is the underlying arrangement of a "multinet," a multi-arrangement with a partition into three or more equinumerous…
The geometry of the moduli space of stable spin curves is studied, with emphasis on its combinatorial properties. In this context, the standard graph theoretic framework is not just a book-keeping device: some purely combinatorial results…
We study supersolvable line arrangements in ${\mathbb P}^2$ over the reals and over the complex numbers, as the first step toward a combinatorial classification. Our main results show that a nontrivial (i.e., not a pencil or near pencil)…