Related papers: Cubic Partial Cubes from Simplicial Arrangements
We combine ideas of Scott and Swarup on good position for almost invariant subsets of a group with ideas of Sageev on constructing cubings from such sets. We construct cubings which are more canonical than in Sageev's original construction.…
It is a classical result that there are $12$ (irreducible) rational cubic curves through $8$ generic points in $\mathbb{P}_{\mathbb{C}}^2$, but little is known about the non-generic cases. The space of $8$-point configurations is…
We give some new explicit examples of putatively optimal projective spherical designs. i.e., ones for which there is numerical evidence that they are of minimal size. These form continuous families, and so have little apparent symmetry in…
We construct nontrivial homomorphisms from the quasi group of some cubic surfaces over $\bbF_{\!p}$ into a group. We show experimentally that the homomorphisms constructed are the only possible ones and that there are no nontrivial…
Various authors have calculated how many pairwise incomparable points can be selected from a partially ordered set. We tackle this question for the family of subsets of a finite set obtained by removing or adding a bounded number of…
A partial cube is a graph having an isometric embedding in a hypercube. Partial cubes are characterized by a natural equivalence relation on the edges, whose classes are called zones. The number of zones determines the minimal dimension of…
We construct a countable family of multi-dimensional continued fraction algorithms, built out of five specific multidimensional continued fractions, and find a wide class of cubic irrational real numbers a so that either (a, a^2) or (a,…
We show how to construct infinite families of explicitly determined cubic number fields whose class group has a subgroup isomorphic to $(\mathbb{Z}/2)^8$ using degree $1$ del Pezzo surfaces. We illustrate the method and provide an example…
In the projective space $\mathrm{PG}(3,q)$, we consider the orbits of lines under the stabilizer group of the twisted cubic. It is well known that the lines can be partitioned into classes every of which is a union of line orbits. All types…
We discover a simple construction of a four-dimensional family of smooth surfaces of general type with $p_g(S)=q(S)=0$, $K^2_S=3$ with cyclic fundamental group $C_{14}$. We use a degeneration of the surfaces in this family to find…
It is well established that a general pair of twisted cubic curves in complex projective space has ten common secant lines. As an initial investigation, we show that the monodromy group of the ten common secant lines over the complex…
A cubic partition is an integer partition wherein the even parts can appear in two colors. In this paper, we introduce the notion of generalized cubic partitions and prove a number of new congruences akin to the classical Ramanujan-type. We…
We classify all cubic systems possessing the maximum number of invariant straight lines (real or complex) taking into account their multiplicities. We prove that there are exactly 23 topological different classes of such systems. For every…
A cubic partition consists of partition pairs $(\lambda,\mu)$ such that $\vert\lambda\vert+\vert\mu\vert=n$ where $\mu$ involves only even integers but no restriction is placed on $\lambda$. This paper initiates the notion of generalized…
We generalize the Rubik's cube, together with its group of configurations, to any abstract regular polytope. After discussing general aspects, we study the Rubik's simplex of arbitrary dimension and provide a complete description of the…
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…
In the 7-vertex triangulation of the torus, the 14 triangles can be partitioned as $T_{1} \sqcup T_{2}$, such that each $T_{i}$ represents the lines of a copy of the Fano plane $PG(2, \mathbb{F}_{2})$. We generalize this observation by…
In this paper, we construct six families of infinite simple conformal superalgebra of finite growth based on our earlier work on constructing vertex operator superalgebras from graded assocaitive algebras. Three subfamilies of these…
We present a study of cubic surfaces from the novel perspective of positive geometry. Our positive geometries have dimension two (the surface minus its 27 lines), dimension three (its complement in 3-space), and dimension four (the moduli…
We compute the fundamental groups of the complements of the family of real conic-line arrangements with up to two conics which are tangent to each other at two points, with an arbitrary number of tangent lines to both conics. All the…