Related papers: Twenty-Seven Lines on a Cubic Surface and Heteroti…
We analyze the nonabelian surface holonomy on a bipartite hypercubic lattice following a proposal in arXiv:1002.4636 [hep-th]. The bipartite structure of the lattice enables us to introduce spike string configurations. These spikes play a…
We present a complete computational classification of the combinatorial types of hyperplane sections, or slices, of the regular cube up to dimension six. For each dimension, we determine the exact number of distinct combinatorial types.…
The present paper deals with lines contained in a smooth complex cubic threefold. It is well-known that the set of lines of the second type on a cubic threefold is a curve on its Fano surface. Here we give a description of the singularities…
We prove the sharp upper bound of at most $52$ lines on a complex K3-surface of degree four with a non-empty singular locus. We also classify the configurations of more than $48$ lines on smooth complex quartics.
A subspace arrangement in a vector space is a finite collection of vector subspaces. Similarly, a configuration of linear spaces in a projective space is a finite collection of linear subspaces. In this paper we study the degree 2 part of…
A (general) polygonal line tiling is a graph formed by a string of cycles, each intersecting the previous at an edge, no three intersecting. In 2022, Matsushita proved the matching complex of a certain type of polygonal line tiling with…
The attempted classification of regular algebras of global dimension four, so-called quantum $\mathbb P^3$s, has been a driving force for modern research in noncommutative algebra. Inspired by the work of Artin, Tate, and Van den Bergh,…
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 study linear systems of surfaces in $\mathbb{P}^3$ singular along general lines. Our purpose is to identify and classify special systems of such surfaces, i.e., those nonempty systems where the conditions imposed by the multiple lines…
A quantum ${\mathbb P}^3$ is a noncommutative analogue of a polynomial ring on four variables, and, herein, it is taken to be a regular algebra of global dimension four. It is well known that if a generic quadratic quantum ${\mathbb P}^3$…
We investigate the conformal geometry of spherically symmetric spacetimes in general without specifying the form of the matter distribution. The general conformal Killing symmetry is obtained subject to a number of integrability conditions.…
Every smooth cubic plane curve has 9 flex points and 27 sextatic points. We study the following question asked by Farb: Is it true that the known algebraic structures give all the possible ways to continuously choose $n$ distinct points on…
Using the topological membrane approach to string theory, we suggest a geometric origin for the heterotic string. We show how different membrane boundary conditions lead to different string theories. We discuss the construction of closed…
We estimate from below the number of lines meeting each of given 4 disjoint smooth closed curves in a given cyclic order in the real projective 3-space and in a given linear order in the Euclidean 3-space. Similarly, we estimate the number…
This note is motivated by the Question 16 of http://cubics.wikidot.com: Which configurations of 15 points in the projective 3-space arise as eigenpoints of a cubic surface? We prove that a general eigenscheme in the projective n-space is…
According to a conjecture of H. Clemens, the dimension of the space of rational curves on a general projective hypersurface should equal the number predicted by a na\"ive dimension count. In the case of a general hypersurface of degree 7 in…
Let $X$ be a smooth projective variety of dimension $n\geq 2$. It is shown that a finite configuration of points on $X$ subject to certain geometric conditions possesses rich inner structure. On the mathematical level this inner structure…
We propose an organizing principle for string theory moduli spaces in six dimensions with $\mathcal{N} = (1,1)$, based on a rank reduction map, into which all known constructions fit. In the case of cyclic orbifolds, which are the main…
We study the tropical lines contained in smooth tropical surfaces in R^3. On smooth tropical quadric surfaces we find two one-dimensional families of tropical lines, like in classical algebraic geometry. Unlike the classical case, however,…
Let X be a smooth cubic hypersurface. We prove that a general cubic surface is isomorphic to a hyperplane section of X .