Related papers: Sextactic and type-9 points on the Fermat cubic an…

Which surfaces can be realized with two-dimensional faces of the five-dimensional cube (the penteract)? How can we visualize them? In recent work, Aveni, Govc, and Roldan, show that there exist 2690 connected closed cubical surfaces up to…

Consider a pair of ordinary elliptic curves $E$ and $E'$ defined over the same finite field $\mathbb{F}_q$. Suppose they have the same number of $\mathbb{F}_q$-rational points, i.e. $|E(\mathbb{F}_q)|=|E'(\mathbb{F}_q)|$. In this paper we…

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…

We show that there are 105 isometric classes of fractal cubes of order 3 with 7 copies possessing one-point intersection property which are subdivided to 5 bi-Lipschitz classes of dendrites and 7 bi-Lipschitz classes of non-dendrites

The evidence of the existence of tetra-quark meson and penta-quark fermion systems has revived interest in the hunt of other multi-quark systems. We explore the possibility of six-quark cluster configurations with the two nucleon…

We show that any Fermat hypercubic is apolar to a trigonal curve, and vice versa. We show also that the Waring number of the polar hypercubic associated to a tetragonal curve of genus $g$ is at most $\lceil 3/2g - 7/2\rceil$, and for a…

In this paper, we explore a new type of global symmetries$-$the fermionic higher-form symmetries. They are generated by topological operators with fermionic parameter, which act on fermionic extended objects. We present a set of field…

We give explicit computations for the variance of the number of points along one-parameter families of cubic curves. We highlight evaluations of variances that involve Jacobsthal sums.

This is a survey on the Fano schemes of linear spaces, conics, rational curves, and curves of higher genera in smooth projective hypersurfaces, complete intersections, Fano threefolds, etc.

Constructions and exploration of plane algebraic curves has received a new push with the development of automated methods, whose algorithms are continuously improved and implemented in various software packages. We use them to explore the…

Let ${\mathcal K}$ denote a smooth conic in the complex projective plane. Pascal's theorem says that, given six points $A,B,C,D,E,F$ on ${\mathcal K}$, the three intersection points $AE \cap BF, AD \cap CF, BD \cap CE$ are collinear. This…

According to a computer search conducted by the author and described in [7], in $Q^+(6, 4)$ there are two types of hyperovals, having 72 and 96 points, respectively. Here we give geometric descriptions for these examples.

Let $X$ be the space of isometry classes of ordered sextuples of points in the hyperbolic plane such that the product of the six corresponding rotations of angle $\pi$ is the identity. This space $X$ is closely related to the…

In this paper we introduce point-ellipse configurations and point-conic configurations. We study some of their basic properties and describe two interesting families of balanced point-ellipse, respectively point-conic $6$-configurations.…

We study a construction, which produces surfaces $Y \subset P_3$ with cusps. For example we obtain surfaces of degree six with 18, 24 or 27 three-divisible cusps. For sextic surfaces in a particular family of up to 30 cusps the codes of…

Recently, W. Barth and S. Rams discussed sextics with up to 30 $A_2$-singularities (also called cusps) and their connection to coding theory [math.AG/0403018]. In the present paper, we find a sextic with 35 cusps within a four-parameter…

We apply the specialization technique based on the decomposition of the diagonal to find an explicit example over $\mathbb{Q}$ of a quadric and cubic hypersurface in $\mathbb{P}^6$ such that their intersection is a smooth stably irrational…

We study the geometry of spaces of planes on smooth complete intersections of three quadrics, with a view toward rationality questions.

The geometry of four-qubit entanglement is investigated. We replace some of the polynomial invariants for four-qubits introduced recently by new ones of direct geometrical meaning. It is shown that these invariants describe four points, six…

We describe how the quadratic Chabauty method may be applied to explicitly determine the set of rational points on modular curves of genus $g>1$ whose Jacobians have Mordell--Weil rank $g$. This extends our previous work on the split Cartan…