Related papers: Sextactic points on the Fermat cubic curve and arr…
In this note we study line and conic arrangements associated to sextactic and type 9 points on the Fermat cubic $F$ and we provide explicit coordinates for each of the 72 type 9 points on $F$.
In this paper we present new results about arrangements of lines and osculating curves associated to the Fermat curves in the projective plane. We first consider the sextactic points on the Fermat curves and show that they are distributed…
In this note, we examine the arrangements of lines and configurations of points that emerge from Fermat (von Dyck) and Komiya-Kuribayashi quartics. These quartics are characterized by having the maximum number of lines of maximal tangency,…
We give optimal lower bounds for the number of sextactic points on a simple closed curve in the real projective plane. Sextactic points are after inflection points the simplest projectively invariant singularities on such curves. Our method…
In this paper we compute the distributions of various markings on smooth cubic surfaces defined over the finite field $\mathbb{F}_q$, for example the distribution of pairs of points, `tritangents' or `double sixes'. We also compute the…
In this paper, we study the algebraic points of degree $4$ over $\mathbb{Q}$ on the Fermat curve $F_5/\mathbb{Q}$ of equation $x^5+y^5+z^5=0$. A geometrical description of these points has been given in 1997 by Klassen and Tzermias. Using…
We present explicit equations for the space of conics in the Fermat quintic threefold $X$, working within the space of plane sections of $X$ with two singular marked points. This space of two-pointed singular plane sections has a birational…
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…
This note is developing, and we will include many additions in the near future. Our purpose here is to highlight that there is plenty of space for a topological development of the Fermat Real Line.
Let $C$ be a curve defined over a number field $K$. A point $P\in C(\overline{\mathbb{Q}})$ is called $K$-quadratic if $[K(P):K]=2$. Let $K$ be a number field such that the rank of the elliptic curves $E_1:\,y^2= x^3 + 4x$ and $E_2:\,y^2=…
In Part I, the present authors introduced the notion of a quasi-Galois point, for investigating the automorphism groups of plane curves. In this second part, the number of quasi-Galois points for smooth plane curves is described. In…
The purpose of this expository paper is to explain how the Fargues-Fontaine curve and its decomposition into a punctured curve and the formal neighborhood of the puncture naturally arise from various forms of topological cyclic homology and…
This paper presents a new characterisation of the Fermat curve, according to the arrangement of Galois points.
In the present article we construct new families of free and nearly free curves starting from a plane cubic curve $C$ and adding some of its hyperosculating conics. We present results that involve nodal cubic curves and the Fermat cubic. In…
A quadric in $\R P^3$ cuts a curve of degree 6 on a cubic surface in $\R P^3$. The papers classifies the nonsingular curves cut in this way on non-singular cubic surfaces up to homeomorphism. Two issues new in the study related to the first…
This is an overview of higher structural constructions in physics. The main motivations of our current attempt are as follows: (i) to provide a brief introduction to derived algebraic geometry, (ii) to understand how derived objects…
The distribution of degree $d$ points on curves is well understood, especially for low degrees. We refine this study to include information on the Galois group in the simplest interesting case: $d = 3$. For curves of genus at least 5, we…
These are the substantially expanded notes of the lectures of JK at the summer school "Higher-Dimensional Geometry over Finite Fields" in G\"ottingen, June 2007. The first part gives an overview of the methods. The main new result is the…
Let $\mathbb{F}_q$ denote the finite field with $q$ elements. In this work, we use characters to give the number of rational points on suitable curves of low degree over $\mathbb{F}_q$ in terms of the number of rational points on elliptic…
We show that even dimensional Fermat cubic hypersurfaces are rational over any field of characteristic different from three by producing explicit rational parametrizations given by polynomials of low degree. As a byproduct of our…