Related papers: Frobenius nonclassicality of generalized Fermat cu…
For Fermat curves $\mathcal{F}:aX^n+bY^n=Z^n$ defined over $\mathbb{F}_q$, we establish necessary and sufficient conditions for $\mathcal{F}$ to be $\mathbb{F}_q$-Frobenius nonclassical with respect to the linear system of plane cubics. In…
For each integer $s\geq 1$, we present a family of curves that are $\mathbb{F}_q$-Frobenius nonclassical with respect to the linear system of plane curves of degree s. In the case $s = 2$, we give necessary and sufficient conditions for…
Let $\mathcal{X}$ be a projective irreducible nonsingular algebraic curve defined over a finite field $\mathbb{F}_q$. This paper presents a variation of the St\"orh-Voloch theory and sets new bounds to the number of…
In 1990, Hefez and Voloch proved that the number of $F_q$-rational points on a nonsingular plane $q$-Frobenius nonclassical curve of degree $d$ is $N = d(q-d+2)$. We address these curves in the singular setting. In particular, we prove that…
Let $\mathcal{G}$ be the projective plane curve defined over $\mathbb{F}_q$ given by $$aX^nY^n-X^nZ^n-Y^nZ^n+bZ^{2n}=0,$$ where $ab\notin\{0,1\}$, and for each $s\in\{2,\ldots,n-1\}$, let $\mathcal{D}_s^{P_1,P_2}$ be the base-point-free…
In this paper, we characterize all curves over $\mathbb{F}_q$ arising from a plane section $$ \mathcal{P} : X_3-e_0X_0-e_1X_1-e_2X_2 = 0 $$ of the Fermat surface $$ \mathcal{S} : X_0^d + X_1^d + X_2^d +X_3^d = 0, $$ where $q = p^{h} = 2d+1$…
In this paper we give an upper bound on the number of rational points on an irreducible curve $C$ of degree $\delta$ defined over a finite field $\mathbb{F}_q$ lying on a Frobenius classical surface $S$ embedded in $\mathbb{P}^3$. This…
We show that the set of F_q-rational points of either certain Fermat curves or certain F_q-Frobenius non-classical plane curves is a complete (k,d)-arc in P^2(F_q), where k and d are respectively the number of F_q-rational points and the…
The number of rational points of a plane non-singular algebraic curve X defined over a finite field is computed, provided that the generic point of X is not an inflexion and that X is Frobenius non-classical with respect to conics.
We use a recent characterization of minimal value set polynomials and $q$- Frobenius nonclassical curves to construct curves that generalize the Hermitian curve. The genus $g$ and the number $N$ of $\mathbb{F}_q$-rational points of the…
We determine the distribution of Galois points for plane curves over a finite field of $q$ elements, which are Frobenius nonclassical for different powers of $q$. This family is an important class of plane curves with many remarkable…
This is an expository paper concerning topics on rational points of curves defined over finite fields based on a paper by St"ohr and Voloch.
A smooth hypersurface over a finite field $\mathbb{F}_q$ is called Frobenius nonclassical if the image of every geometric point under the $q$-th Frobenius endomorphism remains in the unique hyperplane tangent to the point. In this paper, we…
Let $q\geq 5$ be a prime power. In this note, we prove that if a plane curve $\mathcal{X}$ of degree $q - 1$ defined over $\mathbb{F}_q$ without $\mathbb{F}_q$-linear components attains the Sziklai upper bound $(d-1)q+1 = (q - 1)^2$ for the…
We study rational curves on algebraic varieties, especially on normal affine varieties endowed with a $\C^*$-action. For varieties with an isolated singularity, we show that the presence of sufficiently many rational curves outside the…
We show how a type of multi-Frobenius nonclassicality of a curve defined over a finite field $\mathbb{F}_q$ of characteristic $p$ reflects on the geometry of its strict dual curve. In particular, in such cases we may describe all the…
The goal of this article is to initiate the study of estimates of the non-classical Schottky structure in the discrete subgroups of the projective special linear group over the real numbers degree $2$. In fact, in this paper, we have…
Let $\mathcal{F}$ be a plane singular curve defined over a finite field $\mathbb{F}_q$. The linear system of plane curves of a given degree passing through the singularities of $\cF$ provides potentially good bounds for the number of points…
We study the number of rational points of smooth projective curves over finite fields in some relative situations in the spirit of a previous paper from an euclidean point of vue. We prove some kinds of relative Weil bounds, derived from…
In a recent paper, the author and St\"ohr established a bound on the number of iterated Frobenius pullbacks needed to transform a non-smooth non-decomposed point on a regular geometrically integral curve into a rational point. In this note…