Related papers: Remarks on plane maximal curves

We sharpen and generalize the dimension growth bounds for the number of points of bounded height lying on an irreducible algebraic variety of degree $d$, over any global field. In particular, we focus on the affine hypersurface situation by…

Fix positive integers d;m such that $(m^2+4m+6)/6 \leq d < (m^2+4m+6)/3$ (the so-called Range A for space curves). Let G(d;m) be the maximal genus of a smooth and connected curve, of degree d, $C \subset P^3$ such that $h^0(I_C(m-1)) = 0$.…

In this work we present explicit examples of maximal and minimal curves over finite fields in odd characteristic. The curves are of Artin-Schreier type and the construction is closely related to quadratic forms from $\mathbb{F}_{q^n}$ to…

Using Weil descent, we give bounds for the number of rational points on two families of curves over finite fields with a large abelian group of automorphisms: Artin-Schreier curves of the form $y^q-y=f(x)$ with $f\in\Fqr[x]$, on which the…

We provide a tool how one can view a polynomial on the affine plane of bidegree $(a,b)$ - by which we mean that its Newton polygon lies in the triangle spanned by $(a,0)$, $(0,b)$ and the origin - as a curve in a Hirzebruch surface having…

In this paper we deal with the problem of classifying the genera of quotient curves $\mathcal{H}_q/G$, where $\mathcal{H}_q$ is the $\mathbb{F}_{q^2}$-maximal Hermitian curve and $G$ is an automorphism group of $\mathcal{H}_q$. The groups…

In this work we construct a new class of maximal partial spreads in $PG(4,q)$, that we call $q$-added maximal partial spreads. We obtain them by depriving a spread of a hyperplane of some lines and adding $q+1$ lines not of the hyperplane…

A curve has the increasing chord property if for any points $a,b,c,d$ in this order on the curve, the distance of $a,d$ is not smaller than that of $b,c$. Answering a conjecture of Larman and McMullen, Rote proved in 1994 that the arclength…

The author determines the structure of automorphism groups of smooth plane curves of degree at least four. Furthermore, he gives some upper bounds for the order of automorphism groups of smooth plane curves and classifies the cases with…

An abelian variety $A/K$ is heavenly at $\ell$ if the extension $K(A[\ell^\infty])/K(\mu_{\ell^{\infty}}\!)$ is both pro-$\ell$ and unramified away from $\ell$. It is known that for a fixed quadratic field $K$, the number of $K$-isomorphism…

The paper is a contribution to the conjecture of Kobayashi that the complement of a generic curve in the projective plane is hyperbolic, provided the degree is at least five. Previously the authors treated the cases of two quadrics and a…

We give an improved asymptotic upper bound on the number of diagonal Fermat curves $Ax^{\ell}+By^{\ell}=z^{\ell}$ over $\mathbb{F}_{q}$ with no $\mathbb{F}_{q}$-rational points, where $\ell$ is a prime number dividing $q-1$.

Given a prime power q, for every pair of positive integers m and n with m dividing the GCD of n and q-1, we construct a modular curve over F_q that parametrizes elliptic curves over F_q along with F_q-defined points P and Q of order m and…

We introduce a notion of good cohomology for multiple lines in $\mathbb{P}^3$ and we classify multiple lines with good cohomology up to multiplicity 4. In particular, we show that the family of space curves of degree d, not lying on a…

Motivated by previous computations in Garcia, Stichtenoth and Xing (2000) paper ,we discuss the spectrum $\mathbf{M}(q^2)$ for the genera of maximal curves over finite fields of order $q^2$ with $7\leq q\leq 16$. In particular, by using a…

In this note we study rational curves on degree $p^r+1$ Fermat hypersurface in $\PP^{p^r+1}_k$, where $k$ is an algebraically closed field of characteristic $p$. The key point is that the presence of Frobenius morphism makes the behavior of…

Let $X$ be a minuscule homogeneous space, an odd quadric, or an adjoint homogenous space of type different from $A$ and $G_2$. Le $C$ be an elliptic curve. In this paper, we prove that for $d$ large enough, the scheme of degree $d$…

Let $C$ be a curve of genus 2 and $\psi_1:C \lar E_1$ a map of degree $n$, from $C$ to an elliptic curve $E_1$, both curves defined over $\bC$. This map induces a degree $n$ map $\phi_1:\bP^1 \lar \bP^1$ which we call a Frey-Kani covering.…

A curve over a field k is pointless if it has no k-rational points. We show that there exist pointless genus-3 hyperelliptic curves over a finite field F_q if and only if q < 26, that there exist pointless smooth plane quartics over F_q if…

The Dickson-Guralnick-Zieve curve, briefly DGZ curve, defined over the finite field $\mathbb{F}_q$ arises naturally from the classical Dickson invariant of the projective linear group $PGL(3,\mathbb{F}_q)$. The DGZ curve is an (absolutely…