相关论文: Real algebraic curves with large finite number of …

We bound the number of fixed points of an automorphism of a real curve in terms of the genus and the number of connected components of the real part of the curve. Using this bound, we derive some consequences concerning the maximum order of…

We give an upper bound for the degree of rational curves in a family that covers a given birational ruled surface in projective space. The upper bound is stated in terms of the degree, sectional genus and arithmetic genus of the surface. We…

A lower bound on the minimum degree of the plane algebraic curves containing every point in a large point-set $K$ of the Desarguesian plane $PG(2,q)$ is obtained. The case where $K$ is a maximal $(k,n)$-arc is considered to greater extent.

Using an Euclidean approach, we prove a new upper bound for the number of closed points of degree 2 on a smooth absolutely irreducible projective algebraic curve defined over the finite field $\mathbb F\_q$.This bound enables us to provide…

Below we consider the evolutes of plane real-algebraic curves and discuss some of their complex and real-algebraic properties. In particular, for a given degree $d\ge 2$, we provide lower bounds for the following four numerical invariants:…

The problem of constructing curves with many points over finite fields has received considerable attention in the recent years. Using the class field theory approach, we construct new examples of curves ameliorating some of the known…

It is known for a long time that a nonsingular real algebraic curve of degree 2k in the projective plane cannot have more than 7/2*k^2-9/4*k+3/2$ even ovals. We show here that this upper bound is asymptotically sharp, that is to say we…

We introduce and begin the topological study of real rational plane curves, all of whose inflection points are real. The existence of such curves is a corollary of results in the real Schubert calculus, and their study has consequences for…

We give a construction of singular curves with many rational points over finite fields. This construction enables us to prove some results on the maximum number of rational points on an absolutely irreducible projective algebraic curve…

We study a particular plane curve over a finite field whose normalization is of genus 0. The number of rational points of this curve achieves the Aubry-Perret bound for rational curves. The configuration of its rational points and a…

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…

Given a real algebraic curve, embedded in projective space, we study the computational problem of deciding whether there exists a hyperplane meeting the curve in real points only. More generally, given any divisor on such a curve, we may…

The defect of a curve over a finite field is the difference between the number of rational points on the curve and the Weil-Serre upper bound for the number of points on the curve. We present algorithms for constructing curves of genus 5,…

We are interested in shapes of real algebraic curves in the plane and regions surrounded by them: they are named refined algebraic domains by the author. As characteristic finite sets, we consider points contained in two curves and the sets…

Many classical results in algebraic geometry arise from investigating some extremal behaviors that appear among projective varieties not lying on any hypersurface of fixed degree. We study two numerical invariants attached to such…

The paper gives topological as well as rigid isotopy classification of smooth irreducible algebraic curves in the real projective 3-space for the case when the degree of the curve is at most six and its genus is at most one.

Given a minimal surface equipped with a generically finite map to an Abelian variety, we give an optimal bound on the canonical degree of a rational or an elliptic curve. As a corollary, we obtain the finiteness of rational and elliptic…

We use class field theory to search for curves with many rational points over small finite fields. By going through abelian covers of curves of small genus we find a number of new curves. In particular, we settle the question of how many…

We describe the topology of singular real algebraic curves in a smooth surface. We enumerate and bound in terms of the degree the number of topological types of singular algebraic curves in the real projective plane.

In this survey, we discuss the problem of the maximum number of points of curves of genus 1,2 and 3 over finite fields