Related papers: Intersection of curves in projective 4 space

We give necessary and sufficient criteria for a smooth Enriques surface S in P^r to be scheme-theoretically an intersection of quadrics. Moreover we prove in many cases that, when S contains plane cubic curves, the intersection of the…

The {\it crossing number} of a graph $G$ is the minimum number of pairwise intersections of edges in a drawing of $G$. Motivated by the recent work [Faria, L., Figueiredo, C.M.H. de, Sykora, O., Vrt'o, I.: An improved upper bound on the…

Let F be a smooth surface in a smooth projective threefold T, and let X=2F be the first infinitesimal neighborhood of X in T. A locally Cohen-Macaulay curve C in X gives rise to two effective divisors on F, namely the curve part P of the…

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 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 first show that the union of a projective curve with one of its extremal secant lines satisfies the linear general position principle for hyperplane sections. We use this to give an improved approximation of the Betti numbers of curves…

Let $P$ be a set of $n$ points in the plane, and let $\mathcal C$ be a collection of $n$ simple $k$-intersecting curves, meaning that every two distinct curves of $\mathcal C$ meet in at most $k$ points. A classical theorem of Pach and…

We consider an incidence problem in $\mathbb{R}^4$ which asks, for a set of $L$ lines and a set of $S$ planes in general position, what the maximum number of line-plane incidences is. A line-plane incidence is defined as a point where a…

Paul Erdos asked if, among sufficiently many points in general position, there are always $k$ points such that all the circles through $3$ of these $k$ points have different radii. He later proved that this is indeed the case. However, he…

A point P on a smooth hypersurface X of degree d in an N-dimensional projective space is called a star point if and only if the intersection of X with the embedded tangent space T_P(X) is a cone with vertex P. This notion is a…

Edoukou, Ling and Xing in 2010, conjectured that in \mathbb{P}^n(\mathbb{F}_{q^2}), n \geq 3, the maximum number of common points of a non-degenerate Hermitian variety \mathcal{U}_n and a hypersurface of degree d is achieved only when the…

We determine necessary conditions for ample divisors in arbitrary genus as well as for very ample divisors in genus 2 and 3. We also compute the intersection numbers $\lambda^9$ and $\lambda_{g-1}^3$ in genus 4. The latter number is…

Consider the smooth projective models C of curves y^2=f(x) with f(x) in Z[x] monic and separable of degree 2g+1. We prove that for g >= 3, a positive fraction of these have only one rational point, the point at infinity. We prove a lower…

In this paper we consider smooth affine elliptic plane curves having one place at infinity. We identify them with elliptic projective plane curves having only one cusp as their singular points and meeting with the line at infinity only at…

We establish the sharp estimate <<_d N^{2/d} for the number of rational points of height at most N on an irreducible projective curve of degree d. We deduce this from a result for general hypersurfaces that is sensitive to the coefficients…

Combining $2$-descent techniques with Riemann-Roch and B\'ezout's theorems, we give an upper bound on the number of rational points of bounded height on elliptic and hyperelliptic curves over function fields of characteristic $\neq 2$. We…

This paper is devoted to understanding curves $X$ over a number field $k$ that possess infinitely many solutions in extensions of $k$ of degree at most $d$; such solutions are the titular low degree points. For $d=2,3$ it is known (by the…

We show a weak form of the function field version of Oesterle's abc conjecture. It asserts that, if $B$ is a complex projective connected curve, the number of intersection points, counted without multiplicities, of a fixed divisor $D$ of…

In this paper, we present a novel approach to the problem of merging of B\'ezier curves with respect to the $L_2$-norm. We give illustrative examples to show that the solution of the conventional merging problem may not be suitable for…

Let $\cC$ be a set of curves in the plane such that no three curves in $\cC$ intersect at a single point and every pair of curves in $\cC$ intersect at exactly one point which is either a crossing or a touching point. J\'anos Pach…