Related papers: Intersection of curves in projective 4 space

Given n general points p_1, p_2,..., p_n \in P^r, it is natural to ask whether there is a curve of given degree d and genus g passing through them; by counting dimensions a natural conjecture is that such a curve exists if and only if \[n…

By a curve in R^d we mean a continuous map gamma:I -> R^d, where I is a closed interval. We call a curve gamma in R^d at most k crossing if it intersects every hyperplane at most k times (counted with multiplicity). The at most d crossing…

We consider two mixed curve $C,C'\subset {\Bbb C}^2$ which are defined by mixed functions of two variables $\bf z=(z_1,z_2)$. We have shown in \cite{MC}, that they have canonical orientations. If $C$ and $C'$ are smooth and intersect…

In this note, we investigate the maximal number of intersection points of a line with the contour of hypersurface amoebas in $\mathbb{R}^n$. We define the latter number to be the $\mathbb{R}$-degree of the contour. We also investigate the…

For prime degree hypersurfaces of dimension at least 3, Mori asked if every smooth proper limit is still a hypersurface. Interestingly in dimensions 1 and 2, this is not the case. For example, Griffin constructed explicit families of…

We obtain a coarse relationship between geometric intersection numbers of curves and the sum of their subsurface projection distances with explicit quasi-constants. By using this relationship, we give applications in the studies of the…

Fix integers $r\geq 4$ and $i\geq 2$ (for $r=4$ assume $i\geq 3$). Assuming that the rational number $s$ defined by the equation $\binom{i+1}{2}s+(i+1)=\binom{r+i}{i}$ is an integer, we prove an upper bound for the genus of a reduced and…

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 main goal of this work is to prove that every entire curve in a smooth hypersurface of degree greater than or equal to 97 in the complex projective space of dimension 4 must satisfy an algebraic differential equation of order 3. A…

We bound the genus of a projective curve lying on a complete intersection surface in terms of its degree and the degrees of the defining equations of the surface on which it lies.

We study the intersection theory of punctured pseudoholomorphic curves in $4$-dimensional symplectic cobordisms. We first study the local intersection properties of such curves at the punctures. We then use this to develop topological…

This research monograph focuses on the arithmetic, over number fields, of surfaces fibred into curves of genus 1 over the projective line, and of intersections of two quadrics in projective space. The first half takes up and develops…

We prove an incidence theorem for points and planes in the projective space $\mathbb P^3$ over any field $\mathbb F$, whose characteristic $p\neq 2.$ An incidence is viewed as an intersection along a line of a pair of two-planes from two…

We introduce a new formalism and a number of new results in the context of geometric computational vision. The classical scope of the research in geometric computer vision is essentially limited to static configurations of points and lines…

In this paper we examine different problems regarding complete intersection varieties of high degree in a complex projective space. First we show how one can deduce hyperbolicity for generic complete intersection of high multidegree and…

We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of the number of such orbits of curves with a bounded number of self-intersections, as the complexity of the…

We study intersections of exceptional curves on del Pezzo surfaces of degree 1, motivated by questions in arithmetic geometry. Outside characteristics 2 and 3, at most 10 exceptional curves can intersect in a point. We classify the…

Recent research on computing the diameter of geometric intersection graphs has made significant strides, primarily focusing on the 2D case where truly subquadratic-time algorithms were given for simple objects such as unit-disks and…

The geometric intersection number of a curve on a surface is the minimal number of self-intersections of any homotopic curve, i.e. of any curve obtained by continuous deformation. Given a curve $c$ represented by a closed walk of length at…

On a weighted projective surface $\mathbb{P}(a,b,c)$ with $\min(a,b,c)\leq 4$, we compute lower bounds for the {\em effective threshold} of an ample divisor, in other words, the highest multiplicity a section of the divisor can have at a…