Related papers: Counting conics in complete intersections

In this paper, we compute the number of self-intersections of a plane projection of a generic complete intersection curve defined by polynomials with the given support. Moreover, we discuss the tropical counterpart of this problem.

We compute the expectation of the number of linear spaces on a random complete intersection in $p$-adic projective space. Here "random" means that the coefficients of the polynomials defining the complete intersections are sampled uniformly…

We classify the singular loci of real surfaces in three-space that contain two circles through each point. We characterize how a circle in such a surface meets this loci as it moves in its pencil and as such provide insight into the…

This paper addresses the numerical computation of critical angles between two convex cones in finite-dimensional Euclidean spaces. We present a novel approach to computing these critical angles by reducing the problem to finding stationary…

Cographs--defined most simply as complete graphs with colored lines--both dualize and generalize ordinary graphs, and promise a comparably wide range of applications. This article introduces them by examples, catalogues, and elementary…

We give geometric ways to determine the cross ratio of conics in a pencil using elementary projective geometry.

We determine the conditions resulting from equating the area sums of alternative sectors in a circle generated by four, two, and three straight lines, respectively, that connect opposite points on its circumference while passing through a…

Let $S$ be a finite set of points in the plane and let $\mathcal{T}(S)$ be the set of intersection points between pairs of lines passing through any two points in $S$. We characterize all configurations of points $S$ such that iteration of…

We derive various inequalities involving the intersection number of the curves contained in geodesics and tight geodesics in the curve graph. While there already exist such inequalities on tight geodesics, our method applies in the setting…

For a given triangle $\triangle ABC$, we define two sequences of points on line $BC$ and provide their generalizations to real functions such that centers of circumscribed circles around $A$ and adjacent points in subsequences generate a…

We give a formula for counting the triangles in a picture consisting of the three sides of a triangle and some cevians. This lets us prove statements that are claimed without proof in the Online Encyclopedia of Integer Sequences and some…

We solve a technical problem related to adeles on an algebraic surface. Given a finite set of natural numbers up to two, one associates an adelic group. We show that this operation commutes with taking intersections if the surface is…

We get sharp degree bound for generic smoothness and connectedness of the space of conics in low degree complete intersections which generalizes the old work about Fano scheme of lines on Hypersurfaces.

This paper first gives a brief overview over some interesting descriptions of conic sections, showing formulations in the three geometric algebras of Euclidean spaces, projective spaces, and the conformal model of Euclidean space. Second…

Closed form expressions are given for computing the parameters and vectors that identify and define the $n-1$ dimensional conic section that results from the intersection of a hyperplane with an $n$-dimensional conic section: cone,…

Let $S$ be a set of $n$ points in general position in the plane. Join every pair of points in $S$ with a straight line segment. Let $\overline{cr}(S)$ be number of pairs of these edges that intersect in their interior. Suppose that this…

We determine the structure of the finite groups with the property that every cyclic subgroup is the intersection of maximal subgroups, comparing this property with the one where all proper subgroups are intersections of maximal subgroups.

We construct special conics configurations from some points configurations which are the singularities of the dual of a quartic curve.

We generalize the joints problem to sets of varieties and prove almost sharp bound on the number of joints. As a special case, given a set of $N$ $2$-planes in $\mathbb{R}^6$, the number of points at which three $2$-planes intersect and…

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…