Related papers: Conics meeting eight lines over perfect fields
This is a survey on the Fano schemes of linear spaces, conics, rational curves, and curves of higher genera in smooth projective hypersurfaces, complete intersections, Fano threefolds, etc.
After having investigated the real conic sections and their isoptic curves in the hyperbolic plane $\bH^2$ we consider the problem of the isoptic curves of generalized conic sections in the extended hyperbolic plane. This topic is widely…
In this paper, we study the restrictions on the number $m$ of conic-line curves in special pencils. The most general result we obtain is the relation between upper bounds on $m$ and the number $p$ of concurrent lines in these pencils. We…
Three lines are concurrent if they intersect at a single point. In this paper I prove that if $F$ is a bounded family of compact connected sets in the plane, such that every three sets in $F$ can be pierced by a single line, then there…
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.
We investigate the lines tangent to four triangles in R^3. By a construction, there can be as many as 62 tangents. We show that there are at most 162 connected components of tangents, and at most 156 if the triangles are disjoint. In…
For an even set of points in the plane, choose a max-sum matching, that is, a perfect matching maximizing the sum of Euclidean distances of its edges. For each edge of the max-sum matching, consider the ellipse with foci at the edge's…
We give geometric ways to determine the cross ratio of conics in a pencil using elementary projective geometry.
Using equivariant geometry, we find a universal formula that computes the number of times a general cubic surface arises in a family. As applications, we show that the PGL(4) orbit closure of a generic cubic surface has degree 96120, and…
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…
Finding the intersection of two conics is a commonly occurring problem. For example, it occurs when identifying patterns of craters on the lunar surface, detecting the orientation of a face from a single image, or estimating the attitude of…
For each $1\leq n\leq6$ we present formulas for the number of $n-$nodal curves in an $n-$dimensional linear system on a smooth, projective surface. This yields in particular the numbers of rational curves in the system of hyperplane…
Projective spaces for finite-dimensional vector spaces over general fields are considered. The geometry of these spaces and the theory of line bundles over these spaces is presented. Particularly, the space of global regular sections of…
For polyhedral convex cones in ${\mathbb R}^d$, we give a proof for the conic kinematic formula for conic curvature measures, which avoids the use of characterization theorems. For the random cones defined as typical cones of an isotropic…
Let $E:y^2=x^3+ax+b$ be an elliptic curve defined over $\mathbb{Q}$. We compute certain twists of the classical modular curves $X(8)$. Searching for rational points on these twists enables us to find non-trivial pairs of $8$-congruent…
We discuss the theorem on the existence of six points on a convex closed plane curve in which the curve has a contact of order six with the osculating conic. (This is the ``projective version'' of the well known four vertices theorem for a…
Linear Geometry studies geometric properties which can be expressed via the notion of a line. All information about lines is encoded in a ternary relation called a line relation. A set endowed with a line relation is called a liner. So,…
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 purpose of this paper is to show that for a complete intersection curve $C$ in projective space (other than a few stated exceptions), any morphism $f: C \to \mathbb{P}^r$ satisfying $\text{deg}\, f^*\mathcal{O}_{\mathbb{P}^r}(1)…
We find a new Zariski pair with non-isomorphic fundamental groups that consists of degree $ 8 $ conic-line arrangements. Each arrangement has three conics and two lines. We use the Zariski-van Kampen Theorem and some known Coxeter groups to…