Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A characterization of the family of secant lines to a hyperbolic quadric in PG(3,q), q odd

Puspendu Pradhan, Bikramaditya Sahu

Abstract

We give a combinatorial characterization of the family of lines of P G(3, q) which meet a hyperbolic quadric in two points (the so called secant lines) using their intersection properties with the points and planes of PG(3,q).

A characterization of the family of secant lines to a hyperbolic quadric in PG(3,q), q odd

Abstract

We give a combinatorial characterization of the family of lines of P G(3, q) which meet a hyperbolic quadric in two points (the so called secant lines) using their intersection properties with the points and planes of PG(3,q).

Paper Structure

This paper contains 6 sections, 28 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Combinatorial results
  3. Tangent planes
  4. Secant planes
  5. Black points
  6. Proof of Theorem \ref{['q-odd']}

Key Result

Theorem 1.1

Let $\mathcal{S}$ be a family of lines of $PG(3,q)$, $q$ odd, for which the following properties are satisfied: Then either $\mathcal{S}$ is the set of all secant lines with respect to a hyperbolic quadric in $PG(3,q)$, or the set of points each of which is contained in $q^2$ lines of $\mathcal{S}$ form a line $l$ of $PG(3,q)$ and $\mathcal{S}$ is a hypothetical family of $\dfrac{q^4+q^3+2q^2}{2}

Theorems & Definitions (50)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Corollary 2.3
  • Corollary 2.4
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • ...and 40 more