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On $m$-ovoids of $Q^+(7,q)$ with $q$ odd

Sam Adriaensen, Jan De Beule, Giovanni Giuseppe Grimaldi, Jonathan Mannaert

Abstract

In this paper, we provide a construction of $(q+1)$-ovoids of the hyperbolic quadric $Q^+(7,q)$, $q$ an odd prime power, by glueing $(q+1)/2$-ovoids of the elliptic quadric $Q^-(5,q)$. This is possible by controlling some intersection properties of (putative) $m$-ovoids of elliptic quadrics. It yields eventually $(q+1)$-ovoids of $Q^+(7,q)$ not coming from a $1$-system. Secondly, we also construct $m$-ovoids for $m \in \{ 2,4,6,8,10\}$ in $Q^+(7,3)$. Therefore we first investigate how to construct spreads of $\pg(3,q)$ that have as many secants to an elliptic quadric as possible.

On $m$-ovoids of $Q^+(7,q)$ with $q$ odd

Abstract

In this paper, we provide a construction of -ovoids of the hyperbolic quadric , an odd prime power, by glueing -ovoids of the elliptic quadric . This is possible by controlling some intersection properties of (putative) -ovoids of elliptic quadrics. It yields eventually -ovoids of not coming from a -system. Secondly, we also construct -ovoids for in . Therefore we first investigate how to construct spreads of that have as many secants to an elliptic quadric as possible.
Paper Structure (8 sections, 21 theorems, 23 equations, 1 table)

This paper contains 8 sections, 21 theorems, 23 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finite classical polar spaces
  4. m-ovoids and l-systems of finite classical polar spaces
  5. Spreads of PG(3, q) and ovoids of Q+(5, q)
  6. Intersection patterns for m-ovoids
  7. Constructing ( q+1)-ovoids of Q+(7, q)
  8. Spreads without tangents to Q-(3, q)

Key Result

Proposition 2.1

Let $\mathcal{P}_{r,e}$ be a polar space in $\mathrm{PG}(n,q)$.

Theorems & Definitions (29)

  • Proposition 2.1
  • Proposition 2.2: Ball
  • Theorem 2.3: HirsThas
  • Theorem 2.4
  • Definition 2.5
  • Proposition 2.6
  • Lemma 2.7: Bamb
  • Definition 2.8
  • Definition 2.9
  • Theorem 2.10: ST1994
  • ...and 19 more