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Some non-existence results on $m$-ovoids in classical polar spaces

Jan De Beule, Jonathan Mannaert, Valentino Smaldore

Abstract

In this paper we develop non-existence results for $m$-ovoids in the classical polar spaces $Q^-(2r+1,q), W(2r-1,q)$ and $H(2r,q^2)$ for $r>2$. In [4] a lower bound on $m$ for the existence of $m$-ovoids of $H(4,q^2)$ is found by using the connection between $m$-ovoids, two-character sets, and strongly regular graphs. This approach is generalized in [3] for the polar spaces $Q^-(2r+1,q), W(2r-1,q)$ and $H(2r,q^2)$, $r>2$. In [1] an improvement for the particular case $H(4,q^2)$ is obtained by exploiting the algebraic structure of the collinearity graph, and using the characterization of an $m$-ovoid as an intruiging set. In this paper, we use an approach based on geometrical and combinatorial arguments, inspired by the results from [10], to improve the bounds from [3].

Some non-existence results on $m$-ovoids in classical polar spaces

Abstract

In this paper we develop non-existence results for -ovoids in the classical polar spaces and for . In [4] a lower bound on for the existence of -ovoids of is found by using the connection between -ovoids, two-character sets, and strongly regular graphs. This approach is generalized in [3] for the polar spaces and , . In [1] an improvement for the particular case is obtained by exploiting the algebraic structure of the collinearity graph, and using the characterization of an -ovoid as an intruiging set. In this paper, we use an approach based on geometrical and combinatorial arguments, inspired by the results from [10], to improve the bounds from [3].
Paper Structure (5 sections, 18 theorems, 60 equations, 1 figure, 6 tables)

This paper contains 5 sections, 18 theorems, 60 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weighted $m$-ovoids in $\mathcal{P}_{r,e}'$
  4. A lower bound on $m$ for $m$-ovoids in $\mathcal{P}'_{r,e}$
  5. New non-existence conditions on $m$-ovoids

Key Result

Theorem 1.1

Let $q > 2$ and $r \geq 3$. Suppose that $\mathcal{O}$ is an $m$-ovoid in one of the following polar spaces, $Q^-(2r+1,q)$ ($e=2$), $W(2r-1,q)$ ($e=1$) or $H(2r,q)$ ($q$ square, $e=\frac{3}{2})$. If (a) $r\geq 4$, or, (b) $e\in\{1,\frac{3}{2}\}$ and $(r,q,e)\not= (3,3,1)$. Then This bound asymptotically converges to

Figures (1)

  • Figure 1: Double counting argument from Theorem \ref{['Th1:counting']}.

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Remark 2.6
  • ...and 31 more