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Classifying pseudo-ovals, translation generalized quadrangles, and elation Laguerre planes of small order

Giusy Monzillo, Tim Penttila, Alessandro Siciliano

Abstract

We provide classification results for translation generalized quadrangles of order less or equal to $64$, and hence, for all incidence geometries related to them. The results consist of the classification of all pseudo-ovals in $PG(3n-1,2)$, for $n=3,4$, and that of the pseudo-ovals in $PG(3n-1,q)$, for $n=5,6$, such that one of the associated projective planes is Desarguesian.

Classifying pseudo-ovals, translation generalized quadrangles, and elation Laguerre planes of small order

Abstract

We provide classification results for translation generalized quadrangles of order less or equal to , and hence, for all incidence geometries related to them. The results consist of the classification of all pseudo-ovals in , for , and that of the pseudo-ovals in , for , such that one of the associated projective planes is Desarguesian.
Paper Structure (10 sections, 22 theorems, 2 equations)

This paper contains 10 sections, 22 theorems, 2 equations.

Table of Contents

  1. Introduction and known results
  2. Generalized quadrangles
  3. Pseudo-ovals
  4. Laguerre planes
  5. Wild subspaces
  6. Machinery and computational results
  7. Case $n=3$
  8. Case $n=4$
  9. Case $n=5$
  10. Case $n=6$

Key Result

Proposition 1.1

Let $\mathcal{O}_1$ and $\mathcal{O}_2$ be two elementary $n$-dimensional pseudo-ovals in ${\rm PG}(3n-1,q)$ arising from the ovals $\widehat{\mathcal{O}}_1$ and $\widehat{\mathcal{O}}_2$ in ${\rm PG}(2,q^n)$, respectively. Then, $\mathcal{O}_1$ and $\mathcal{O}_2$ are projectively equivalent if and

Theorems & Definitions (32)

  • Proposition 1.1
  • proof
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • ...and 22 more