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Triple O'Nan Configurations in Buekenhout-Metz Unitals of Odd Order

Wen-Ai Jackson, Peter Wild

Abstract

An O'Nan configuration in a unital is a set of four lines forming a quadrilateral, where the six intersections of pairs of lines are points of the unital. In 2019 Feng and Li elegantly construct O'Nan configurations in Buekenhout-Metz unitals, in particular, for odd order unitals. We extend their work by showing the existence of Triple O'Nan configurations (a configuration containing three distinct O'Nan configurations) in these odd order Buekenhout-Metz unitals.

Triple O'Nan Configurations in Buekenhout-Metz Unitals of Odd Order

Abstract

An O'Nan configuration in a unital is a set of four lines forming a quadrilateral, where the six intersections of pairs of lines are points of the unital. In 2019 Feng and Li elegantly construct O'Nan configurations in Buekenhout-Metz unitals, in particular, for odd order unitals. We extend their work by showing the existence of Triple O'Nan configurations (a configuration containing three distinct O'Nan configurations) in these odd order Buekenhout-Metz unitals.
Paper Structure (19 sections, 13 theorems, 72 equations, 2 figures)

This paper contains 19 sections, 13 theorems, 72 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Triple O'Nan configurations in Buekenhout-Metz unitals
  3. Coordinates of the orthogonal BM unitals
  4. The automorphism group of the BM unital, odd $q$
  5. Equivalence of BM unitals
  6. Main results of paper
  7. The O'Nan construction of Feng and Li
  8. BM-special Triple O'Nan configurations in BM unitals for $q$ odd
  9. Construction of BM-special Triple O'Nans for $q$ odd
  10. Existence Conditions for BM-special Triple O'Nans, for $q$ odd
  11. Non-existence of BM-special Triple O'Nans for $q=3,5$
  12. Existence of BM-special O'Nans for $q>5$ odd, $a$ non-square: the Conic Unital
  13. BM-special Triple O'Nans for $q>5$ odd, $a$ non-square, $q\equiv 1\pmod 4$
  14. BM-special Triple O'Nans for $q>3$ odd, $a$ non-square, $q\equiv 3\pmod 4$
  15. Existence of BM-special Triple O'Nans for $q$ odd, $a$ square
  16. ...and 4 more sections

Key Result

Theorem 4

The $q$ odd O'Nan construction of Feng and Li FengLi2019 cannot be extended to a Triple O'Nan.

Figures (2)

  • Figure 1: BM-special Triple O'Nan
  • Figure 2: Extended Triple O'Nan

Theorems & Definitions (24)

  • Definition 1
  • Conjecture 2
  • Definition 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • proof
  • Lemma 7
  • Corollary 8
  • proof
  • ...and 14 more