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A Note on Fano Planes in Orthogonal Buekenhout-Metz Unitals of Even 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 orthogonal and Tits Buekenhout-Metz unitals. We extend their work by extending their construction to a Fano plane embedded in the orthogonal Buekenhout-Metz unital of even order. We deduce that there exist O'Nan configurations in orthogonal Buekenhout-Metz unitals different to those of Feng and Li, and make a conjecture about Fano planes embedded in orthogonal Buekenhout-Metz unitals.

A Note on Fano Planes in Orthogonal Buekenhout-Metz Unitals of Even 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 orthogonal and Tits Buekenhout-Metz unitals. We extend their work by extending their construction to a Fano plane embedded in the orthogonal Buekenhout-Metz unital of even order. We deduce that there exist O'Nan configurations in orthogonal Buekenhout-Metz unitals different to those of Feng and Li, and make a conjecture about Fano planes embedded in orthogonal Buekenhout-Metz unitals.
Paper Structure (15 sections, 4 theorems, 27 equations)

This paper contains 15 sections, 4 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Buekenhout-Metz unitals
  3. Field properties for even $q$
  4. Coordinates of the orthogonal BM unitals
  5. The automorphism group of the orthogonal BM unital, even $q$
  6. The equivalence of orthogonal BM unitals, even $q$
  7. Fano planes in Buekenhout-Metz unitals
  8. Contribution of this paper
  9. Main results for BM-unitals, even $q$
  10. Existence of Fano planes in even order orthogonal BM unitals
  11. Number of BM-special Fano planes
  12. Tits Unitals
  13. Discussion
  14. Examples
  15. Conclusion

Key Result

Lemma 2

For even $q$, if ${\mathcal{U}}_{a,b}$ is any non-classical orthogonal BM unital, then without loss of generality we may assume

Theorems & Definitions (10)

  • Conjecture 1
  • Lemma 2
  • proof
  • Definition 3
  • Theorem 4
  • Theorem 5
  • Corollary 6
  • proof
  • Example 1
  • Example 2