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Quasi-Hermitian Varieties and Their Barlotti--Cofman Representation

Angela Aguglia, Viola Siconolfi

Abstract

Quasi-Hermitian varieties arise as higher-dimensional generalizations of non-classical unitals, including the Buekenhout--Metz (BM) and Buekenhout--Tits (BT) families. After reviewing known constructions and structural properties, we determine explicitly the BC representation of BM and BT quasi-Hermitian varieties in $\mathrm{PG}(3,q^2)$ inside $\mathrm{PG}(6,q)$. We prove that BM varieties correspond to quadratic cones with hyperbolic base, whereas BT varieties give rise to non-quadratic cones, and we describe the associated configuration of spread elements in the section at infinity. These results provide a geometric interpretation of the non-classical nature of BM and BT varieties within the BC framework.

Quasi-Hermitian Varieties and Their Barlotti--Cofman Representation

Abstract

Quasi-Hermitian varieties arise as higher-dimensional generalizations of non-classical unitals, including the Buekenhout--Metz (BM) and Buekenhout--Tits (BT) families. After reviewing known constructions and structural properties, we determine explicitly the BC representation of BM and BT quasi-Hermitian varieties in inside . We prove that BM varieties correspond to quadratic cones with hyperbolic base, whereas BT varieties give rise to non-quadratic cones, and we describe the associated configuration of spread elements in the section at infinity. These results provide a geometric interpretation of the non-classical nature of BM and BT varieties within the BC framework.
Paper Structure (20 sections, 15 theorems, 97 equations)

This paper contains 20 sections, 15 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Quasi-Polar Spaces and Historical Background
  3. Quasi-Hermitian Varieties and Known Constructions
  4. BM and BT quasi-Hermitian varieties
  5. BM quasi-Hermitian varieties
  6. BT quasi-Hermitian varieties
  7. The isomorphism issue
  8. Barlotti-Cofman representation of $\mathrm{PG}(r,q^n)$
  9. BC representation of Hermitian varieties
  10. Coordinates in BC representation of $\mathrm{PG}(3, q^2)$
  11. BC representation of BM quasi-Hermitian varieties, odd characteristics
  12. BC Representation of $\mathcal{B}_{a,b}$ (odd case).
  13. Representation of $\mathcal{M}_{ab}$ (odd case).
  14. BC representation of BM quasi-Hermitian varieties, even characteristics
  15. BC representation of $\mathcal{B}_{ab}$ (even case).
  16. ...and 5 more sections

Key Result

Theorem 3.1

Let $q=p^n\ge 4$ be a prime power. The number of projectively inequivalent BM unitals in $\mathrm{PG}(2,q^2)$ is where $\Phi$ is Euler's totient function and $n_0$ is the odd part of $n$ if $p>2$, while $n_0=0$ if $p=2$.

Theorems & Definitions (19)

  • Theorem 3.1: BEE
  • Theorem 3.2: AG
  • Theorem 3.3: AGMS
  • Theorem 3.4: AM
  • Theorem 4.1
  • Proposition 5.1
  • proof
  • Theorem 5.2
  • Proposition 5.3
  • proof
  • ...and 9 more