{
  "id": "2209.10863",
  "version": "v1",
  "published": "2022-09-22T08:50:03.000Z",
  "updated": "2022-09-22T08:50:03.000Z",
  "title": "On the Equivalence, Stabilisers, and Feet of Buekenhout-Tits Unitals",
  "authors": [
    "Jake Faulkner",
    "Geertrui Van de Voorde"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "This paper addresses a number of problems concerning Buekenhout-Tits unitals in $PG(2,q^2)$, where $q = 2^{e+1}$ and $e \\geq 1$. We show that all Buekenhout-Tits unitals are $PGL$-equivalent (addressing an open problem in [S. Barwick and G. L. Ebert. Unitals in projective planes. Springer Monographs in Mathematics. Springer, New York, 2008.]), explicitly describe their $P\\Gamma L$-stabiliser (expanding Ebert's work in [G.L. Ebert. Buekenhout-Tits unitals. J. Algebraic. Combin. 6.2 (1997), 133-140], and show that lines meet the feet of points no on $\\ell_\\infty$ in at most four points. Finally, we show that feet of points not on $\\ell_\\infty$ are not always a $\\{0,1,2,4\\}$-set, in contrast to what happens for Buekenhout-Metz unitals [N. Abarz\\'ua, R. Pomareda, and O. Vega. Feet in orthogonal-Buekenhout-Metz unitals. Adv. Geom. 18.2 (2018), 229-236].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-22T08:50:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stabiliser",
      "equivalence",
      "problems concerning buekenhout-tits unitals",
      "expanding eberts work",
      "springer monographs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}