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Codes Cryptogr."],"published-print":{"date-parts":[[2025,2]]},"abstract":"Abstract<\/jats:title>\n This paper addresses a number of problems concerning Buekenhout-Tits unitals in \n \n $${{\\,\\textrm{PG}\\,}}(2, q^2)$$<\/jats:tex-math>\n \n \n \n \n PG<\/mml:mtext>\n \n <\/mml:mrow>\n (<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n \n q<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, where \n \n $$q = 2^{2e + 1}$$<\/jats:tex-math>\n \n \n q<\/mml:mi>\n =<\/mml:mo>\n \n 2<\/mml:mn>\n \n 2<\/mml:mn>\n e<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and \n \n $$e \\ge 1$$<\/jats:tex-math>\n \n \n e<\/mml:mi>\n \u2265<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. We show that all Buekenhout-Tits unitals are equivalent under \n \n $${{\\,\\textrm{PGL}\\,}}(3, q^2)$$<\/jats:tex-math>\n \n \n \n \n PGL<\/mml:mtext>\n \n <\/mml:mrow>\n (<\/mml:mo>\n 3<\/mml:mn>\n ,<\/mml:mo>\n \n q<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> [addressing an open problem in Barwick and Ebert (Unitals in Projective Planes. Springer Monographs in Mathematics. Springer, New York, 2008)], explicitly describe their stabiliser in \n \n $$\\textrm{P}\\Gamma \\textrm{L}(3, q^2)$$<\/jats:tex-math>\n \n \n P<\/mml:mtext>\n \u0393<\/mml:mi>\n L<\/mml:mtext>\n (<\/mml:mo>\n 3<\/mml:mn>\n ,<\/mml:mo>\n \n q<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> [expanding Ebert\u2019s work in Ebert (J Algebraic Comb 6(2):133\u2013140, 1997)], and show that lines meet the feet of points not on \n \n $$\\ell _{\\infty }$$<\/jats:tex-math>\n \n \n \u2113<\/mml:mi>\n \u221e<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> in at most four points. Finally, we show that feet of points not on \n \n $$\\ell _{\\infty }$$<\/jats:tex-math>\n \n \n \u2113<\/mml:mi>\n \u221e<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> are not always a \n \n $$\\{0, 1, 2, 4\\}$$<\/jats:tex-math>\n \n \n {<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n 4<\/mml:mn>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-set, in contrast to what happens for Buekenhout-Metz unitals Abarz\u00faa et al (Adv Geom 18(2):229\u2013236, 2018).<\/jats:p>","DOI":"10.1007\/s10623-023-01234-4","type":"journal-article","created":{"date-parts":[[2023,5,19]],"date-time":"2023-05-19T08:01:53Z","timestamp":1684483313000},"page":"297-307","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On the equivalence, stabilisers, and feet of Buekenhout-Tits unitals"],"prefix":"10.1007","volume":"93","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2427-8181","authenticated-orcid":false,"given":"Jake","family":"Faulkner","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4957-6911","authenticated-orcid":false,"given":"Geertrui","family":"Van de Voorde","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,5,18]]},"reference":[{"issue":"2","key":"1234_CR1","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1515\/advgeom-2017-0050","volume":"18","author":"N Abarz\u00faa","year":"2018","unstructured":"Abarz\u00faa N., Pomareda R., Vega O.: Feet in orthogonal-Buekenhout-Metz unitals. 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