{"pk":64860,"title":"Quasi-polar spaces","subtitle":null,"abstract":"Quasi-polar spaces are sets of points having the same intersection numbers with respect to hyperplanes as classical polar spaces. Non-classical examples of quasi-quadrics have been constructed using a technique called {\\em pivoting} in a paper by De Clerck, Hamilton, O'Keefe and Penttila. We introduce a more general notion of pivoting, called switching, and also extend this notion to Hermitian polar spaces.  The main result of this paper studies the switching technique in detail by showing that, for \\(q\\geq 4\\), if we modify the points of a hyperplane of a polar space to create a quasi-polar space, the only thing that can be done is pivoting. The cases \\(q=2\\) and \\(q=3\\) play a special role for parabolic quadrics and are investigated in detail. Furthermore, we give a construction for quasi-polar spaces obtained from pivoting multiple times.  Finally, we focus on the case of parabolic quadrics in even characteristic and determine under which hypotheses the existence of a nucleus (which was included in the definition given in the De Clerck-Hamilton-O'Keefe-Penttila paper) is guaranteed.\n \nMathematics Subject Classifications: 51E20\n \nKeywords: Projective geometry, quadrics, hyperplanes, quasi-quadrics, intersection numbers","language":"en","license":{"name":"Creative Commons Attribution 4.0","short_name":"CC BY 4.0","text":"Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.","url":"https://creativecommons.org/licenses/by/4.0"},"keywords":[{"word":"Projective geometry"},{"word":"quadrics"},{"word":"hyperplanes"},{"word":"quasi-quadrics"},{"word":"intersection numbers"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/8ff3j88m","frozenauthors":[{"first_name":"Jeroen","middle_name":"","last_name":"Schillewaert","name_suffix":"","institution":"School of Mathematics, University of Auckland, Auckland, New Zealand","department":""},{"first_name":"Geertrui","middle_name":"","last_name":"Van de Voorde","name_suffix":"","institution":"School of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand","department":""}],"date_submitted":"2022-10-11T15:55:23Z","date_accepted":"2022-10-11T15:55:23Z","date_published":"2022-10-15T07:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64860/galley/49670/download/"}]}