Quasipolar spaces
Abstract
Quasipolar spaces are sets of points having the same intersection numbers with respect to hyperplanes as classical polar spaces. Nonclassical examples of quasiquadrics have been constructed using a technique called pivoting [5]. 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 >= 4, if we modify the points of a hyperplane of a polar space to create a quasipolar 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 quasipolar 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 [5]) is guaranteed.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.12710
 Bibcode:
 2021arXiv210912710S
 Keywords:

 Mathematics  Combinatorics;
 51E20