Quasi-polar spaces

Quasi-polar spaces are sets of points having the same intersection numbers with respect to hyperplanes as classical polar spaces. Non-classical examples quasi-quadrics have been constructed using a technique called {\em pivoting} in paper by De Clerck, Hamilton, O'Keefe and Penttila. We introduce more general notion pivoting, switching, also extend this Hermitian The main result studies switching detail showing that, for \(q\geq 4\), if we modify hyperplane space create quasi-polar space, only thing that can be done is pivoting. cases \(q=2\) \(q=3\) play special role parabolic quadrics investigated detail. Furthermore, give construction obtained from pivoting multiple times. Finally, focus on case even characteristic determine under which hypotheses existence nucleus (which was included definition given Clerck-Hamilton-O'Keefe-Penttila paper) guaranteed.Mathematics Subject Classifications: 51E20Keywords: Projective geometry, quadrics, hyperplanes, quasi-quadrics,