A note on quasi-Hermitian varieties and singular quasi-quadrics

Quasi-quadrics were introduced by Penttila, De Clerck, O’Keefe and Hamilton in [2]. They are defined as point sets which have the same intersection numbers with respect to hyperplanes as non-singular quadrics. We extend this definition in two ways. The first extension is to quasi-Hermitian varieties, which are point sets which have the same intersection numbers with respect to hyperplanes as non-singular Hermitian varieties. The second one is to singular quasi-quadrics, i.e. point sets K which have the same intersection numbers with respect to hyperplanes as singular quadrics. Our starting point was to investigate whether every singular quasi-quadric is a cone over a non-singular quasi-quadric. This question is tackled in the case of a point set K with the same intersection numbers with respect to hyperplanes as a point over an ovoid.