Ovoids of the Hermitian surface in odd characteristic

Abstract

We construct a new ovoid of the polar space arising from the Hermitian surface of PG(3,# 2 ) with q ^ 5 odd. The automorphism group Γ of such an ovoid has a normal cyclic subgroup Φ of order ^ (q + 1) such that Γ/Φ ^ PGL(2, q). Furthermore, Γ has three orbits on the ovoid, one of size q + 1 and two of size \q(q -\)(q + 1).

Caps on Hermitian varieties and maximal curves

“…The size of the corresponding cap K of U 3 is q 2 + 1 + 1 2 q(q − 1) 2 = 1 2 (q 3 + q + 2). The cap K is incomplete, since it is contained in an ovoid of U 3 ; see [13].…”

A Geometric Construction for Some Ovoids of the Hermitian Surface

“…When O happens to be an ovoid, it will be called an ovoid of type (1) or (2) according as O contains ∆ 1 or ∆ 2 . Examples of ovoids of type (1) were constructed in [4].…”

“…Existence of non-classical ovoids of H(3, q 2 ) has been known since 1994, see [10]. However, a thorough work on the subject has begun only recently, prompted by the discovery of new large families [2,4].…”

“…In fact, the non-classical ovoids of [2,4] are multiply derivable. When the replacement of O ∩ by H ∩ , as described above, is an ovoid O , we say that O is the derived ovoid of O, by its replaceable (q + 1)-secant .…”

“…Clearly, the existence and nature of replaceable sets depends heavily on the nature of O. The ovoids found in [4] are multiply derivable.…”