Transitive ovoids of the Hermitian surface of PG\((3,q ^{2}\)), \(q\) even
An ovoid of the Hermitian surface \(H=H(3,q^2)\) in \(PG(3,q^2)\) is a set of \(q^3+1\) points of \(H\) intersecting every line of \(H\) in exactly one point. The classical ovoid of \(H\) is a planar section \(H(2,q^2)\) of \(H\) with a non-tangent plane. The authors study the ovoids of \(H\) stabilized by a subgroup of PGU\((4,q^2)\), acting transitively on the ovoid. Their results show that there are two such transitive ovoids. Apart from the classical ovoid, there is one other example. This latter example is obtained by using a cyclic spread of a Hermitian curve \(H(2,q^2)\). A cyclic Singer group of \(PG(2,q^2)\) contains a cyclic subgroup of order \(q^2-q+1\). Using this cyclic group, it is possible to partition the \(q^3+1\) points of \(H(2,q^2)\) into \(q^2-q+1\) disjoint Baer sublines, which is called a cyclic spread. By embedding \(H(2,q^2)\) into \(H\), it is then possible to construct the other example of a transitive ovoid of \(H\).
scientific article; zbMATH DE number 1888441
scientific article