Characterization of Non-Singular Hyperplanes of H (s, q2) in PG (s, q2)

Abstract

In this paper, we present a combinatorial characterization of the non-singular hyperplanes defined with respect to a non-singular hermitian variety H (s, q2) in the projective space PG (s, q2) where s ≥ 3 and q > 2. By analyzing the intersection numbers of hyperplanes with points and co-dimension 2 subspaces, we establish necessary and sufficient conditions for a collection of hyperplanes to be the set of all non-singular hyperplanes of a hermitian variety. This approach extends previous characterizations of hermitian varieties based on intersection properties, providing a purely combinatorial method for identifying the known set of hyperplanes.