Characterization of non-singular hyperplanes of Hs,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 Hs,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.