On the equivalence of certain quasi‐Hermitian varieties

By Aguglia et al., new quasi-Hermitian varieties ℳ α , β ${{\rm{ {\mathcal M} }}}_{\alpha ,\beta }$ in PG ( r q 2 ) $\text{PG}(r,{q}^{2})$ depending on a pair of parameters $\alpha $ from the underlying field GF $\text{GF}({q}^{2})$ have been constructed. In present paper we study structure lines contained and consequently determine projective equivalence classes such for $q$ odd = 3 $r=3$ . As byproduct, also prove that collinearity graph is connected with diameter ≡ 1 mod 4 $q\equiv 1\,(\mathrm{mod}\,4)$