On the equivalence of certain quasi-Hermitian varieties

By Aguglia et al., new quasi-Hermitian varieties M alpha,beta ${{\rm{ {\mathcal M} }}}_{\alpha ,\beta }$ in PG(r,q2) $\text{PG}(r,{q}<^>{2})$ depending on a pair of parameters alpha,beta $\alpha ,\beta $ from the underlying field GF(q2) $\text{GF}({q}<^>{2})$ have been constructed. In the present paper we study the structure of the lines contained in M alpha,beta ${{\rm{ {\mathcal M} }}}_{\alpha ,\beta }$ and consequently determine the projective equivalence classes of such varieties for q $q$ odd and r=3 $r=3$. As a byproduct, we also prove that the collinearity graph of M alpha,beta ${{\rm{ {\mathcal M} }}}_{\alpha ,\beta }$ is connected with diameter 3 for q equivalent to 1(mod4) $q\equiv 1\,(\mathrm{mod}\,4)$.