On the equivalence of certain quasi-Hermitian varieties

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