Irregular cusps of ball quotients

We study the branch divisors on the boundary of the canonical toroidal compactification of ball quotients. We show a criterion, the low slope cusp form trick, for proving that ball quotients are of general type. Moreover, we classify when irregular cusps exist in the case of the discriminant kernel and construct concrete examples for some arithmetic subgroups. As another direction of study, when a complex ball is embedded into a Hermitian symmetric domain of type IV, we determine when regular or irregular cusps map to regular or irregular cusps studied by Ma.     

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 $q\equiv 1\left(\mathrm{mod}4\right)$.