Some remarks on atypical intersections

In this paper we show how some known weak forms of the Zilber-Pink conjecture can be strengthened by combining them with the Mordell-Lang conjecture or its variants. We illustrate this idea by proving some theorems on atypical intersections in the semiabelian and modular settings. Given a “finitely generated” set Γ with a certain structure, we consider Γ-special subvarieties-weakly special subvarieties containing a point of Γ-and show that every variety V contains only finitely many maximal Γ-atypical subvarieties, i.e. atypical intersections of V with Γ-special varieties the weakly special closures of which are Γ-special.