# Some Remarks on Atypical Intersections

## TL;DR

This paper strengthens weak forms of the Zilber--Pink conjecture by integrating them with the Mordell--Lang conjecture, proving finiteness results for atypical intersections in semiabelian and modular contexts.

## Contribution

It introduces a method to enhance weak Zilber--Pink conjecture forms using Mordell--Lang, establishing finiteness of maximal atypical intersections in specific algebraic settings.

## Key findings

- Finiteness of maximal Γ-atypical subvarieties in given varieties.
- Extension of weak Zilber--Pink conjecture using Mordell--Lang conjecture.
- Application to semiabelian and modular varieties.

## Abstract

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 $\Gamma$ with a certain structure, we consider $\Gamma$-special subvarieties -- weakly special subvarieties containing a point of $\Gamma$ -- and show that every variety $V$ contains only finitely many maximal $\Gamma$-atypical subvarieties, i.e. atypical intersections of $V$ with $\Gamma$-special varieties the weakly special closures of which are $\Gamma$-special.