Effective atypical intersections and applications to orbit closures

TL;DR

This paper introduces a unified approach to atypical intersections, providing new proofs of finiteness results, and develops algorithms for computing atypical orbit closures and special loci, with applications to Hodge theory.

Contribution

It offers a new framework for atypical intersections, proves finiteness of orbit closures, and presents implementable algorithms for computing these structures.

Findings

Proved finiteness of maximal atypical orbit closures in translation surfaces.

Developed an algorithm to compute all maximal atypical orbit closures.

Provided an effective proof of the geometric Zilber-Pink conjecture for variations of mixed Hodge structures.

Abstract

We propose a unifying setting for dealing with monodromically atypical intersections that goes beyond the usual Zilber-Pink conjecture. In particular we obtain a new proof of finiteness of the maximal atypical orbit closures in each stratum of translation surfaces $\Omega \mathcal{M}_g (\kappa)$, as given by Eskin, Filip, and Wright. We also describe a concrete algorithm, implementable in principle on a computer, which provably computes all maximal orbit closures which are 'atypical' in a sense described by Filip. The same methods also give a general algorithm for computing atypical special loci associated to systems of differential equations, and in particular give an effective and o-minimal free proof of the geometric Zilber-Pink conjecture for variations of mixed Hodge structures.