Effective atypical intersections and applications to orbit closures

2024/06/24 by Gregorio Baldi, Baldi, Gregorio, David Urbanik +1 · 2 citations
Engineering · #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics #Manufacturing Process and Optimization #Number Theory (math.NT) #Optimization and Packing Problems

paper · pdf · doi:10.48550/arxiv.2406.16628

openalex publication_date 2024/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ΩM_g (κ), 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.

Cited by

• Unlikely intersections in Shimura varieties and beyond: a survey
• Murphy's law in non-abelian Hodge theory

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