Effective atypical intersections and applications to orbit closures
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912486469402624 |
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| author | Baldi, Gregorio Urbanik, David |
| author_facet | Baldi, Gregorio Urbanik, David |
| contents | 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 $Ω\mathcal{M}_g (κ)$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_16628 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Effective atypical intersections and applications to orbit closures Baldi, Gregorio Urbanik, David Algebraic Geometry Dynamical Systems Number Theory 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 $Ω\mathcal{M}_g (κ)$, 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. |
| title | Effective atypical intersections and applications to orbit closures |
| topic | Algebraic Geometry Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2406.16628 |