Finite special containment formulation of the Zilber–Pink conjecture

Let S\mathfrak{S} be a semiabelian variety or Y(1)nY(1)^n, and let VSV\subseteq\mathfrak{S} be an algebraic subvariety. An atypical subvariety of VV is an atypical component of an intersection of VV with a special subvariety of S\mathfrak{S}.

Zilber–Pink conjecture, finite-containment formulation. There is a finite collection Σ\Sigma of proper special subvarieties of S\mathfrak{S} such that every atypical subvariety XX of VV is contained in some TΣT\in\Sigma.

This is presented in the source as an equivalent formulation of the preceding Zilber–Pink conjecture. The assertion is open in the general semiabelian and modular settings.

Sources & referencesView supporting material

Primary source

Vahagn Aslanyan, “Some Remarks on Atypical Intersections”, arXiv:1905.00827 (2021).

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