Modular Zilber–Pink conjecture, optimal version

Let nn be a positive integer and let VACnV\subseteq\mathbb{A}_{\mathbb{C}}^{n} be an algebraic variety. For a subvariety XVX\subseteq V, define its defect by

def(X)=dimspcl(X)dimX.\operatorname{def}(X)=\dim\operatorname{spcl}(X)-\dim X.

The subvariety XX is optimal in VV if every subvariety WVW\subseteq V with XWX\subsetneq W satisfies def(X)<def(W)\operatorname{def}(X)<\operatorname{def}(W), and Opt(V)\operatorname{Opt}(V) denotes the set of all optimal subvarieties of VV. Modular Zilber–Pink conjecture, optimal version. The set Opt(V)\operatorname{Opt}(V) is finite. This is a formulation in terms of optimal subvarieties, which generalize maximal atypical components; its resolution is not specified in the source.

Sources & referencesView supporting material

Primary source

Vahagn Aslanyan, Sebastian Eterović and Guy Fowler, “Modular Zilber-Pink for geometrically generic varieties”, arXiv:2509.03160 (2025).

Additional references

4 papers in this index state this conjecture (2018–2025). The statement above is taken from the most recent of them; the others are arXiv:2209.12192, arXiv:1803.04753, arXiv:1803.05895.

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