# Torsion hypersurfaces on abelian schemes and Betti coordinates

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## Abstract

In this paper we extend to arbitrary complex coefficients certain finiteness results on unlikely intersections linked to torsion in abelian surface schemes over a curve, which have been recently proved for the case of algebraic coefficients; in this way we complete the solution of Zilber–Pink conjecture for abelian surface schemes over a curve. As experience has shown also in previous cases, the extension from algebraic to complex coefficients often requires entirely new arguments, whereas simple specialization arguments fail. The outcome gives as a byproduct new finiteness results when the base of the scheme has arbitrary dimension; another consequence is a proof of an expectation of Mazur concerning the structure of the locus in the base when a given section is torsion. Finally, we show the link with an old work of Griffiths and Harris on a higher dimensional extension of a theorem of Poncelet.

## Notes

### Acknowledgements

We express our warmest thanks to an extremely careful referee who detected several inaccuracies, and helped us very much in improving our presentation. The referee also indicated tentative thoughts for later works, for instance suggesting the investigation of the relation between the ranks of the Betti and Kodaira–Spencer maps associated to an abelian scheme with a section. Indeed, a joint work in progress with André explores the role of the Kodaira–Spencer map in this setting. Further warm thanks go to Bertrand for his interest and comments which helped our presentation. We are also grateful to Voisin for sending us her recent preprint [22] and for a helpful correspondence with the third author.

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