The INI has a new website!

This is a legacy webpage. Please visit the new site to ensure you are seeing up to date information.

Skip to content



Néron-Severi groups under specialization

Poonen, B (MIT)
Friday 28 August 2009, 09:30-10:30

Seminar Room 1, Newton Institute


This is joint work with Davesh Maulik and Claire Voisin. We prove that given a smooth proper family X --> B of varieties over an algebraically closed field k of characteristic 0, there exists a closed fiber having the same Picard number as the geometric generic fiber, even if k is countable. In fact, we give two proofs, and they show that the locus on the base where the Picard number jumps is "small" in two different senses. The first proof uses Hodge theory and the actions of geometric monodromy groups and Galois groups to show that the locus is small in a sense related to Hilbert irreducibility. The second proof uses the "p-adic Lefschetz (1,1) theorem" of Berthelot and Ogus to show that in a family of varieties with good reduction at p, the locus is nowhere p-adically dense. Finally, we prove analogous statements for cycles of higher codimension, under the assumption of the variational Hodge conjecture or a p-adic analogue conjectured by M. Emerton.


The video for this talk should appear here if JavaScript is enabled.
If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible.

Back to top ∧