Neron–Severi groups under specialization
Andre used Hodge-theoretic methods to show that in a smooth proper family X → B of varieties over an algebraically closed field k of characteristic zero, there exists a closed fiber having the same Picard number as the geometric generic fiber, even if k is countable. We give a completely different a...
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| Format: | Article |
| Language: | en_US |
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Duke University Press
2013
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| Online Access: | http://hdl.handle.net/1721.1/80706 https://orcid.org/0000-0002-8593-2792 |
| _version_ | 1811087179370201088 |
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| author | Maulik, Davesh Poonen, Bjorn |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Maulik, Davesh Poonen, Bjorn |
| author_sort | Maulik, Davesh |
| collection | MIT |
| description | Andre used Hodge-theoretic methods to show that in a smooth proper family X → B of varieties over an algebraically closed field k of characteristic zero, there exists a closed fiber having the same Picard number as the geometric generic fiber, even if k is countable. We give a completely different approach to André’s theorem, which also proves the following refinement: in a family of varieties with good reduction at p, the locus on the base where the Picard number jumps is p-adically nowhere dense. Our proof uses the “p-adic Lefschetz (1,1)-theorem” of Berthelot and Ogus, combined with an analysis of p-adic power series. We prove analogous statements for cycles of higher codimension, assuming a p-adic analogue of the variational Hodge conjecture, and prove that this analogue implies the usual variational Hodge conjecture. Applications are given to abelian schemes and to proper families of projective varieties. |
| first_indexed | 2024-09-23T13:41:20Z |
| format | Article |
| id | mit-1721.1/80706 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T13:41:20Z |
| publishDate | 2013 |
| publisher | Duke University Press |
| record_format | dspace |
| spelling | mit-1721.1/807062022-10-01T16:30:04Z Neron–Severi groups under specialization Maulik, Davesh Poonen, Bjorn Massachusetts Institute of Technology. Department of Mathematics Poonen, Bjorn Andre used Hodge-theoretic methods to show that in a smooth proper family X → B of varieties over an algebraically closed field k of characteristic zero, there exists a closed fiber having the same Picard number as the geometric generic fiber, even if k is countable. We give a completely different approach to André’s theorem, which also proves the following refinement: in a family of varieties with good reduction at p, the locus on the base where the Picard number jumps is p-adically nowhere dense. Our proof uses the “p-adic Lefschetz (1,1)-theorem” of Berthelot and Ogus, combined with an analysis of p-adic power series. We prove analogous statements for cycles of higher codimension, assuming a p-adic analogue of the variational Hodge conjecture, and prove that this analogue implies the usual variational Hodge conjecture. Applications are given to abelian schemes and to proper families of projective varieties. National Science Foundation (U.S.) (Grant DMS-0841321) National Science Foundation (U.S.) (Grant DMS-1069236) 2013-09-13T12:41:58Z 2013-09-13T12:41:58Z 2012-08 2011-11 Article http://purl.org/eprint/type/JournalArticle 0012-7094 http://hdl.handle.net/1721.1/80706 Maulik, Davesh, and Bjorn Poonen. “Néron–Severi groups under specialization.” Duke Mathematical Journal 161, no. 11 (August 2012): 2167-2206. https://orcid.org/0000-0002-8593-2792 en_US http://dx.doi.org/10.1215/00127094-1699490 Duke Mathematical Journal Creative Commons Attribution-Noncommercial-Share Alike 3.0 http://creativecommons.org/licenses/by-nc-sa/3.0/ application/pdf Duke University Press MIT web domain |
| spellingShingle | Maulik, Davesh Poonen, Bjorn Neron–Severi groups under specialization |
| title | Neron–Severi groups under specialization |
| title_full | Neron–Severi groups under specialization |
| title_fullStr | Neron–Severi groups under specialization |
| title_full_unstemmed | Neron–Severi groups under specialization |
| title_short | Neron–Severi groups under specialization |
| title_sort | neron severi groups under specialization |
| url | http://hdl.handle.net/1721.1/80706 https://orcid.org/0000-0002-8593-2792 |
| work_keys_str_mv | AT maulikdavesh neronseverigroupsunderspecialization AT poonenbjorn neronseverigroupsunderspecialization |