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Mathematische Annalen

, Volume 368, Issue 3–4, pp 1191–1225 | Cite as

Complete intersections: moduli, Torelli, and good reduction

  • A. Javanpeykar
  • D. Loughran
Article
  • 190 Downloads

Abstract

We study the arithmetic of complete intersections in projective space over number fields. Our main results include arithmetic Torelli theorems and versions of the Shafarevich conjecture, as proved for curves and abelian varieties by Faltings. For example, we prove an analogue of the Shafarevich conjecture for cubic and quartic threefolds and intersections of two quadrics.

Mathematics Subject Classification

11G35 14M10 14K30 14J50 14C34 14D23 

Notes

Acknowledgments

We are grateful to Giuseppe Ancona, Jean-Benoît Bost, Martin Bright, Fréderic Campana, Jean-Louis Colliot-Thélène, Bas Edixhoven, Jochen Heinloth, Marc Hindry, David Holmes, Ben Moonen, Laurent Moret-Bailly, Duco van Straten, Lenny Taelman, Olivier Wittenberg, and Kang Zuo for helpful discussions on several parts of this paper. Special thanks go to Olivier Benoist for very helpful discussions on complete intersections and level structure, Yohan Brunebarbe for his help on period maps and the proof of Theorem 2.8, and Angelo Vistoli for answering our questions on stacks and his help in proving Proposition 2.12. We are very grateful to the anonymous referee for useful comments. The first named author gratefully acknowledges the support of SFB/Transregio 45.

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Authors and Affiliations

  1. 1.Institut für MathematikJohannes Gutenberg-Universität MainzMainzGermany
  2. 2.Institut für Algebra, Zahlentheorie und Diskrete MathematikLeibniz Universität HannoverHannoverGermany

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