Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Bertini theorems and Lefschetz pencils over discrete valuation rings, with applications to higher class field theory

Authors: Uwe Jannsen and Shuji Saito
Journal: J. Algebraic Geom. 21 (2012), 683-705
Published electronically: February 21, 2012
MathSciNet review: 2957692
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Abstract | References | Additional Information

Abstract: We show the existence of good hyperplane sections for schemes over discrete valuation rings with good or (quasi-)semi-stable reduction, and the existence of good Lefschetz pencils for schemes with good reduction or ordinary quadratic reduction. As an application, we prove that the reciprocity map introduced for smooth projective varieties over local fields $ K$ by Bloch, Kato and Saito is an isomorphism after $ \ell $-adic completion, if the variety has good or ordinary quadratic reduction and $ \ell \neq \mathrm {char}(K)$.

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Additional Information

Uwe Jannsen
Affiliation: Fakultät für Mathematik, Universität Regensburg, Universitätsstr. 31, 93040 Regensburg, Germany

Shuji Saito
Affiliation: Graduate School of Mathematics, University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8914, Japan
Address at time of publication: Interactive Research Center of Science, Graduate School of Science and Engineering, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro, Tokyo 152-8551 Japan

Received by editor(s): November 9, 2009
Received by editor(s) in revised form: March 9, 2010
Published electronically: February 21, 2012
Additional Notes: The first author was supported by DFG Research Group FOR 570 ‘Algebraic Cycles and L-Functions’. The second author was supported by JSPS Grant-in-Aid, Scientific Research B-18340003 and Scientific Research S-19104001

American Mathematical Society