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
DOI: https://doi.org/10.1090/S1056-3911-2012-00570-0
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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  • [Bl] S. Bloch, Higher Algebraic $ K$-theory and class field theory for arithmetic surfaces, Ann. of Math. 114 (1981), 229-265. MR 632840 (83m:14025)
  • [Ha] R. Hartshorne, Algebraic Geometry, Graduate Texts in Math. 52, Springer, New York 1977. MR 0463157 (57:3116)
  • [HW] C. Haesemeyer, C. Weibel, Norm varieties and the chain lemma (after Markus Rost), Algebraic topology, 95130, Abel Symp., 4, Springer, Berlin, 2009. MR 2597737 (2011f:19002)
  • [JS] U. Jannsen, S. Saito, Kato Homology of Arithmetic Schemes and Higher Class Field Theory over Local Fields, Documenta Math. Extra Volume: Kazuya Kato's Fiftieth Birthday (2003) 479-538. MR 2046606 (2005c:11087)
  • [Jou] J.-P. Jouanolou, Théorèmes de Bertini et Applications, Progress in Math. 42, Birkhäuser, Basel 1983. MR 725671 (86b:13007)
  • [KS1] K. Kato and S. Saito, Unramified class field theory of arithmetic surfaces, Ann. of Math., 118 (1985), 241-275. MR 717824 (86c:14006)
  • [MS] A.S. Merkurjev and A.A. Suslin, $ K$-cohomology of Severi-Brauer Varieties and the norm residue homomorphism, Math. USSR Izvestiya 21 (1983), 307-340.
  • [Po] B. Poonen, Bertini theorems over finite fields, Ann. of Math. (2) 160 (2004), no. 3, 1099-1127. MR 2144974 (2006a:14035)
  • [Ro] M. Rost, Norm varieties and algebraic cobordism, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 77-85, Higher Ed. Press, Beijing, 2002. Errata, ibid., Vol. I (Beijing, 2002), 649. MR 1957022 (2003m:19003)
  • [Sai] S. Saito, Class field theory for curves over local fields, J. Number Theory 21 (1985), 44-80. MR 804915 (87g:11075)
  • [Sat] K. Sato, Non-divisible cycles on surfaces over local fields, J. Number Theory 114 (2005), no. 2, 272-297. MR 2167971 (2006g:19006)
  • [SJ] A. Suslin, Andrei, S. Joukhovitski, Norm varieties. J. Pure Appl. Algebra 206 (2006), no. 1-2, 245-276. MR 2220090 (2008a:14015)
  • [V1] V. Voevodsky, Motivic cohomology with $ Z/2$-coefficients, Publ. Math. Inst. Hautes Études Sci. No. 98 (2003), 59-104. MR 2031199 (2005b:14038b)
  • [V2] V. Voevodsky, On motivic cohomology with $ \mathbb{Z}/\ell $-coefficients, http://arxiv.org/
    abs/0805.4430, Annals of Math. (2) 174 (2011), no. 1, 401-438. MR 2811603
  • [V3] V. Voevodsky, Motivic Eilenberg-MacLane spaces, http://arxiv.org/abs/0805.4432, Publ. Math. IHES No. 112 (2010), 199. MR 2737977
  • [W] C. Weibel, The norm residue isomorphism theorem, J. Topol. 2 (2009), no. 2, 346-372. MR 2529300 (2011a:14039)
  • [SGA 7] P. Deligne, N. Katz Groupes de Monodromie en Géométrie Algébrique, I: Lect. Notes in Math. 288, Springer, Berlin, 1972, II: Lect. Notes in Math. 340, Springer, Berlin, 1973. MR 0354657 (50:7135)


Additional Information

Uwe Jannsen
Affiliation: Fakultät für Mathematik, Universität Regensburg, Universitätsstr. 31, 93040 Regensburg, Germany
Email: uwe.jannsen@mathematik.uni-r.de

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
Email: sshuji@msb.biglobe.ne.jp

DOI: https://doi.org/10.1090/S1056-3911-2012-00570-0
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