Conic bundle fourfolds with nontrivial unramified Brauer group
Authors:
Asher Auel, Christian Böhning, Hans-Christian Graf von Bothmer and Alena Pirutka
Journal:
J. Algebraic Geom. 29 (2020), 285-327
DOI:
https://doi.org/10.1090/jag/743
Published electronically:
October 22, 2019
MathSciNet review:
4069651
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We derive a formula for the unramified Brauer group of a general class of rationally connected fourfolds birational to conic bundles over smooth threefolds. We produce new examples of conic bundles over $\mathbb {P}^3$ where this formula applies and which have nontrivial unramified Brauer group. The construction uses the theory of contact surfaces and, at least implicitly, matrix factorizations and symmetric arithmetic Cohen–Macaulay sheaves, as well as the geometry of special arrangements of rational curves in $\mathbb {P}^2$. We also prove the existence of universally $\operatorname {CH}_0$-trivial resolutions for the general class of conic bundle fourfolds we consider. Using the degeneration method, we thus produce new families of rationally connected fourfolds whose very general member is not stably rational.
References
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- David Eisenbud, Sorin Popescu, and Charles Walter, Lagrangian subbundles and codimension 3 subcanonical subschemes, Duke Math. J. 107 (2001), no. 3, 427–467. MR 1828297, DOI https://doi.org/10.1215/S0012-7094-01-10731-X
- Philippe Gille and Tamás Szamuely, Central simple algebras and Galois cohomology, Cambridge Studies in Advanced Mathematics, vol. 101, Cambridge University Press, Cambridge, 2006. MR 2266528
- Édouard Goursat, Étude des surfaces qui admettent tous les plans de symétrie d’un polyèdre régulier, Ann. Sci. École Norm. Sup. (3) 4 (1887), 159–200 (French). MR 1508797
- Alexander Grothendieck, Le groupe de Brauer. III. Exemples et compléments, Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math., vol. 3, North-Holland, Amsterdam, 1968, pp. 88–188 (French). MR 244271
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- Brendan Hassett, Alena Pirutka, and Yuri Tschinkel, Stable rationality of quadric surface bundles over surfaces, Acta Math. 220 (2018), no. 2, 341–365. MR 3849287, DOI https://doi.org/10.4310/ACTA.2018.v220.n2.a4
- Brendan Hassett, Alena Pirutka, and Yuri Tschinkel, A very general quartic double fourfold is not stably rational, Algebr. Geom. 6 (2019), no. 1, 64–75. MR 3904799, DOI https://doi.org/10.14231/ag-2019-004
- Brendan Hassett and Yuri Tschinkel, On stable rationality of Fano threefolds and del Pezzo fibrations, J. Reine Angew. Math. 751 (2019), 275–287. MR 3956696, DOI https://doi.org/10.1515/crelle-2016-0058
- V. A. Iskovskikh, On the rationality problem for conic bundles, Duke Math. J. 54 (1987), no. 2, 271–294. MR 899398, DOI https://doi.org/10.1215/S0012-7094-87-05416-0
- Bruno Kahn, On the Scharlau transfer, Rocky Mountain J. Math. 19 (1989), no. 3, 741–747. Quadratic forms and real algebraic geometry (Corvallis, OR, 1986). MR 1043246, DOI https://doi.org/10.1216/RMJ-1989-19-3-741
- Bruno Kahn, Lower $\scr H$-cohomology of higher-dimensional quadrics, Arch. Math. (Basel) 65 (1995), no. 3, 244–250. MR 1344022, DOI https://doi.org/10.1007/BF01195094
- Bruno Kahn, Markus Rost, and R. Sujatha, Unramified cohomology of quadrics. I, Amer. J. Math. 120 (1998), no. 4, 841–891. MR 1637963
- Kazuya Kato, A Hasse principle for two-dimensional global fields, J. Reine Angew. Math. 366 (1986), 142–183. With an appendix by Jean-Louis Colliot-Thélène. MR 833016, DOI https://doi.org/10.1515/crll.1986.366.142
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- V. G. Sarkisov, Birational automorphisms of conic bundles, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 4, 918–945, 974 (Russian). MR 587343
- V. G. Sarkisov, On conic bundle structures, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 2, 371–408, 432 (Russian). MR 651652
- Eugenio G. Togliatti, Una notevole superficie de 5$^o$ ordine con soli punti doppi isolati, Vierteljschr. Naturforsch. Ges. Zürich 85 (1940), no. Beiblatt (Festschrift Rudolf Fueter), 127–132 (Italian). MR 4492
- Burt Totaro, Hypersurfaces that are not stably rational, J. Amer. Math. Soc. 29 (2016), no. 3, 883–891. MR 3486175, DOI https://doi.org/10.1090/S0894-0347-2015-00840-2
- Claire Voisin, Unirational threefolds with no universal codimension $2$ cycle, Invent. Math. 201 (2015), no. 1, 207–237. MR 3359052, DOI https://doi.org/10.1007/s00222-014-0551-y
- Ernst Witt, Über ein Gegenbeispiel zum Normensatz, Math. Z. 39 (1935), no. 1, 462–467 (German). MR 1545510, DOI https://doi.org/10.1007/BF01201366
- A. A. Zagorskiĭ, Three-dimensional conic bundles, Mat. Zametki 21 (1977), no. 6, 745–758 (Russian). MR 463181
References
- Jón Kr. Arason, Cohomologische invarianten quadratischer Formen, J. Algebra 36 (1975), no. 3, 448–491 (French). MR 0389761, DOI https://doi.org/10.1016/0021-8693%2875%2990145-3
- M. Artin and D. Mumford, Some elementary examples of unirational varieties which are not rational, Proc. London Math. Soc. (3) 25 (1972), 75–95. MR 0321934, DOI https://doi.org/10.1112/plms/s3-25.1.75
- Asher Auel, Jean-Louis Colliot-Thélène, and Raman Parimala, Universal unramified cohomology of cubic fourfolds containing a plane, Brauer groups and obstruction problems, Progr. Math., vol. 320, Birkhäuser/Springer, Cham, 2017, pp. 29–55. MR 3616006
- A. Auel, C. Böhning, H.-C. Graf v. Bothmer, and A. Pirutka, M2 files for conic bundles over threefolds with nontrivial unramified Brauer group, 2016, available at http://www.math.uni-hamburg.de/home/bothmer/M2/conicBundles/.
- A. Auel, Chr. Böhning, A. Bigazzi, and H.-Chr. Graf v. Bothmer, Universal triviality of the Chow group of 0-cycles and the Brauer group, Internat. Math. Res. Notices, rnz171, 2019, https://doi.org/10.1093/imrn/rnz171.
- A. Auel, Chr. Böhning, A. Bigazzi, and H.-Chr. Graf v. Bothmer, Unramified Brauer groups of conic bundle threefolds in characteristic two, arXiv:1806.02668 [math.AG], 2018.
- Arnaud Beauville, Variétés de Prym et jacobiennes intermédiaires, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 3, 309–391 (French). MR 0472843
- Arnaud Beauville, Jean-Louis Colliot-Thélène, Jean-Jacques Sansuc, and Peter Swinnerton-Dyer, Variétés stablement rationnelles non rationnelles, Ann. of Math. (2) 121 (1985), no. 2, 283–318 (French). MR 786350, DOI https://doi.org/10.2307/1971174
- Arnaud Beauville, Determinantal hypersurfaces, Michigan Math. J. 48 (2000), 39–64. MR 1786479, DOI https://doi.org/10.1307/mmj/1030132707
- Arnaud Beauville, A very general quartic double fourfold or fivefold is not stably rational, Algebr. Geom. 2 (2015), no. 4, 508–513. MR 3403239, DOI https://doi.org/10.14231/AG-2015-022
- Spencer Bloch and Arthur Ogus, Gersten’s conjecture and the homology of schemes, Ann. Sci. École Norm. Sup. (4) 7 (1974), 181–201 (1975). MR 0412191
- Christian Böhning and Hans-Christian Graf von Bothmer, Degenerations of Gushel-Mukai fourfolds, with a view towards irrationality proofs, Eur. J. Math. 4 (2018), no. 3, 802–826. MR 3851118, DOI https://doi.org/10.1007/s40879-018-0227-z
- F. Catanese, Babbage’s conjecture, contact of surfaces, symmetric determinantal varieties and applications, Invent. Math. 63 (1981), no. 3, 433–465. MR 620679, DOI https://doi.org/10.1007/BF01389064
- Jean-Louis Colliot-Thélène and Manuel Ojanguren, Variétés unirationnelles non rationnelles: au-delà de l’exemple d’Artin et Mumford, Invent. Math. 97 (1989), no. 1, 141–158 (French). MR 999316, DOI https://doi.org/10.1007/BF01850658
- Jean-Louis Colliot-Thélène, Cycles algébriques de torsion et $K$-théorie algébrique, Arithmetic algebraic geometry (Trento, 1991) Lecture Notes in Math., vol. 1553, Springer, Berlin, 1993, pp. 1–49 (French). MR 1338859, DOI https://doi.org/10.1007/BFb0084728
- J.-L. Colliot-Thélène, Birational invariants, purity and the Gersten conjecture, $K$-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992) Proc. Sympos. Pure Math., vol. 58, Amer. Math. Soc., Providence, RI, 1995, pp. 1–64. MR 1327280
- Jean-Louis Colliot-Thélène and Alena Pirutka, Hypersurfaces quartiques de dimension 3: non-rationalité stable, Ann. Sci. Éc. Norm. Supér. (4) 49 (2016), no. 2, 371–397 (French, with English and French summaries). MR 3481353, DOI https://doi.org/10.24033/asens.2285
- Igor V. Dolgachev, Classical algebraic geometry: A modern view, Cambridge University Press, Cambridge, 2012. MR 2964027
- Igor V. Dolgachev and Vasily A. Iskovskikh, Finite subgroups of the plane Cremona group, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, Progr. Math., vol. 269, Birkhäuser Boston, Inc., Boston, MA, 2009, pp. 443–548. MR 2641179, DOI https://doi.org/10.1007/978-0-8176-4745-2_11
- David Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), no. 1, 35–64. MR 570778, DOI https://doi.org/10.2307/1999875
- David Eisenbud, Sorin Popescu, and Charles Walter, Lagrangian subbundles and codimension 3 subcanonical subschemes, Duke Math. J. 107 (2001), no. 3, 427–467. MR 1828297, DOI https://doi.org/10.1215/S0012-7094-01-10731-X
- Philippe Gille and Tamás Szamuely, Central simple algebras and Galois cohomology, Cambridge Studies in Advanced Mathematics, vol. 101, Cambridge University Press, Cambridge, 2006. MR 2266528
- Édouard Goursat, Étude des surfaces qui admettent tous les plans de symétrie d’un polyèdre régulier, Ann. Sci. École Norm. Sup. (3) 4 (1887), 159–200 (French). MR 1508797
- Alexander Grothendieck, Le groupe de Brauer. III. Exemples et compléments, Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math., vol. 3, North-Holland, Amsterdam, 1968, pp. 88–188 (French). MR 244271
- Brendan Hassett, Andrew Kresch, and Yuri Tschinkel, Stable rationality and conic bundles, Math. Ann. 365 (2016), no. 3-4, 1201–1217. MR 3521088, DOI https://doi.org/10.1007/s00208-015-1292-y
- Brendan Hassett, Alena Pirutka, and Yuri Tschinkel, Stable rationality of quadric surface bundles over surfaces, Acta Math. 220 (2018), no. 2, 341–365. MR 3849287, DOI https://doi.org/10.4310/ACTA.2018.v220.n2.a4
- Brendan Hassett, Alena Pirutka, and Yuri Tschinkel, A very general quartic double fourfold is not stably rational, Algebr. Geom. 6 (2019), no. 1, 64–75. MR 3904799
- Brendan Hassett and Yuri Tschinkel, On stable rationality of Fano threefolds and del Pezzo fibrations, J. Reine Angew. Math. 751 (2019), 275–287. MR 3956696, DOI https://doi.org/10.1515/crelle-2016-0058
- V. A. Iskovskikh, On the rationality problem for conic bundles, Duke Math. J. 54 (1987), no. 2, 271–294. MR 899398, DOI https://doi.org/10.1215/S0012-7094-87-05416-0
- Bruno Kahn, On the Scharlau transfer, Quadratic forms and real algebraic geometry (Corvallis, OR, 1986), Rocky Mountain J. Math. 19 (1989), no. 3, 741–747. MR 1043246, DOI https://doi.org/10.1216/RMJ-1989-19-3-741
- Bruno Kahn, Lower $\mathcal {H}$-cohomology of higher-dimensional quadrics, Arch. Math. (Basel) 65 (1995), no. 3, 244–250. MR 1344022, DOI https://doi.org/10.1007/BF01195094
- Bruno Kahn, Markus Rost, and R. Sujatha, Unramified cohomology of quadrics. I, Amer. J. Math. 120 (1998), no. 4, 841–891. MR 1637963
- Kazuya Kato, A Hasse principle for two-dimensional global fields, with an appendix by Jean-Louis Colliot-Thélène, J. Reine Angew. Math. 366 (1986), 142–183. MR 833016, DOI https://doi.org/10.1515/crll.1986.366.142
- Stephen Lichtenbaum, Duality theorems for curves over $p$-adic fields, Invent. Math. 7 (1969), 120–136. MR 0242831, DOI https://doi.org/10.1007/BF01389795
- Alena Pirutka, Varieties that are not stably rational, zero-cycles and unramified cohomology, Algebraic geometry: Salt Lake City 2015, Proc. Sympos. Pure Math., vol. 97, Amer. Math. Soc., Providence, RI, 2018, pp. 459–483. MR 3821181
- Aleksandr Pukhlikov, Birationally rigid varieties, Mathematical Surveys and Monographs, vol. 190, American Mathematical Society, Providence, RI, 2013. MR 3060242
- V. G. Sarkisov, Birational automorphisms of conic bundles, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 4, 918–945, 974 (Russian). MR 587343
- V. G. Sarkisov, On conic bundle structures, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 2, 371–408, 432 (Russian). MR 651652
- Eugenio G. Togliatti, Una notevole superficie de 5$^o$ ordine con soli punti doppi isolati, Vierteljschr. Naturforsch. Ges. Zürich 85 85 (1940), Beiblatt (Festschrift Rudolf Fueter), 127–132 (Italian). MR 0004492
- Burt Totaro, Hypersurfaces that are not stably rational, J. Amer. Math. Soc. 29 (2016), no. 3, 883–891. MR 3486175, DOI https://doi.org/10.1090/jams/840
- Claire Voisin, Unirational threefolds with no universal codimension $2$ cycle, Invent. Math. 201 (2015), no. 1, 207–237. MR 3359052, DOI https://doi.org/10.1007/s00222-014-0551-y
- Ernst Witt, Über ein Gegenbeispiel zum Normensatz, Math. Z. 39 (1935), no. 1, 462–467 (German). MR 1545510, DOI https://doi.org/10.1007/BF01201366
- A. A. Zagorskiĭ, Three-dimensional conic bundles, Mat. Zametki 21 (1977), no. 6, 745–758 (Russian). MR 0463181
Additional Information
Asher Auel
Affiliation:
Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755
MR Author ID:
932786
Email:
asher.auel@dartmouth.edu
Christian Böhning
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Email:
C.Boehning@warwick.ac.uk
Hans-Christian Graf von Bothmer
Affiliation:
Fachbereich Mathematik der Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany
MR Author ID:
724323
Email:
hans.christian.v.bothmer@uni-hamburg.de
Alena Pirutka
Affiliation:
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012
MR Author ID:
934651
Email:
pirutka@cims.nyu.edu
Received by editor(s):
April 22, 2017
Received by editor(s) in revised form:
April 4, 2019
Published electronically:
October 22, 2019
Additional Notes:
The first author was partially supported by the NSA grant H98230-16-1-032
Article copyright:
© Copyright 2019
University Press, Inc.
