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Stable rationality and conic bundles

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We study stable rationality properties of conic bundles over rational surfaces.

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References

  1. Abramovich, D., Corti, A., Vistoli, A.: Twisted bundles and admissible covers. Comm. Algebra 31, 3547–3618 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abramovich, D., Graber, T., Vistoli, A.: Gromov–Witten theory of Deligne–Mumford stacks. Am. J. Math. 130, 1337–1398 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abramovich, D., Olsson, M., Vistoli, A.: Tame stacks in positive characteristic. Ann. Inst. Fourier (Grenoble) 58, 1057–1091 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Abramovich, D., Olsson, M., Vistoli, A.: Twisted stable maps to tame Artin stacks. J. Algebraic Geom. 20, 399–477 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Abramovich, D., Vistoli, A.: Compactifying the space of stable maps. J. Am. Math. Soc. 15, 27–75 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Alper, J.: Good moduli spaces for Artin stacks. Ann. Inst. Fourier (Grenoble) 63, 2349–2402 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Artin, M., Mumford, D.: Some elementary examples of unirational varieties which are not rational. Proc. Lond. Math. Soc. 25(3), 75–95 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  8. Auel, A., Colliot-Thélène, J.-L., Parimala, R.: Universal unramified cohomology of cubic fourfolds containing a plane, to appear in Brauer groups and obstruction problems: moduli spaces and arithmetic. Palo Alto, CA (2013)

  9. Beauville, A.: Variétés de Prym et jacobiennes intermédiaires. Ann. Sci. École Norm. Sup. 10(4), 309–391 (1977)

    MathSciNet  MATH  Google Scholar 

  10. Beauville, A.: Le groupe de monodromie des familles universelles d’hypersurfaces et d’intersections complètes, Complex analysis and algebraic geometry (Göttingen, 1985), Lecture Notes in Math. 1194, pp. 8-18. Springer, Berlin (1986)

  11. Beauville, A.: A very general quartic double fourfold or fivefold is not stably rational. Algebraic Geom. (to appear)

  12. Beauville, A.: A very general sextic double solid is not stably rational. arXiv:1411.7484

  13. Behrend, K., Manin, Yu.: Stacks of stable maps and Gromov–Witten invariants. Duke Math. J. 85, 1–60 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bruzzo, U., Sala, F.: Framed sheaves on projective stacks. Adv. Math. 272, 20–95 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cadman, C.: Using stacks to impose tangency conditions on curves. Am. J. Math. 129, 405–427 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Colliot-Thélène, J.-L.: Birational invariants, purity and the Gersten conjecture. In: K-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), Proc. Sympos. Pure Math., 58, Part 1, Am. Math. Soc., pp. 1-64, Providence, RI (1995)

  17. Colliot-Thélène, J.-L., Ojanguren, M.: Variétés unirationnelles non rationnelles: au-delà de l’exemple d’Artin et Mumford. Invent. Math. 97(1), 141–158 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  18. Colliot-Thélène, J.-L., Pirutka, A.: Hypersurfaces quartiques de dimension 3: non rationalité stable. Ann. Sci. École Norm. Sup. (4) (to appear)

  19. Colliot-Thélène, J.-L., Pirutka, A.: Revêtements cycliques qui ne sont pas stablement rationnels. arXiv:1506.00420

  20. de Jong, A.J.: The period-index problem for the Brauer group of an algebraic surface. Duke Math. J. 123, 71–94 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Inst. Hautes Études Sci. Publ. Math. 36, 75–109 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  22. Fulton, W.: Intersection Theory, 2nd edn. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  23. Grothendieck, A.: Le groupe de Brauer, I-III. In: Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math. 3, pp. 46-188, North-Holland, Amsterdam (1968)

  24. Hassett, B., Kresch, A., Tschinkel, Y.: On the moduli of degree 4 Del Pezzo surfaces. Dev. Moduli Theory (Kyoto, 2013) (to appear)

  25. Iskovskikh, V.A.: On the rationality problem for conic bundles. Duke Math. J. 54, 271–294 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kollár, J.: Nonrational hypersurfaces. J. Am. Math. Soc. 8, 241–249 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kresch, A.: Flattening stratification and the stack of partial stabilizations of prestable curves. Bull. Lond. Math. Soc. 45, 93–102 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Laumon, G., Moret-Bailly, L.: Champs Algébriques. Springer, Berlin (2000)

    Google Scholar 

  29. Lieblich, M.: Period and index in the Brauer group of an arithmetic surface. J. Reine Angew. Math. 659, 1–41 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lieblich, M.: On the ubiquity of twisted sheaves. In: Bogomolov, F., Hassett, B., Tschinkel, Y. (eds.) Birational Geometry, Rational Curves, and Arithmetic, pp. 205–227. Springer, New York (2013)

    Chapter  Google Scholar 

  31. Mella, M.: On the unirationality of 3-fold conic bundles. arXiv:1403.7055

  32. Merkurjev, A.: Unramified elements in cycle modules. J. Lond. Math. Soc. 78(2), 51–64 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  33. Milne, J.S.: Étale Cohomology. Princeton University Press, Princeton (1980)

    MATH  Google Scholar 

  34. Nironi, F.: Grothendieck duality for Deligne-Mumford stacks. arXiv:0811.1955

  35. O’Grady, K.: Moduli of vector bundles on surfaces. In: Algebraic Geometry (Santa Cruz, 1995), Proc. Symposia Pure Math. 62, Part 1, Am. Math. Soc. pp. 101-126, Providence, RI (1996)

  36. Rost, M.: Chow groups with coefficients. Doc. Math. 1, 319–393 (1996)

    MathSciNet  MATH  Google Scholar 

  37. Rydh, D.: Families of cycles and the Chow scheme. Ph. D. thesis, KTH, Stockholm (2008)

  38. Rydh, D.: Étale dévissage, descent and pushouts of stacks. J. Algebra 331, 194–223 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. Saltman, D.J.: Retract rational fields and cyclic Galois extensions. Israel Math. J. 47, 165–215 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  40. Sarkisov, V.G.: Birational automorphisms of conic bundles. Izv. Akad. Nauk SSSR Ser. Mat. 44, 918–945 (1980)

    MathSciNet  MATH  Google Scholar 

  41. Sarkisov, V.G.: On conic bundle structures. Izv. Akad. Nauk SSSR Ser. Mat. 46, 371–408 (1982)

    MathSciNet  MATH  Google Scholar 

  42. Shokurov, V.V.: Prym varieties: theory and applications. Izv. Akad. Nauk SSSR Ser. Mat. 47, 785–855 (1983)

    MathSciNet  Google Scholar 

  43. Totaro, B.: Hypersurfaces that are not stably rational. arXiv:1502.04040

  44. Vistoli, A.: Grothendieck topologies, fibered categories and descent theory. In: Fundamental algebraic geometry, pp. 1-104, Math. Surveys Monogr. 123, Am. Math. Soc., Providence, RI (2005)

  45. Voisin, C.: Unirational threefolds with no universal codimension 2 cycle. Invent. Math. (to appear)

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Acknowledgments

The first author was supported by NSF Grants 1148609 and 1401764. The second author was supported by the Swiss National Science Foundation. The third author was supported by NSF Grant 1160859. We are grateful to I. Cheltsov, J.-L. Colliot-Thélène, L. Katzarkov, A. Pirutka, and B. Totaro for comments and suggestions.

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Correspondence to Yuri Tschinkel.

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Hassett, B., Kresch, A. & Tschinkel, Y. Stable rationality and conic bundles. Math. Ann. 365, 1201–1217 (2016). https://doi.org/10.1007/s00208-015-1292-y

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  • DOI: https://doi.org/10.1007/s00208-015-1292-y

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