Skip to main content Accessibility help
×
Home

Derived categories and rationality of conic bundles

  • Marcello Bernardara (a1) (a2) and Michele Bolognesi (a3)

Abstract

We show that a standard conic bundle over a minimal rational surface is rational and its Jacobian splits as the direct sum of Jacobians of curves if and only if its derived category admits a semiorthogonal decomposition by exceptional objects and the derived categories of those curves. Moreover, such a decomposition gives the splitting of the intermediate Jacobian also when the surface is not minimal.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Derived categories and rationality of conic bundles
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Derived categories and rationality of conic bundles
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Derived categories and rationality of conic bundles
      Available formats
      ×

Copyright

References

Hide All
[AGP05]d’Almeida, J., Gruson, L. and Perrin, N., Courbes de genre 5 munies d’une involution sans point fixe, J. Lond. Math. Soc. (2) 72 (2005), 545570.
[AM98]del Angel, P. L. and Müller-Stach, S., Motives of uniruled 3-folds, Compositio Math. 112 (1998), 116.
[AM72]Artin, M. and Mumford, D., Some elementary examples of unirational varieties which are not rational, Proc. Lond. Math. Soc. (3) 25 (1972), 7595.
[Bea77a]Beauville, A., Prym varieties and Schottky problem, Invent. Math. 41 (1977), 149196.
[Bea77b]Beauville, A., Variétés de Prym et jacobiennes intermédiaires, Ann. Sci. Éc. Norm. Supér. (4) 10 (1977), 309391.
[Bea00]Beauville, A., Determinantal hypersurfaces, Michigan Math. J. 48 (2000), 3964.
[Bei84]Beilinson, A. A., The derived category of coherent sheaves on ${ \mathbb{P} }^{n} $, Selecta Math. Sov. 34 (1984), 233237.
[BF83]Beltrametti, M. and Francia, P., Conic bundles on non-rational surfaces, in Algebraic geometry—open problems, Ravello 1982, Lecture Notes in Mathematics, vol. 997 (Springer, 1983), 3489.
[Bel85]Beltrametti, M., On the Chow group and the intermediate Jacobian of a conic bundle, Ann. Mat. Pura Appl. (4) 41 (1985), 331351.
[Ber07]Bernardara, M., Fourier–Mukai transforms of curves and principal polarizations, C. R. Math. Acad. Sci. Paris 345 (2007), 203208.
[BMMS12]Bernardara, M., Macrí, E., Mehrotra, S. and Stellari, P., A categorical invariant for cubic threefolds, Adv. Math. 229 (2012), 770803.
[Bol09]Bolognesi, M., A conic bundle degenerating on the Kummer surface, Math. Z. 261 (2009), 149168.
[Bon90]Bondal, A., Representations of associative algebras and coherent sheaves, Math. USSR, Izv. 34 (1990), 2342.
[BK90]Bondal, A. and Kapranov, M., Representable functors, Serre functors, and reconstructions, Math. USSR, Izv. 35 (1990), 519541.
[BO95]Bondal, A. and Orlov, D., Semiorthogonal decomposition for algebraic varieties, Preprint (1995), arXiv:alg-geom/9506012.
[Cas98]Casnati, G., Covers of algebraic varieties III: the discriminant of a cover of degree 4 and the trigonal construction, Trans. Amer. Math. Soc. 350 (1998), 13591378.
[CE96]Casnati, G. and Ekedahl, T., Covers of algebraic varieties. I. A general structure theorem, covers of degree $3, 4$ and Enriques surfaces, J. Algebraic Geom. 5 (1996), 439460.
[CG72]Clemens, C. H. and Griffiths, P. A., The intermedite Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281356.
[Don92]Donagi, R., The fibers of the Prym map, in Curves, Jacobians, and Abelian varieties, Proc. AMS-IMS-SIAM Jt. Summer Res. Conf. Schottky Probl., Contemp. Math. 136 (1992), 55125.
[vGee05]van Geemen, B., Some remarks on Brauer groups of $K 3$ surfaces, Adv. Math. 197 (2005), 222247.
[Har92]Harris, J., Algebraic geometry. A first course, Graduate Texts in Mathematics, vol. 133 (Spinger, Berlin, 1992).
[Huy06]Huybrechts, D., Fourier–Mukai transforms in algebraic geometry, Oxford Mathematical Monographs (The Clarendon Press, Oxford University Press, Oxford, 2006).
[IM00]Iliev, A. and Markushevich, D., The Abel–Jacobi map for cubic threefold and periods of Fano threefolds of degree 14, Doc. Math. 5 (2000), 2347.
[Isk82]Iskovskikh, V. A., Congruences of conics in ${P}^{3} $ (Russian–English summary), Vestnik Moskov. Univ. Ser. I Mat. Mekh. 121 (1982), 5762 (English translation in Moscow Univ. Math. Bull. 37 (1982), 67–73).
[Isk87]Iskovskikh, V. A., On the rationality problem for conic bundles, Duke Math. J. 54 (1987), 271294.
[Kap88]Kapranov, M. M., On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math. 92 (1988), 479508.
[Kat09]Katzarkov, L., Generalized homological mirror symmetry, superschemes and nonrationality, in Special metrics and supersymmetry. Lectures given in the workshop on geometry and physics: special metrics and supersymmetry, Bilbao, 29–31 May 2008, AIP Conference Proceedings, vol. 1093 (American Institute of Physics, Melville, NY, 2009), 92131.
[Kat10]Katzarkov, L., Generalized homological mirror symmetry and rationality questions, in Cohomological and geometric approaches to rationality problems, Progress in Mathematics, vol. 282 (Birkhäuser, Boston, MA, 2010), 163208.
[Kuz04]Kuznetsov, A., Derived categories of cubic and ${V}_{14} $ threefolds, Proc. Steklov Inst. Math. 246 (2004), 171194 (English translation of Tr. Mat. Inst. Steklova 246 (2004), 183–207).
[Kuz08]Kuznetsov, A., Derived categories of quadric fibrations and intersections of quadrics, Adv. Math. 218 (2008), 13401369.
[Kuz09]Kuznetsov, A., Derived categories of Fano threefolds, Proc. Steklov Inst. Math. 264 (2009), 110122.
[Kuz10]Kuznetsov, A., Derived categories of cubic fourfolds, in Cohomological and geometric approaches to rationality problems, Progress in Mathematics, vol. 282 (Birkhäuser, Boston, MA, 2010), 163208.
[Mer81]Merkur’ev, A. S., On the norm residue symbol of degree 2, Dokl. Akad. Nauk SSSR 261 (1981), 542547 (English translation in Soviet Math. Doklady 24 (1981), 546–551).
[Mum74]Mumford, D., Prym varieties I, in Contribution to analysis (Academic Press, New York, NY, 1974).
[Mur90]Murre, J. P., On the motive of an algebraic surface, J. Reine Angew. Math. 409 (1990), 190204.
[NS09]Nagel, J. and Saito, M., Relative Chow–Künneth decomposition for conic bundles and Prym varieties, Int. Math. Res. Not. IMRN (2009), 29783001.
[Orl93]Orlov, D. O., Projective bundles, monoidal transformations and derived categories of coherent sheaves, Izv. Math. 41 (1993), 133141.
[Orl03]Orlov, D. O., Derived categories of coherent sheaves and equivalences between them, Russian Math. Surveys 58 (2003), 511591.
[Orl05]Orlov, D. O., Derived categories of coherent sheaves and motives, Russian Math. Surveys 60 (2005), 12421244.
[Pan84]Panin, I. A., Rationality of bundles of conics with degenerate curve of degree five and even theta-characteristic, J. Sov. Math. 24 (1984), 449452 (Russian original in Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 103 (1980), 100–106).
[Rec74]Recillas, S., Jacobians of curves with ${ g}_{4}^{1} $’s are the Pryms of trigonal curves, Bol. Soc. Mat. Mexicana (2) 19 (1974), 913.
[Sar81]Sarkisov, V. G., Birational automorphisms of conic bundles, Math. USSR Izv. 17 (1981), 177202.
[Sar82]Sarkisov, V. G., On conic bundle structures, Math. USSR Izv. 20 (1982), 355390.
[Sch94]Scholl, A. J., Classical motives, in Motives, Seattle, WA, 20 July–2 August 1991, Proceedings of Symposia in Pure Mathematics, vol. 55.1 (Part 1) (American Mathematical Society, Providence, RI, 1994), 163187.
[Sho84]Shokurov, V. V., Prym varieties: theory and applications, Math. USSR Izv. 23 (1984), 83147.
[Ver87]Verra, A., The fibre of the Prym map in genus three, Math. Ann. 276 (1987), 433448.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

Derived categories and rationality of conic bundles

  • Marcello Bernardara (a1) (a2) and Michele Bolognesi (a3)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed