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Fano varieties of cubic fourfolds containing a plane

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Abstract

We prove that the Fano variety of lines of a generic cubic fourfold containing a plane is isomorphic to a moduli space of twisted stable complexes on a K3 surface. On the other hand, we show that the Fano varieties are always birational to moduli spaces of twisted stable coherent sheaves on a K3 surface. The moduli spaces of complexes and of sheaves are related by wall-crossing in the derived category of twisted sheaves on the corresponding K3 surface.

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References

  1. Arcara, D., Bertram, A.: Bridgeland-stable moduli spaces for K-trivial surfaces (with an appendix by M. Lieblich). J. Eur. Math. Soc. arXiv:0708.2247 (to appear)

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

    MathSciNet  MATH  Google Scholar 

  3. Beauville A.: Variétés Kähleriennes dont la première classe de Chern est nulle. J. Differ. Geom. 18, 755–782 (1983)

    MathSciNet  MATH  Google Scholar 

  4. Beauville A., Donagi R.: La variété des droites d’une hypersurface cubique de dimension 4. C. R. Acad. Sci. Paris Ser. I Math. 301, 703–706 (1985)

    MathSciNet  MATH  Google Scholar 

  5. Bernardara M., Macrì E., Mehrotra S., Stellari P.: A categorical invariant for cubic threefolds. Adv. Math. 229, 770–803 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bridgeland T.: Stability conditions on K3 surfaces. Duke Math. J. 141, 241–291 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bridgeland T.: Stability conditions on triangulated categories. Ann. Math. 166, 317–346 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Căldăraru, A.: Derived categories of twisted sheaves on Calabi-Yau manifolds, PhD-thesis, Cornell University (2000)

  9. Canonaco A., Stellari P.: Twisted Fourier–Mukai functors. Adv. Math. 212, 484–503 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. van Geemen B.: Some remarks on Brauer groups of K3 surfaces. Adv. Math. 197, 222–247 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Happel, D., Reiten, I., Smal  S.O.: Tilting in abelian categories and quasitilted algebras. Mem. Am. Math. Soc. 120 (1996)

  12. Hassett B.: Some rational cubic fourfolds. J. Algebra Geom. 8, 103–114 (1999)

    MathSciNet  MATH  Google Scholar 

  13. Hassett B.: Special cubic fourfolds. Compositio Math. 120, 1–23 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hassett B., Tschinkel Y.: Flops on holomorphic symplectic fourfolds and determinantal cubic hypersurfaces. J. Inst. Math. Jussieu 9, 125–153 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hassett B., Várilly-Alvarado A., Varilly P.: Transcendental obstructions to weak approximation on general K3 surfaces. Adv. Math. 228, 1377–1404 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Huybrechts, D.: A global Torelli theorem for hyperkähler manifolds (after Verbitsky). Séminaire Bourbaki. arXiv:1106.5573

  17. Huybrechts, D.: Fourier–Mukai transforms in algebraic geometry. Oxford Mathematical Monographs (2006)

  18. Huybrechts D.: The Kähler cone of a compact hyperkähler manifold. Math. Ann. 326, 499–513 (2003)

    MathSciNet  MATH  Google Scholar 

  19. Huybrechts D., Macrì E., Stellari P.: Stability conditions for generic K3 surfaces. Compositio Math. 144, 134–162 (2008)

    Article  MATH  Google Scholar 

  20. Huybrechts D., Stellari P.: Equivalences of twisted K3 surfaces. Math. Ann. 332, 901–936 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Huybrechts D., Stellari P.: Proof of Căldăraru’s conjecture. The 13th MSJ Inter. Research Inst.—Moduli Spaces and Arithmetic Geometry. Adv. Stud. Pure Math. 45, 31–42 (2006)

    MathSciNet  Google Scholar 

  22. Hulek K., Gritsenko V., Sankaran G.K.: Moduli spaces of irreducible symplectic manifolds. Compositio Math. 146, 404–434 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Inaba M.: Smoothness of the moduli space of complexes of coherent sheaves on an abelian or a projective K3 surface. Adv. Math. 227, 1399–1412 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kuznetsov, A.: Derived categories of cubic fourfolds. In: Cohomological and Geometric Approaches to Rationality Problems, Progr. Math. vol. 282. Birkhäuser, Boston, pp. 219–243 (2010)

  25. Kuznetsov A.: Derived categories of quadric fibrations and intersections of quadrics. Adv. Math. 218, 1340–1369 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kuznetsov A., Manivel L., Markushevich D.: Abel–Jacobi maps for hypersurfaces and non-commutative Calabi-Yau’s. Commun. Contemp. Math. 12, 373–416 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kuznetsov A., Markushevich D.: Symplectic structures on moduli spaces of sheaves via the Atiyah class. J. Geom. Phys. 59, 843–860 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Markman E.: On the monodromy of moduli spaces of sheaves on K3 surfaces. J. Algebraic Geom. 17, 29–99 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Markman E.: Integral constraints on the monodromy group of the hyperkähler resolution of a symmetric product of a K3 surface. Int. J. Math. 21, 169–223 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Milne J.S.: Étale Cohomology. Princeton Mathematical Series, vol. 33. Princeton University Press, Princeton (1980)

    Google Scholar 

  31. Nikulin V.V.: Integral symmetric bilinear forms and some of their applications. Math. USSR Izvestija 14, 103–167 (1980)

    Article  MATH  Google Scholar 

  32. Orlov D.: Projective bundles, monoidal transformations, and derived categories of coherent sheaves. Russian Acad. Sci. Izv. Math. 41, 133–141 (1993)

    MathSciNet  Google Scholar 

  33. Toda Y.: Moduli stacks and invariants of semistable objects on K3 surfaces. Adv. Math. 217, 2736–2781 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Verbitsky, M.: A global Torelli theorem for hyperkähler manifolds. arXiv:0908.4121

  35. Voisin C.: Théorème de Torelli pour les cubiques de \({{\mathbb{P}}^5}\) . Invent. Math. 86, 577–601 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  36. Yoshioka K.: Moduli spaces of twisted sheaves on a projective variety, in: Moduli spaces and arithmetic geometry. Adv. Stud. Pure Math. 45, 1–30 (2006)

    MathSciNet  Google Scholar 

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Author information

Author notes

  1. Emanuele Macrì

    Present address: Department of Mathematics, The Ohio State University, 231 W18th Avenue, Columbus, OH, 43210, USA

Authors and Affiliations

  1. Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT, 84112, USA

    Emanuele Macrì

  2. Mathematical Institute, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Emanuele Macrì

  3. Dipartimento di Matematica “F. Enriques”, Università Degli Studi di Milano, Via Cesare Saldini 50, 20133, Milan, Italy

    Paolo Stellari

Authors
  1. Emanuele Macrì
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  2. Paolo Stellari
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Correspondence to Paolo Stellari.

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Macrì, E., Stellari, P. Fano varieties of cubic fourfolds containing a plane. Math. Ann. 354, 1147–1176 (2012). https://doi.org/10.1007/s00208-011-0776-7

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  • DOI: https://doi.org/10.1007/s00208-011-0776-7

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