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Derived category of toric varieties with small Picard number

  • Research Article
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Central European Journal of Mathematics

Abstract

This paper aims to construct a full strongly exceptional collection of line bundles in the derived category D b(X), where X is the blow up of ℙn−r×ℙr along a multilinear subspace ℙn−r−1×ℙr−1 of codimension 2 of ℙn−r×ℙr. As a main tool we use the splitting of the Frobenius direct image of line bundles on toric varieties.

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References

  1. Batyrev V.V., On the classification of smooth projective toric varieties, Tohoku Math. J., 1991, 43(4), 569–585

    Article  MathSciNet  MATH  Google Scholar 

  2. Beilinson A.A., Coherent sheaves on ℙn and problems of linear algebra, Funct. Anal. Appl., 1978, 12(3), 214–216

    Article  MathSciNet  Google Scholar 

  3. Bondal A.I., Representations of associative algebras and coherent sheaves, Math. USSR-Izv., 1990, 34(1), 23–42

    Article  MathSciNet  MATH  Google Scholar 

  4. Bondal A.I., Derived categories of toric varieties, In: Convex and Algebraic Geometry, Oberwolfach, January 29–February 4, 2006, Oberwolfach Rep., 2006, 3(5), 284–286

    Google Scholar 

  5. Bondal A., Orlov D., Derived categories of coherent sheaves, In: Proceedings of the International Congress of Mathematicians, 2, Beijing, August 20–28, 2002, Higher Education Press, Beijing, 2002, 47–56

    Google Scholar 

  6. Costa L., Miró-Roig R.M., Tilting sheaves on toric varieties, Math. Z., 2004, 248(4), 849–865

    Article  MathSciNet  MATH  Google Scholar 

  7. Costa L., Miró-Roig R.M., Derived categories of projective bundles, Proc. Amer. Math. Soc., 2005, 133(9), 2533–2537

    Article  MathSciNet  MATH  Google Scholar 

  8. Costa L., Miró-Roig R.M., Frobenius splitting and Derived category of toric varieties, Illinois J. Math., 2010, 54(2), 649–669

    MathSciNet  MATH  Google Scholar 

  9. Costa L., Miró-Roig R.M., Derived category of fibrations, Math. Res. Lett., 2011, 18(3), 425–432

    MathSciNet  MATH  Google Scholar 

  10. Fulton W., Introduction to Toric Varieties, Ann. of Math. Stud., 131, Princeton University Press, Princeton, 1993

    MATH  Google Scholar 

  11. Funch Thomsen J., Frobenius direct images of line bundles on toric varieties, J. Algebra, 2000, 226(2), 865–874

    Article  MathSciNet  MATH  Google Scholar 

  12. Gorodentsev A.L., Kuleshov S.A., Helix theory, Mosc. Math. J., 2004, 4(2), 377–440

    MathSciNet  MATH  Google Scholar 

  13. Hille L., Perling M., Exceptional sequences of invertible sheaves on rational surfaces, Compos. Math., 2011, 147(4), 1230–1280

    Article  MathSciNet  MATH  Google Scholar 

  14. Kapranov M.M., On the derived category of coherent sheaves on Grassmann manifolds, Math. USSR-Izv., 1985, 24(1), 183–192

    Article  MATH  Google Scholar 

  15. Kapranov M.M., On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math., 1988, 92(3), 479–508

    Article  MathSciNet  MATH  Google Scholar 

  16. Kawamata Y., Derived categories of toric varieties, Michigan Math. J., 2006, 54(3), 517–535

    Article  MathSciNet  MATH  Google Scholar 

  17. King A., Tilting bundles on some rational surfaces, preprint available at http://www.maths.bath.ac.uk/~masadk/papers/

  18. Kuznetsov A.G., Derived category of Fano threefolds, Proc. Steklov Inst. Math., 2009, 264(1), 110–122

    Article  MathSciNet  Google Scholar 

  19. Lason M., Michałek M., On the full, strongly exceptional collections on toric varieties with Picard number three, Collect. Math., 2011, 62(3), 275–296

    Article  MathSciNet  MATH  Google Scholar 

  20. Oda T., Convex Bodies and Algebraic Geometry, Ergeb. Math. Grenzgeb., 15, Springer, Berlin, 1988

    MATH  Google Scholar 

  21. Orlov D.O., Projective bundles, monoidal transformations, and derived categories of coherent sheaves, Russian Acad. Sci. Izv. Math., 1993, 41(1), 133–141

    MathSciNet  Google Scholar 

  22. Samokhin A., Some remarks on the derived categories of coherent sheaves on homogeneous spaces, J. Lond. Math. Soc., 2007, 76(1), 122–134

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Facultat de Matemàtiques, Departament d’Algebra i Geometria, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007, Barcelona, Spain

    Laura Costa & Rosa Maria Miró-Roig

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  1. Laura Costa
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  2. Rosa Maria Miró-Roig
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Correspondence to Laura Costa.

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Costa, L., Miró-Roig, R.M. Derived category of toric varieties with small Picard number. centr.eur.j.math. 10, 1280–1291 (2012). https://doi.org/10.2478/s11533-012-0060-4

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  • DOI: https://doi.org/10.2478/s11533-012-0060-4

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