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Geometriae Dedicata

, Volume 108, Issue 1, pp 1–14 | Cite as

Some Remarks about the FM-partners of K3 Surfaces with Picard Numbers 1 and 2

  • Paolo Stellari
Article

Abstract

In this paper we describe some results about K3 surfaces with Picard number 1 and 2. In particular, we give a new simple proof of a theorem due to Oguiso which shows that, given an integer N, there is a K3 surface with Picard number 2 and at least N non-isomorphic FM-partners. We describe also the Mukai vectors of the moduli spaces associated to the FM-partners of K3 surfaces with Picard number 1.

K3 surfaces Fourier–Mukai partners 

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References

  1. 1.
    Barth, W., Peters, C. and Van de Ven, A.: Compact Complex Surfaces, Springer-Verlag, Berlin, 1984Google Scholar
  2. 2.
    Cassels, J. W. S.: Rational Quadratic Forms, Academic Press, New York, 1978.Google Scholar
  3. 3.
    Hosono, S., Lian, B. H., Oguiso, K. and Yau, S. T.: Autoequivalences of derived categoryof a K3 surface and monodromy transformations, to appear in J. Algebra. Geom. Google Scholar
  4. 4.
    Hosono, S., Lian, B. H., Oguiso, K. and Yau, S. T.: Fourier—Mukai number of a K3 surface, math.AG/0202014.Google Scholar
  5. 5.
    Hosono, S., Lian, B. H., Oguiso, K. and Yau, S. T.: Fourier—Mukai partners of a K3 surface of Picard number one, In: Vector Bundles and Representation Theory (Columbia, 2002), Contemp. Math. 322, Amer. Math. Soc., Providence, 2003, pp. 43–55.MathSciNetGoogle Scholar
  6. 6.
    Hosono, S., Lian, B. H., Oguiso, K. and Yau, S.-T.: Kummer structures on a K3 surface — An old question of T. Shioda, Duke Math. J. 120 (2003), 635–647.CrossRefMathSciNetGoogle Scholar
  7. 7.
    Iwaniec, H.: Almost-primes represented by quadratic polynomials, Invent. Math. 47 (1978), 171–188.CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Morrison, D. R.: On K3 surfaces with large Picard number, Invent. Math. 75 (1984), 105–121.CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Mukai, S.: Duality of polarized K3 surfaces, In: New Trends in Algebraic Geometry (Warwick, 1996), London Math. Soc. Lecture Note Ser. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 311–326.Google Scholar
  10. 10.
    Mukai, S.: On the moduli space of bundles on K3 surfaces, I, In: Vector Bundles on Algebraic Varieties, Bombay, 1984.Google Scholar
  11. 11.
    Mukai, S.: Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent. Math. 77 (1984), 101–116.MATHMathSciNetGoogle Scholar
  12. 12.
    Nikulin, V. V.: Integral symmetric bilinear forms and some of their applications, Math. USSR Izv. 14 (1980), 103–167.CrossRefMATHGoogle Scholar
  13. 13.
    Oguiso, K.: K3 surfaces via almost-prime, Math. Res. Lett. 9 (2002), 47–63 (also in math.AG/0110282).MATHMathSciNetGoogle Scholar
  14. 14.
    Orlov, D.: Equivalences of derived categories and K3 surfaces, J. Math. Sci. 84 (1997), 1361–1381.MATHMathSciNetGoogle Scholar
  15. 15.
    Scattone, F.: On the compacti cation of moduli spaces for algebraic K3 surfaces, Mem. Amer. Math. Soc. 70 (1987) 374.MathSciNetGoogle Scholar
  16. 16.
    Shioda, T.: The period map of abelian surfaces, J. Fac. Sci. Univ. Tokyo, Sect IA 25 (1978), 47–59.MATHMathSciNetGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Paolo Stellari
    • 1
  1. 1.Department of MathematicsUniversità degli Studi di MilanoMilanItaly

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