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, Volume 130, Issue 1, pp 113–120 | Cite as

Remarks on the nef cone on symmetric products of curves

  • F. BastianelliEmail author


Let C be a very general curve of genus g and let C (2) be its second symmetric product. This paper concerns the problem of describing the convex cone \({Nef\,(C^{(2)})_{\mathbb{R}}}\) of all numerically effective \({\mathbb{R}}\) -divisors classes in the Néron–Severi space \({N^1(C^{(2)})_{\mathbb{R}}}\) . In a recent work, Julius Ross improved the bounds on \({Nef\,(C^{(2)})_{\mathbb{R}}}\) in the case of genus five. By using his techniques and by studying the gonality of the curves lying on C (2), we give new bounds on the nef cone of C (2) when C is a very general curve of genus 5 ≤ g ≤ 8.

Mathematics Subject Classification (2000)

14C20 14H10 14Q10 


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  1. Arbarello E., Cornalba M., Griffiths P.A., Harris J.: Geometry of Algebraic Curves, vol. I. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles in Mathematical Sciences], 267. Springer-Verlag, New York (1985)Google Scholar
  2. Ciliberto C., Kouvidakis A.: On the symmetric product of a curve with general moduli. Geom. Dedicata 78(3), 327–343 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  3. Ein, L., Lazarsfeld, R.: Seshadri constants on smooth surfaces. Astérisque 218, 177–186 (1993). Journées de Géométrie Algébrique d’Orsay (Orsay, 1992)Google Scholar
  4. Griffiths P.A., Harris J.: Principles of Algebraic Geometry. Pure and Applied Mathematics. Wiley Interscience, New York (1978)Google Scholar
  5. Knutsen, A.L., Syzdek, W., Szemberg, T.: Moving curves and Seshadri constants. Math. Res. Lett. (to appear)Google Scholar
  6. Kouvidakis A.: Divisors on symmetric products of curves. Trans. Am. Math. Soc. 337(1), 117–128 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  7. Lazarsfeld, R.: Positivity in Algebraic Geometry, vol. I. Ergebnisse der Mathematik und ihrer Grenzebiete (3), 48, Springer-Verlag, Berlin (2004)Google Scholar
  8. Pirola G.P.: Curves on generic Kummer varieties. Duke Math. J. 59(3), 701–708 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  9. Ross J.: Seshadri constants on symmetric products of curves. Math. Res. Lett. 14(1), 63–75 (2007)zbMATHMathSciNetGoogle Scholar
  10. Strycharz-Szemberg B., Szemberg T.: Remarks on the Nagata conjecture. Serdica Math. J. 30(2–3), 405–430 (2004)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di PaviaPaviaItaly

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