Advertisement

manuscripta mathematica

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

Remarks on the nef cone on symmetric products of curves

  • F. BastianelliEmail author
Article

Abstract

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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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

Personalised recommendations