Skip to main content

A note on gonality of curves on general hypersurfaces

Abstract

This short paper concerns the existence of curves with low gonality on smooth hypersurfaces \(X\subset \mathbb {P}^{n+1}\). After reviewing a series of results on this topic, we report on a recent progress we achieved as a product of the Workshop Birational geometry of surfaces, held at University of Rome “Tor Vergata” on January 11th–15th, 2016. In particular, we obtained that if \(X\subset \mathbb {P}^{n+1}\) is a very general hypersurface of degree \(d\geqslant 2n+2\), the least gonality of a curve \(C\subset X\) passing through a general point of X is \(\mathrm {gon}(C)=d-\left\lfloor \frac{\sqrt{16n+1}-1}{2}\right\rfloor \), apart from some exceptions we list.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Alzati, A., Pirola, G.P.: On abelian subvarieties generated by symmetric correspondences. Math. Z. 205, 333–342 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Ballico, E.: On the gonality of curves in \(\mathbb{P}^n\). Comment. Math. Univ. Carolin. 38, 177–186 (1997)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Ballico, E.: Gonality of reduced curves on a smooth surface with generically spanned anticanonical line bundle. Int. J. Pure Appl. Math. 55, 105–108 (2009)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Basili, B.: Indice de Clifford des intersections complètes de l’espace. Bull. Soc. Math. France 124, 61–95 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Bastianelli, F.: On symmetric products of curves. Trans. Am. Math. Soc. 364, 2493–2519 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Bastianelli, F., Cortini, R., De Poi, P.: The gonality theorem of Noether for hypersurfaces. J. Algebraic Geom. 23, 313–339 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Bastianelli, F., De Poi, P., Ein, L., Lazarsfeld, R. Ullery, B.: Measures of irrationality for hypersurfaces of large degree. Compositio Math. (2016) (to appear)

  8. 8.

    Borcea, C.: Deforming varieties of k-planes of projective complete intersections. Pacific J. Math. 143, 25–36 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Chiantini, L., Lopez, A.F.: Focal loci of families and the genus of curves on surfaces. Proc. Am. Math. Soc. 127, 3451–3459 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Choi, Y., Kim, S.: Gonality and Clifford index of projective curves on ruled surfaces. Proc. Am. Math. Soc. 140, 393–402 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Ciliberto, C.: Osservazioni su alcuni classici teoremi di unirazionalità per ipersuperficie e complete intersezioni algebriche proiettive. Ricerche Mat. 29, 175–191 (1980)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Ciliberto, C.: Alcune applicazioni di un classico procedimento di Castelnuovo, Seminari di geometria, 1982–1983 (Bologna, 1982/1983), pp. 17–43. University of Stud. Bologna, Bologna (1984)

  13. 13.

    Ciliberto, C., Flamini, F., Zaidenberg, M.: Genera of curves on a very general surface in \(\mathbb{P}^3\). Int. Math. Res. Not. IMRN 22, 12177–12205 (2015)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Ciliberto, C., Flamini, F., Zaidenberg, M.: Gaps for geometric genera. Arch. Math. (Basel) 106, 531–541 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Ciliberto, C., Knutsen, A.L.: On \(k\)-gonal loci in Severi varieties on general K3 surfaces and rational curves on hyperkähler manifolds. J. Math. Pures Appl. 101, 473–494 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Clemens, H.: Curves on generic hypersurfaces. Ann. Sci. École Norm. Sup. 19, 629–636 (1986)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Debarre, O., Manivel, L.: Sur la varieté des espaces lineaires contenus dans une intersection complete. Math. Ann. 312, 549–574 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    De Poi, P.: On first order congruences of lines of \(\mathbb{P}^4\) with a fundamental curve. Manuscripta Math. 106, 101–116 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Ein, L.: Subvarieties of generic complete intersections. Invent. Math. 94, 163–169 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Ein, L.: Subvarieties of generic complete intersections II. Math. Ann. 289, 465–471 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Farkas, G.: Brill-Noether loci and the gonality stratification of \(M_g\). J. Reine Angew. Math. 539, 185–200 (2001)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Flamini, F., Knutsen, A.L., Pacienza, G.: Singular curves on a K3 surface and linear series on their normalizations. Int. J. Math. 18, 671–693 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Flamini, F., Knutsen, A.L., Pacienza, G.: On families of rational curves in the Hilbert square of a surface (with an appendix by Edoardo Sernesi). Michigan Math. J. 58, 639–682 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Flamini, F., Knutsen, A.L., Pacienza, G., Sernesi, E.: Nodal curves with general moduli on K3 surfaces. Comm. Algebra 36, 3955–3971 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Harris, J., Mazur, B., Pandharipande, R.: Hypersurfaces of low degree. Duke Math. J. 95, 125–160 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Harris, J., Roth, M., Starr, J.: Rational curves on hypersurfaces of low degree. J. Reine Angew. Math. 571, 73–106 (2004)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Hartshorne, R.: Clifford index of ACM curves in \(\mathbb{P}^3\). Milan J. Math. 70, 209–221 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Hartshorne, R., Schlesinger, E.: Gonality of a general ACM curve in \(\mathbb{P}^3\). Pacific J. Math. 251, 269–313 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Kawaguchi, R.: The gonality and the Clifford index of curves on a toric surface. J. Algebra 449, 660–686 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Knutsen, A.L., Lelli—Chiesa, M., Mongardi, G.: Severi Varieties and Brill-Noether theory of curves on abelian surfaces. J. Reine Angew. Math. (to appear)

  31. 31.

    Knutsen, A.L., Lopez, A.F.: Brill-Noether theory for curves on Enriques surfaces, I: the positive cone and gonality. Math. Z. 261, 659–690 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Knutsen, A.L., Lopez, A.F.: Brill-Noether theory of curves on Enriques surfaces, II The Clifford index. Manuscripta Math. 147, 193–237 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Kollár, J.: Rational Curves on Algebraic Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3, vol. 32. Springer-Verlag, Berlin (1996)

    Book  Google Scholar 

  34. 34.

    Lazarsfeld, R.: Brill-Noether-Petri without degenerations. J. Differ. Geom. 23, 299–307 (1986)

    MathSciNet  Article  MATH  Google Scholar 

  35. 35.

    Lopez, A.F., Pirola, P.: On the curves through a general point of a smooth surface in \(\mathbb{P}^3\). Math. Z. 19, 93–106 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    Marcucci, V.O.: On the genus of curves in a Jacobian variety. Ann. Sc. Norm. Super. Pisa Cl. Sci. 12, 735–754 (2013)

  37. 37.

    Martens, G.: The gonality of curves on a Hirzebruch surface. Arch. Math. (Basel) 67, 349–352 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  38. 38.

    Mori, S., Mukai, S.: The uniruledness of the moduli space of curves of genus 11, in Algebraic geometry (Tokyo/Kyoto). In: Lecture Notes in Mathematics, vol. 1016, pp. 334–353. Springer-Verlag, Berlin (1983)

  39. 39.

    Morin, U.: Sull’insieme degli spazi lineari contenuti in una ipersuperficie algebrica, Atti Accad. Naz. Lincei, Rend., Cl. Sci. Fis. Mat. Nat. 24, 188–190 (1936)

  40. 40.

    Mumford, D.: Prym varieties I. In: Alfors, L.V., et al. (eds.) Contributions to analysis, pp. 325–350. Academic Press, New York (1974)

    Chapter  Google Scholar 

  41. 41.

    Naranjo, J.C., Pirola, G.P.: On the genus of curves in the generic Prym variety. Indag. Math. (N.S.) 5, 101–105 (1994)

  42. 42.

    Pacienza, G.: Rational curves on general projective hypersurfaces. J. Algebraic Geom. 12, 245–267 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  43. 43.

    Pacienza, G.: Subvarieties of general type on a general projective hypersurface. Trans. Am. Math. Soc. 356, 2649–2661 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  44. 44.

    Pirola, G.P.: Curves on generic Kummer varieties. Duke Math. J. 59, 701–708 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  45. 45.

    Predonzan, A.: Intorno agli \(S_k\) giacenti sulla varietà intersezione completa di più forme, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. 5, 238–242 (1948)

    MathSciNet  MATH  Google Scholar 

  46. 46.

    Ramponi, M.: Gonality and Clifford index of curves on elliptic K3 surfaces with Picard number two. Arch. Math. (Basel) 106, 355–362 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  47. 47.

    Segre, B.: Intorno agli Sk che appartengono alle forme generali di dato ordine. Atti Accad. Naz. Lincei, Rend., Cl. Sci. Fis. Mat. Nat. 4, 261–265, 341–346 (1948)

  48. 48.

    Takahashi, T.: Galois morphism computing the gonality of a nonsingular projective curve on a Hirzebruch surface. J. Pure Appl. Algebra 216, 12–19 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  49. 49.

    Voisin, C.: On a conjecture of Clemens on rational curves on hypersurfaces. J. Differ. Geom. 44, 200–213 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  50. 50.

    Voisin, C.: A correction on “On a conjecture of Clemens on rational curves on hypersurfaces”. J. Differ. Geom. 49, 601–611 (1998)

    Article  MATH  Google Scholar 

  51. 51.

    Xu, G.: Subvarieties of general hypersurfaces in projective space. J. Differ. Geom. 39, 139–172 (1994)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Flaminio Flamini.

Additional information

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC Grant Agreement No. 307119, MIUR FIRB 2012 “Spazi di moduli e applicazioni”, MIUR PRIN 2010–2011 “Geometria delle varietà algebriche”, and INdAM (GNSAGA).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bastianelli, F., Ciliberto, C., Flamini, F. et al. A note on gonality of curves on general hypersurfaces. Boll Unione Mat Ital 11, 31–38 (2018). https://doi.org/10.1007/s40574-017-0129-x

Download citation

Keywords

  • General Hypersurface
  • Rational Curves
  • Basic Questions Concern
  • Arbitrary Smooth Variation
  • Smooth Quadric