Advertisement

Bollettino dell'Unione Matematica Italiana

, Volume 11, Issue 1, pp 31–38 | Cite as

A note on gonality of curves on general hypersurfaces

  • Francesco Bastianelli
  • Ciro Ciliberto
  • Flaminio Flamini
  • Paola Supino
Article

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.

References

  1. 1.
    Alzati, A., Pirola, G.P.: On abelian subvarieties generated by symmetric correspondences. Math. Z. 205, 333–342 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ballico, E.: On the gonality of curves in \(\mathbb{P}^n\). Comment. Math. Univ. Carolin. 38, 177–186 (1997)MathSciNetzbMATHGoogle 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)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Basili, B.: Indice de Clifford des intersections complètes de l’espace. Bull. Soc. Math. France 124, 61–95 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bastianelli, F.: On symmetric products of curves. Trans. Am. Math. Soc. 364, 2493–2519 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bastianelli, F., Cortini, R., De Poi, P.: The gonality theorem of Noether for hypersurfaces. J. Algebraic Geom. 23, 313–339 (2014)MathSciNetCrossRefzbMATHGoogle 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) Google Scholar
  8. 8.
    Borcea, C.: Deforming varieties of k-planes of projective complete intersections. Pacific J. Math. 143, 25–36 (1990)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)Google Scholar
  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)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Ciliberto, C., Flamini, F., Zaidenberg, M.: Gaps for geometric genera. Arch. Math. (Basel) 106, 531–541 (2016)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Clemens, H.: Curves on generic hypersurfaces. Ann. Sci. École Norm. Sup. 19, 629–636 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Debarre, O., Manivel, L.: Sur la varieté des espaces lineaires contenus dans une intersection complete. Math. Ann. 312, 549–574 (1998)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ein, L.: Subvarieties of generic complete intersections. Invent. Math. 94, 163–169 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ein, L.: Subvarieties of generic complete intersections II. Math. Ann. 289, 465–471 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Farkas, G.: Brill-Noether loci and the gonality stratification of \(M_g\). J. Reine Angew. Math. 539, 185–200 (2001)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Harris, J., Mazur, B., Pandharipande, R.: Hypersurfaces of low degree. Duke Math. J. 95, 125–160 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Harris, J., Roth, M., Starr, J.: Rational curves on hypersurfaces of low degree. J. Reine Angew. Math. 571, 73–106 (2004)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Hartshorne, R.: Clifford index of ACM curves in \(\mathbb{P}^3\). Milan J. Math. 70, 209–221 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Hartshorne, R., Schlesinger, E.: Gonality of a general ACM curve in \(\mathbb{P}^3\). Pacific J. Math. 251, 269–313 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kawaguchi, R.: The gonality and the Clifford index of curves on a toric surface. J. Algebra 449, 660–686 (2016)MathSciNetCrossRefzbMATHGoogle 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) Google Scholar
  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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kollár, J.: Rational Curves on Algebraic Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3, vol. 32. Springer-Verlag, Berlin (1996)CrossRefGoogle Scholar
  34. 34.
    Lazarsfeld, R.: Brill-Noether-Petri without degenerations. J. Differ. Geom. 23, 299–307 (1986)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)Google Scholar
  37. 37.
    Martens, G.: The gonality of curves on a Hirzebruch surface. Arch. Math. (Basel) 67, 349–352 (1996)MathSciNetCrossRefzbMATHGoogle 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)Google Scholar
  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)Google Scholar
  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)CrossRefGoogle 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)Google Scholar
  42. 42.
    Pacienza, G.: Rational curves on general projective hypersurfaces. J. Algebraic Geom. 12, 245–267 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Pacienza, G.: Subvarieties of general type on a general projective hypersurface. Trans. Am. Math. Soc. 356, 2649–2661 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Pirola, G.P.: Curves on generic Kummer varieties. Duke Math. J. 59, 701–708 (1989)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)Google Scholar
  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)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Voisin, C.: On a conjecture of Clemens on rational curves on hypersurfaces. J. Differ. Geom. 44, 200–213 (1996)MathSciNetCrossRefzbMATHGoogle 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)CrossRefzbMATHGoogle Scholar
  51. 51.
    Xu, G.: Subvarieties of general hypersurfaces in projective space. J. Differ. Geom. 39, 139–172 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Unione Matematica Italiana 2017

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di BariBariItaly
  2. 2.Dipartimento di MatematicaUniversità degli Studi di Roma “Tor Vergata”RomaItaly
  3. 3.Dipartimento di Matematica e FisicaUniversità degli Studi “Roma Tre”RomaItaly

Personalised recommendations