Linear series on a general k-gonal curve

Article

Abstract

This paper is a contribution towards a Brill-Noether theory for the moduli space of smooth &-gonal curves of genusg. Specifically, we prove the existence of certain special divisors on a generalk-gonal curveC of genusg, and we detect an irreducible component of the “expected” dimension in the varietyW r d (C), (r ≤k — 2) of special divisors ofC. The latter induces a new proof of the existence theorem for special divisors on a smooth curve.

References

  1. [1]
    E. Arbarello etM. Cornalba, Su una congettura di Petri.Comment. Math. Helv. 56(1981), 1–38.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    —,Footnotes to a paper of Beniamino Segre.Math. Ann. 256 (1981), 341–362.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    E. Arbarello,M. Cornalba,Ph. Griffiths, andJ. Harris,Geometry of algebraic curves. I. Springer 1985.Google Scholar
  4. [4]
    E. Ballico, A remark on linear series on general &-gonal curves.Boll. U.M.I. (7), 3-A (1989), 195–197.Google Scholar
  5. [5]
    —, On special linear systems on curves.Comm. in Algebra 18 (1990), 279–284.MATHCrossRefGoogle Scholar
  6. [6]
    M. Coppens, Brill-Noether theory for non-special linear systems.Compos. Math. 97 (1995), 17–27.MATHMathSciNetGoogle Scholar
  7. [7]
    —, Brill-Noether theory for non-special linear systems II: Connectedness and irreducibility.Geom. Dedic. 68 (1997), 169–185.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    M. Coppens, C. Keem, andG. Martens, Primitive linear series on curves.manuscr. math. 77 (1992), 237–264.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    —, The primitive length of a general &-gonal curve.Indag. Math., N. S.5 (1994), 145–159.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    M. Coppens andG. Martens, Linear series on 4-gonal curves. To appear in:Math. Nachr. (1999)Google Scholar
  11. [11]
    D. Eisenbud andJ. Harris, Divisors on general curves and cuspidal rational curves.Invent. Math. 74 (1983), 371–418.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    W. Fulton andR. Lazarsfeld, On the connectedness of degeneracy loci and special divisors.Acta Math. 146 (1984), 271–283.CrossRefMathSciNetGoogle Scholar
  13. [13]
    A. Grothendieck,Technique de descente et thèorémes d’existence en géométrie algébrique V. Les schémas de Picard. Sém. Bourbaki232 1961/62.Google Scholar
  14. [14]
    G. Kempf,Schubert methods with an application to algebraic curves. Publ. Math. Centrum, Amsterdam 1972.Google Scholar
  15. [15]
    S. Kleiman andD. Laksov, On the existence of special divisors.Amer. J. Math 94 (1972), 431–436.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    —, Another proof of the existence of special divisors.Acta Math. 132 (1974), 163–176.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    A. Maroni, Le série lineari speciali sulle curve trigonali.Ann. di mat. (4)25 (1946), 341–354.CrossRefMathSciNetGoogle Scholar
  18. [18]
    T. Meis,Die minimale Blätterzahl der Konkretisierung einer kompakten Riemannschen Fläche. Schriftenreihe des Math. Inst. der Univ. Münster16 1960.Google Scholar
  19. [19]
    G. Martens andF.-O. Schreyer, Line bundles and syzygies of trigonal curves.Abh. Math. Sem. Univ. Hamburg 56 (1986), 169–189.MATHMathSciNetCrossRefGoogle Scholar
  20. [20]
    B. Segre, Sui moduli delle curve poligonali, e sopra un complemento al teorema di existenza di Riemann.Math. Ann. 100 (1928), 537–551.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Mathematische Seminar 1999

Authors and Affiliations

  1. 1.Katholieke Industriele Hogeschool der KempenCampus H. I. KempenGeelBelgium
  2. 2.Mathematisches InstitutUniversität Erlangen-NürnbergErlangenGermany

Personalised recommendations