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Linear series on a general k-gonal curve

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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.

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References

  1. E. Arbarello etM. Cornalba, Su una congettura di Petri.Comment. Math. Helv. 56(1981), 1–38.

    Article  MATH  MathSciNet  Google Scholar 

  2. —,Footnotes to a paper of Beniamino Segre.Math. Ann. 256 (1981), 341–362.

    Article  MATH  MathSciNet  Google Scholar 

  3. E. Arbarello,M. Cornalba,Ph. Griffiths, andJ. Harris,Geometry of algebraic curves. I. Springer 1985.

  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. —, On special linear systems on curves.Comm. in Algebra 18 (1990), 279–284.

    Article  MATH  Google Scholar 

  6. M. Coppens, Brill-Noether theory for non-special linear systems.Compos. Math. 97 (1995), 17–27.

    MATH  MathSciNet  Google Scholar 

  7. —, Brill-Noether theory for non-special linear systems II: Connectedness and irreducibility.Geom. Dedic. 68 (1997), 169–185.

    Article  MATH  MathSciNet  Google Scholar 

  8. M. Coppens, C. Keem, andG. Martens, Primitive linear series on curves.manuscr. math. 77 (1992), 237–264.

    Article  MATH  MathSciNet  Google Scholar 

  9. —, The primitive length of a general &-gonal curve.Indag. Math., N. S.5 (1994), 145–159.

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Coppens andG. Martens, Linear series on 4-gonal curves. To appear in:Math. Nachr. (1999)

  11. D. Eisenbud andJ. Harris, Divisors on general curves and cuspidal rational curves.Invent. Math. 74 (1983), 371–418.

    Article  MATH  MathSciNet  Google Scholar 

  12. W. Fulton andR. Lazarsfeld, On the connectedness of degeneracy loci and special divisors.Acta Math. 146 (1984), 271–283.

    Article  MathSciNet  Google Scholar 

  13. A. Grothendieck,Technique de descente et thèorémes d’existence en géométrie algébrique V. Les schémas de Picard. Sém. Bourbaki232 1961/62.

  14. G. Kempf,Schubert methods with an application to algebraic curves. Publ. Math. Centrum, Amsterdam 1972.

    Google Scholar 

  15. S. Kleiman andD. Laksov, On the existence of special divisors.Amer. J. Math 94 (1972), 431–436.

    Article  MATH  MathSciNet  Google Scholar 

  16. —, Another proof of the existence of special divisors.Acta Math. 132 (1974), 163–176.

    Article  MATH  MathSciNet  Google Scholar 

  17. A. Maroni, Le série lineari speciali sulle curve trigonali.Ann. di mat. (4)25 (1946), 341–354.

    Article  MathSciNet  Google Scholar 

  18. T. Meis,Die minimale Blätterzahl der Konkretisierung einer kompakten Riemannschen Fläche. Schriftenreihe des Math. Inst. der Univ. Münster16 1960.

  19. G. Martens andF.-O. Schreyer, Line bundles and syzygies of trigonal curves.Abh. Math. Sem. Univ. Hamburg 56 (1986), 169–189.

    Article  MATH  MathSciNet  Google Scholar 

  20. B. Segre, Sui moduli delle curve poligonali, e sopra un complemento al teorema di existenza di Riemann.Math. Ann. 100 (1928), 537–551.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Katholieke Industriele Hogeschool der Kempen, Campus H. I. Kempen, Kleinhoefstraat 4, B - 2440, Geel, Belgium

    Marc Coppens

  2. Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstr. 1 1/2, D - 91054, Erlangen, Germany

    Gerriet Martens

Authors
  1. Marc Coppens
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  2. Gerriet Martens
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Correspondence to Marc Coppens or Gerriet Martens.

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Coppens, M., Martens, G. Linear series on a general k-gonal curve. Abh.Math.Semin.Univ.Hambg. 69, 347–371 (1999). https://doi.org/10.1007/BF02940885

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  • DOI: https://doi.org/10.1007/BF02940885

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