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Weierstrass points and gap sequences for families of curves

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Arkiv för Matematik

Abstract

The theory of Weierstrass points and gap sequences for linear series on smooth curves is generalized to smooth families of curves with geometrically irreducible fibers, and over an arbitrary base scheme.

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References

  1. Altman, A. B. andKleiman, S. L.,Introduction to Grothendieck Duality Theory,Lecture Notes in Math. 146, Springer-Verlag, Berlin-Heidelberg, 1970.

    Book  MATH  Google Scholar 

  2. Arakelov, S. Yu., Families of algebraic curves with fixed degeneracies,Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 1269–1293 (Russian). English transl.: Math. USSR-Izv.5 (1971), 1277–1302.

    MathSciNet  Google Scholar 

  3. Arbarello, E., Weierstrass points and moduli of curves,Compositio Math. 29 (1974), 325–342.

    MathSciNet  Google Scholar 

  4. Arbarello, E., On subvarieties of the moduli space of curves of genusg defined in terms of Weierstrass points,Atti Acad. Naz. Lincei 15 (1978), 3–20.

    MathSciNet  Google Scholar 

  5. Bourbaki, N.,Algèbre I, Chapitres 1 à 3, Hermann, Paris, 1970.

    Google Scholar 

  6. Coppens, M., Weierstrass points with two prescribed nongaps,Pacific J. Math. 131 (1988), 71–104.

    MathSciNet  Google Scholar 

  7. Cukierman, F., Families of Weierstrass points,Duke Math. J. 58 (1989), 317–346.

    Article  MathSciNet  Google Scholar 

  8. Diaz, S.,Exceptional Weierstrass points and the divisor on moduli space that they define,Mem. Amer. Math. Soc. 327, Amer. Math. Soc., Providence, R. I., 1985.

    Google Scholar 

  9. Diaz, S., Deformations of exceptional Weierstrass points,Proc. Amer. Math. Soc. 96 (1986), 7–10.

    Article  MathSciNet  Google Scholar 

  10. Eisenbud, D. andHarris, J., Divisors on general curves and cuspidal rational curves,Invent. Math. 74 (1983), 371–418.

    Article  MathSciNet  Google Scholar 

  11. Eisenbud, D. andHarris, J., Limit series: Basic theory,Invent. Math. 85 (1986), 337–371.

    Article  MathSciNet  Google Scholar 

  12. Eisenbud, D. andHarris, J., When ramification points meet,Invent. Math. 87 (1987), 485–493.

    Article  MathSciNet  Google Scholar 

  13. Eisenbud, D. andHarris, J., Existence, decomposition, and limits of certain Weierstrass points,Invent. Math. 87 (1987), 495–515.

    Article  MathSciNet  Google Scholar 

  14. Eisenbud, D. andHarris, J., The monodromy of Weierstrass points,Invent. Math. 90 (1987), 333–341.

    Article  MathSciNet  Google Scholar 

  15. Eisenbud, D. andHarris, J., The Kodaira dimension of the moduli space of curves of genus≥23,Invent. Math. 90 (1987), 359–387.

    Article  MathSciNet  Google Scholar 

  16. Galbura, G., II wronskiano di un sistema di sezione,Ann. Mat. Pura Appl. 98 (1974), 349–355.

    Article  MathSciNet  Google Scholar 

  17. Gatto, L.,Weight sequences versus gap sequences at singular points of Gorenstein curves, Rapporto n 19/93 (Preprint), Politecnico di Torino, Italy, 1993.

    Google Scholar 

  18. Griffiths, P. andHarris, J.,Principles of Algebraic Geometry, John Wiley & Sons, New York, 1978.

    MATH  Google Scholar 

  19. Grothendieck, A., Rédigés avec la collaboration deDieudonné, J.,Éléments de Géométrie Algébrique IV 4, Inst. Hautes Études Sci. Publ. Math.32, 1967.

  20. Hartshorne, R.,Algebraic Geometry,Graduate Texts in Math. 52, Springer-Verlag, New York, 1977.

    MATH  Google Scholar 

  21. Hasse, H. andSchmidt, F. K., Noch eine Begründung der Theorie der höheren Differentialquotienten in einem algebraischen Funktionenkörper einer Unbestimmten,J. Reine Angew. Math. 177 (1937), 215–237.

    Google Scholar 

  22. Kleiman, S. L. andLønsted, K., Basics on families of hyperelliptic curves.,Compositio Math. 38 (1979), 83–111.

    MathSciNet  Google Scholar 

  23. Laksov, D., Weierstrass points on curves,Astérisque 87–88 (1981), 221–247.

    MathSciNet  Google Scholar 

  24. Laksov, D., Wronskians and Plücker formulas for linear systems on curves,Ann. Sci. École Norm. Sup. 17 (1984), 45–66.

    MathSciNet  Google Scholar 

  25. Laksov, D. andThorup, A., The Brill-Segre formula for families of curves, inEnumerative Algebraic Geometry: Proceedings of the 1989 Zeuthen Symposium (Kleiman, S. L. and Thorup, A., eds.), pp. 131–148,Contemporary Math. 123, Amer. Math. Soc., Providence, R. I., 1991.

    Google Scholar 

  26. Lax, R., Weierstrass points on the universal curve,Math. Ann. 216 (1975), 35–42.

    Article  MathSciNet  Google Scholar 

  27. Lax, R. andWidland, C., Gap sequences at a singularity,Pacific J. Math. 150 (1991), 111–122.

    MathSciNet  Google Scholar 

  28. Matsumura, H.,Commutative Algebra, Math. Lecture Notes Series, Benjamin, New York, 1970.

    MATH  Google Scholar 

  29. Matsumura, H.,Commutative Ring Theory, Advanced Mathematics8, Cambridge University Press, Cambridge, 1990.

    Google Scholar 

  30. Matzat, B. H.,Ein Vortrag über Weierstrasspunkte, Karlsruhe, 1975.

  31. Piene, R., Numerical Characters of a curve in Projectiven-space, inReal and Complex Singularities—Oslo 1976 (Holm, P., ed.), pp. 475–495, Sijthoff and Noordhoff, Alphen aan den Rijn, 1977.

    Google Scholar 

  32. Rauch, H. E., Weierstrass points, branch points, and muduli of Riemann surfaces,Comm. Pure Appl. Math. 12 (1959), 543–560.

    MathSciNet  Google Scholar 

  33. Rim, D. S. andVitulli, M., Weierstrass points and monomial curves,J. Algebra 48 (1977), 454–476.

    Article  MathSciNet  Google Scholar 

  34. Schmidt, F. K., Die Wronskische Determinante in beliebigen differenzierbaren Funktionenkörper,Math. Z. 45 (1939), 62–74.

    Article  MathSciNet  Google Scholar 

  35. Schmidt, F. K., Zur arithmetischen Theorie der algebraischen Funktionen II,Math. Z. 45 (1939), 75–96.

    Article  MathSciNet  Google Scholar 

  36. Segre, C., Introduzione alla geometria sopra un ente algebrice simplicemente infinito,Ann. Mat. Pura Appl. 22 (Ser II) (1894), 4–142.

    Google Scholar 

  37. Serre, J.-P.,Groupes algébriques et corps de classes, Herman, Paris, 1959.

    MATH  Google Scholar 

  38. Thorup, A. andKleiman, S., Complete Bilinear Forms, inAlgebraic Geometry, Sundance 1986 (Holme, A. and Speiser, R., eds.),Lecture Notes in Math. 1311, pp. 253–320, Springer-Verlag, Berlin-Heidelberg, 1988.

    Chapter  Google Scholar 

  39. Widland, C. andLax, R., Weierstrass points on Gorenstein curves,Pacific J. Math. 142 (1990), 197–208.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, KTH, S-100 44, Stockholm, Sweden

    Dan Laksov

  2. Matematisk Institut, Universitetsparken 5, DK-2100, København Ø, Denmark

    Anders Thorup

Authors
  1. Dan Laksov
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  2. Anders Thorup
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Partially supported by The Göran Gustafsson Foundation for Research in Natural Sciences and Medicine

Supported in part by the Danish Natural Science Research Council, grant 11-7428

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Laksov, D., Thorup, A. Weierstrass points and gap sequences for families of curves. Ark. Mat. 32, 393–422 (1994). https://doi.org/10.1007/BF02559578

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  • DOI: https://doi.org/10.1007/BF02559578

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