, Volume 54, Issue 1, pp 37–59 | Cite as

Weierstrass points and their impact in the study of algebraic curves: a historical account from the “Lückensatz” to the 1970s

  • Andrea Del CentinaEmail author


In this note we give a historical account of the origin and the development of the concept of Weierstrass point. We also explain how Weierstrass points have contributed to the study of compact Riemann surfaces and algebraic curves in the century from Weierstrass’ statement of the gap theorem to the 1970s. In particular, we focus on the seminal work of Hürwitz that raised questions which are at the center of contemporary research on this topic.


Riemann surfaces Algebraic curves Weierstrass point Origin and development of the concept Impact in the study of algebraic curves 

Mathematics Subject Classification (2000)

01A 14H 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Accola, R.D.M.: On generalized Weierstrass points on Riemann surfaces. In: Modular functions in Analysis and Number theory. Lectures Notes in Mathematics and Statistics, pp. 1–19. Pittsburg (1983)Google Scholar
  2. 2.
    Accola, R.D.M.: Topics in the theory of Riemann surfaces. In: Lectures Notes in Mathematics, vol. 1595. Springer, Berlin (1955)Google Scholar
  3. 3.
    Ahlfors, L.: The complex analytic structure of the space of closed Riemann surfaces. In: Analytic functions, pp. 45–66. Princeton, NJ (1960)Google Scholar
  4. 4.
    Ahlfors L. and Sario L. (1960). Riemann Surfaces. Princeton University Press, Princeton zbMATHGoogle Scholar
  5. 5.
    Arbarello E. (1974). Weierstrass points and moduli of curves. Compos. Math. 29: 325–342 MathSciNetzbMATHGoogle Scholar
  6. 6.
    Baker, H.F.: Abel’s theorem and the allied theory of theta functions. Cambridge (1897)Google Scholar
  7. 7.
    Bers, L.: Riemann surfaces. Courant Institut of Math. Sciences, NYU (1958)Google Scholar
  8. 8.
    Bers, L.: The space of Riemann surfaces. In: Proceedings of the International Congress of Mathematicians, pp. 349–361. Edinburg (1958)Google Scholar
  9. 9.
    Brill, A., Noether, M.: Über die algebraischen Functionen und ihre Anwendung in der Geometrie. Math. Ann. VII (1873)Google Scholar
  10. 10.
    Buchweitz, R.O.: On Zariski’s criterion for equisingularity and non smoothable monomial curves (1980, preprint)Google Scholar
  11. 11.
    Cayley, A.: Sextactic points of a plane curve. Lond. Philos. Trans. 155 (1865)Google Scholar
  12. 12.
    Chevalley, C.: Introduction to the theory of algebraic functions of one variable. In: Mathematical Surveys, vol. 6. American Mathematical Society, NY (1961)Google Scholar
  13. 13.
    Chisini O. (1924). Intorno alla dimostrazione di un teorema di Noether. Boll. Un. Mat. It. 3(197): 197–200 MathSciNetGoogle Scholar
  14. 14.
    Cipolla I. (1905). Sul numero dei punti di Weierstrass fra loro distinti di una curva algebrica. Rend. R. Ac. Lincei, 5a, serie XIV: 210–214 Google Scholar
  15. 15.
    Coolidge J.L. (1931). A treatise on algebraic plane curve. Clarendon Press, Oxford Google Scholar
  16. 16.
    De Jonquières J. (1866). Mémoire sur les contacts multiples d’ordre quelconque des courbes... Jour. für reine und angew. Math. 66: 289–321 zbMATHCrossRefGoogle Scholar
  17. 17.
    Deligne P. and Mumford D. (1969). The irreducibility of the space of stable curves of given genus. Publ. Math. I.H.E.S. 36: 75–110 MathSciNetzbMATHGoogle Scholar
  18. 18.
    Diaz S. (1984). Tangent spaces in moduli via deformations with applications to Weierstrass points. Duke Math. J. 51: 905–922 CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Diaz S. (1984). Moduli of curves with two exceptional Weierstrass points. J. Differ. Geom. 20: 471–478 zbMATHGoogle Scholar
  20. 20.
    Edge W.L. (1950). A plane quartic with eight undulation points. Edinb. Math. Proc. 8(2): 147–162 MathSciNetzbMATHGoogle Scholar
  21. 21.
    Eisenbud D. and Harris J. (1987). Existence, decomposition and limits of certain Weierstrass points. Invent. math. 87: 495–515 CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Farkas H.M. (1966). Special divisors and analytic subvarieties of Teichmüller space. Am. J. Math. 88: 881–901 CrossRefMathSciNetGoogle Scholar
  23. 23.
    Farkas H.M. (1969). Weierstrass points and analytic submanifolds of Teichmüller space. Proc. Am. Math. Soc. 20: 35–38 CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Farkas, H.M.: Remarks on automorphisms of compact Riemann surfaces. In: Discontinous groups and Riemann surfaces. Proc. Conf. Univ. Maryland, College Park 1973, pp. 121–144. AMS, vol. 79, Princeton University Press, Princeton (1974)Google Scholar
  25. 25.
    Farkas H.M. and Kra I. (1980). Riemann surfaces, GTM 71. Springer, Berlin Google Scholar
  26. 26.
    Fulton W. (1969). Hürwitz scheme and irreducibility of moduli of algebraic curves. Ann. Math. 90: 542–575 CrossRefMathSciNetGoogle Scholar
  27. 27.
    Galbura G. (1974). Il wronskiano di un sistema di sezioni di un fibrato vettoriale di rango 1 sopra una curva algebrica ed il relativo divisore di Brill-Severi. Ann. di Mat. Pura e Appl. 98: 349–355 CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Guerrero, I.: Automorphisms of compact Riemann surfaces and Weierstrass points. In: Proceedings of “Riemann surfaces and related topics”. Stony Brook 1978, pp. 215–224. Princeton University Press, Princeton (1980)Google Scholar
  29. 29.
    Gunning R.C. (1966). Lectures on Riemann surfaces. Princeton Ac. Press, Princeton zbMATHGoogle Scholar
  30. 30.
    Hartshorne R. (1977). Algebraic geometry, GTM 52. Spinger, Berlin Google Scholar
  31. 31.
    Haure and M.M. (1896). Recherches sur les points de Weierstrass d’une curbe plane algébriques. Ann. École Nor. Sup. 13: 115–196 Google Scholar
  32. 32.
    Hensel, K., Landsberg, G.: Theorie der algebraischen Funktionen einer variabeln und ihre anwendung auf algebraische Kurven und abelsche Integrale. Leipzig (1902)Google Scholar
  33. 33.
    Horiuchi R. (1969). A note on a paper of Farkas. Proc. Jpn Acad. 45: 859–860 MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Hürwitz A. (1893). Über algebraische Gebilde mit eindeutigen Trasformationen in sich. Math. Ann. 41: 391–430 Google Scholar
  35. 35.
    Jenkins J.A. (1974). Some remars on Weierstrass points. Proc. Am. Math. Soc. 44: 121–122 CrossRefMathSciNetzbMATHGoogle Scholar
  36. 36.
    Kato T. (1979). On Weierstrass points whose first non-gaps are three. Jour. für reine und angew. Math. 316: 99–109 Google Scholar
  37. 37.
    Kato T. (1979). Non-hyperelliptic Weierstrass points of maximal weight. Math. Ann. 239: 141–147 CrossRefMathSciNetzbMATHGoogle Scholar
  38. 38.
    Kato T. (1980). Weierstrass normal form of a Riemann surface and its applications (in Japanese). Sûgaku 32: 73–75 Google Scholar
  39. 39.
    Komeda J. (1983). On Weierstrass points whose first non-gaps are four. J. Reine Angew. Math. 341: 68–86 MathSciNetzbMATHGoogle Scholar
  40. 40.
    Kuribayshi I. and Komya K. (1977). On Weierstrass points of non-hyperelliptic compact Riemann surfaces of genus three. Hiroshima Math. J. 7: 743–768 MathSciNetGoogle Scholar
  41. 41.
    Laksov, D.: Weierstrass points on curves. In: Young tableaux and Schur functions in algebra and geometry, Torun 1980. Astérisque 87–88, pp. 221–242 (1981)Google Scholar
  42. 42.
    Laufer, H.: On generalized Weierstrass points and rings with no prime elements. In: Riemann surfaces and related problems, 1978. Ann. of Math. Studies. Princeton University Press, Princeton (1981)Google Scholar
  43. 43.
    Lax R.F. (1975). Weierstrass points of the universal curve. Math. Ann. 216: 34–42 CrossRefMathSciNetGoogle Scholar
  44. 44.
    Lax R.F. (1980). Gap sequences and moduli in genus 4. Math. Zeitschrift 175: 67–75 CrossRefMathSciNetzbMATHGoogle Scholar
  45. 45.
    Lewittes J. (1963). Automorphisms of compact Riemann surfaces. Am. J. Math. 85: 734–752 CrossRefMathSciNetzbMATHGoogle Scholar
  46. 46.
    Macbeath A.M. (1961). On a theorem of Hürwitz. Proc. Glasgow Math. Soc. 5: 90–96 MathSciNetzbMATHGoogle Scholar
  47. 47.
    MacDonald I. (1962). Symmetric products of an algebraic curve. Topology 1: 319–343 CrossRefMathSciNetzbMATHGoogle Scholar
  48. 48.
    Maclaclhan C. (1969). A bound for the number of automorphisms of a compact Riemann surface. J. Lond. Math. Soc. 44: 265–272 CrossRefGoogle Scholar
  49. 49.
    Masoni U. (1882). Sopra alcune curve del quarto ordine dotate di punti di ondulazione. Rend. R. Acc. Napoli 21: 45–69 Google Scholar
  50. 50.
    Mayer, A.: Compactification of the variety of moduli of curves. Notes of the Inst. for Adv. Study, pp. 6–15 (1969)Google Scholar
  51. 51.
    Miranda, R.: Algebraic curves and Riemann surfaces. In: Graduate Studies in Mathematics, vol. 5. AMS, Providence (1995)Google Scholar
  52. 52.
    Mumford D. (1975). Curves and their Jacobians. University of Michigan Press, Ann Arbor zbMATHGoogle Scholar
  53. 53.
    Mumford, D., Fogarty, J., Kirwan, F.: Geometric invariant theory. In: Erg. der Math. und ihrer Grenzgebiete, vol. 34. Springer, Berlin (1994)Google Scholar
  54. 54.
    Neeman A. (1984). The distribution of Weierstrass points on a compact Riemann surface. Ann. Math. 120: 317–328 CrossRefMathSciNetGoogle Scholar
  55. 55.
    Noether, M.: Note über die algebraischen Curven, welche eine Schaar eindeutiger Transformationen in sich zulassen. Math. Ann. XX:59–62 (1882)Google Scholar
  56. 56.
    Noether, M.: Nachtrag zur “Note über die algebraischen Curven, welche eine Schaar eindeutiger Transformationen in sich zulassen. Math. Ann. XXI:138–140 (1883)Google Scholar
  57. 57.
    Noether M. (1884). Beweis und Erweiterung eines algebraisch-functionen-theoretischen Satzes des Herrn Weierstrass. Jour. für reine und angew. Math. 97: 224–229 CrossRefGoogle Scholar
  58. 58.
    Olsen B.A. (1972). On higher order Weierstrass points. Ann. Math. 95: 357–364 CrossRefMathSciNetGoogle Scholar
  59. 59.
    Petersson H. (1950). Über Weirstrasspunkte und die expliziten Darstellungen der automorphen Formen von reeller Dimension. Math. Z. 52: 32–59 CrossRefMathSciNetGoogle Scholar
  60. 60.
    Pflaum U. (1987). The canonical constellation of k–Weierstrass points. Manusc. Math. 59: 21–34 CrossRefMathSciNetzbMATHGoogle Scholar
  61. 61.
    Pinkham, H.C.: Deformation of algebraic varieties with G m action. Astérisque 20, Paris Soc. Mathématique de France (1974)Google Scholar
  62. 62.
    Poincaré H. (1884). Sur un théorème de M. Fuchs. C. R. Ac. Sc Paris 99: 75–77 Google Scholar
  63. 63.
    Rauch H.E. (1955). On the transcendental moduli of algebraic Riemann surfaces. Proc. Natl. Acad. Sci. USA 41: 42–48 CrossRefMathSciNetzbMATHGoogle Scholar
  64. 64.
    Rauch H.E. (1955). On moduli in conformal mapping. Proc. Natl. Acad. Sci. USA 41: 176–180 CrossRefMathSciNetzbMATHGoogle Scholar
  65. 65.
    Rauch H.E. (1959). Weierstrass points, branch points and moduli of Riemann surfaces. Comm. Pure Appl. Math. 12: 543–560 CrossRefMathSciNetGoogle Scholar
  66. 66.
    Rauch, H.E.: Variational methods in the problem of the moduli of Riemann surfaces. In: Contributions to Functions Theory, pp. 17–40. Tata Inst. Bombay (1960)Google Scholar
  67. 67.
    Riemann B. (1857). Theorie der Abelschen Functionen. Jour. für reine und angew. Math. 54: 115–155 zbMATHCrossRefGoogle Scholar
  68. 68.
    Rim D.S. and Vitulli D.S. (1977). Weierstrass points and monomial curves. J. Algebra 48: 454–476 CrossRefMathSciNetzbMATHGoogle Scholar
  69. 69.
    Roch G. (1866). Über Theta-Functionen vielfacher Argumente. Jour. für reine und angew. Math. 66: 177–184 zbMATHCrossRefGoogle Scholar
  70. 70.
    Rosati C. (1924). Sopra un teorema di Noether. Boll. Un. Mat. It. 3 197: 162–167 Google Scholar
  71. 71.
    Schmidt F.K. (1939). Zur arithmetischen Theorie der algebraischen Functionen II. Math. Zeitschrift 45: 75–96 CrossRefGoogle Scholar
  72. 72.
    Schottky F.H. (1877). Über die conforme Abbildung mehrfach zusammenhängender ebene Flächen. Jour. für reine und angew. Math. 83: 300–351 CrossRefGoogle Scholar
  73. 73.
    Schwarz A. (1879). Über diejenigen algebraischen Gleichungen zwischen zwei veränderlichen... Jour. für reine und angew. Math. 87: 139–145 CrossRefGoogle Scholar
  74. 74.
    Segre, C.: Introduzione alla geometria sopra un ente algebrico semplicemente infinito. In: Annali Mat. Pura Appl. serie II, tomo, vol. XXII, pp. 42–142 (1894)Google Scholar
  75. 75.
    Segre C. (1899). Intorno ai punti di Weierstass di una curva algebrica. Atti Reale Acc. Lincei, Rendiconti serie V VIII: 89–91 Google Scholar
  76. 76.
    Severi, F.: Trattato di Geometria algebrica, I. Bologna (1926)Google Scholar
  77. 77.
    Springer, G.: Introduction to Riemann surfaces. Addison-Wesley, Reading (1957)Google Scholar
  78. 78.
    Vainsencher I. (1981). Counting divisors with prescribed singularities. Trans. Am. Math. Soc. 267: 399–422 CrossRefMathSciNetzbMATHGoogle Scholar
  79. 79.
    Vermeulen, A.M.: Weierstrass points of weight two on curves of genus three. Thesis Universitait van Amsterdam (1983)Google Scholar
  80. 80.
    Veronese G. (1881). Behandlugung der projectivischen Verhältnisse der Räume von verschiedenen Dimensionen... Math. Ann. XIX: 161–234 CrossRefMathSciNetGoogle Scholar
  81. 81.
    Weierstrass, K.: Mathematische Werke, 7 vols (1894–1927) Berlin, reprint by G. Olms, Hildesheim (1967)Google Scholar
  82. 82.
    Weyl H. (1955). The concept of Riemann surface. Addison-Wesley, Reading Google Scholar
  83. 83.
    Wieman A. (1897). Zur Theorie der endlichen Gruppen von birationalen Trasformationen in der Ebene. Math. Ann. 98: 195–210 Google Scholar

Copyright information

© Università degli Studi di Ferrara 2008

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di FerraraFerraraItaly

Personalised recommendations