On Weierstrass Gaps at Several Points

  • Wanderson Tenório
  • Guilherme TizziottiEmail author


We consider the problem of determining Weierstrass gaps and pure Weierstrass gaps at several points. Using the notion of relative maximality in generalized Weierstrass semigroups due to Delgado (Proc Am Math Soc 108(3):627–631, 1990), we present a description of these elements which generalizes the approach of Homma and Kim (J Pure Appl Algebra 162(2–3):273–290, 2001) given for pairs. Through this description, we study the gaps and pure gaps at several points on a certain family of curves with separated variables.


Weierstrass semigroup Generalized Weierstrass semigroup Pure gaps Curves with separated variables 



The authors thank the referee for the useful comments that have improved the presentation of this work. The author W. Tenório was supported by PNDP/CAPES and the author G. Tizziotti was supported by FAPEMIG (Brazil), Grant APQ-01661-17.


  1. Arbarello, E., Cornalba, M., Griffiths, P., Harris, J.: Geometry of Algebraic Curves. Springer, Berlin (1985)CrossRefGoogle Scholar
  2. Bartoli, D., Quoos, L., Zini, G.: Algebraic geometric codes on many points from Kummer extensions. (2008). arXiv:1606.04143v1 (preprint)
  3. Beelen, P., Tutas, N.: A generalization of the Weierstrass semigroup. J. Pure Appl. Algebra 207(2), 243–260 (2006)MathSciNetCrossRefGoogle Scholar
  4. Carvalho, C., Kato, T.: On Weierstrass semigroup and sets: a review with new results. Geom. Dedicata 139, 139–195 (2009)MathSciNetCrossRefGoogle Scholar
  5. Carvalho, C., Torres, F.: On Goppa codes and Weierstrass gaps at several points. Des. Codes Cryptogr. 35(2), 211–225 (2005)MathSciNetCrossRefGoogle Scholar
  6. Delgado, F.: The semigroup of values of a curve singularity with several branches. Manuscr. Math. 59(3), 347–374 (1987)MathSciNetCrossRefGoogle Scholar
  7. Delgado, F.: The symmetry of the Weierstrass generalized semigroups and affine embeddings. Proc. Am. Math. Soc. 108(3), 627–631 (1990)MathSciNetCrossRefGoogle Scholar
  8. Duursma, I.M., Park, S.: Delta sets for divisors supported in two points. Finite Fields Appl. 18(5), 865–885 (2012)MathSciNetCrossRefGoogle Scholar
  9. Fulton, W.: Algebraic Curves: An Introduction to Algebraic Geometry. Addison Wesley, Boston (1969)zbMATHGoogle Scholar
  10. Homma, M.: The Weierstrass semigroup of a pair of points on a curve. Arch. Math. 67, 337–348 (1996)MathSciNetCrossRefGoogle Scholar
  11. Homma, M., Kim, S.J.: Goppa codes with Weierstrass pairs. J. Pure Appl. Algebra 162(2–3), 273–290 (2001)MathSciNetCrossRefGoogle Scholar
  12. Hu, C., Yang, S.: Multi-point codes over Kummer extensions. Des. Codes Cryptogr. 86(1), 211–230 (2018)MathSciNetCrossRefGoogle Scholar
  13. Hu, C., Yang, S.: Pure Weierstrass gaps from a quotient of the Hermitian curve. Finite Fields Appl. 50, 251–271 (2018)MathSciNetCrossRefGoogle Scholar
  14. Kim, S.J.: On the index of the Weierstrass semigroup of a pair of points on a curve. Arch. Math. 62, 73–82 (1994)MathSciNetCrossRefGoogle Scholar
  15. Matthews, G.L.: The Weierstrass Semigroup of an m-Tuple of Points on a Hermitian Curve, 12–24, Lecture Notes in Computer Science, p. 2948. Springer, Berlin (2004)Google Scholar
  16. Moyano-Fernández, J.J., Tenório, W., Torres, F.: Generalized Weierstrass semigroups and their Poincaré series. (2017). arXiv:1706.03733 (preprint)
  17. Tenório, W., Tizzioti, G.: Generalized Weierstrass semigroups and Riemann–Roch spaces for certain curves with separated variables. (2017). arXiv:1709.00263 (preprint)

Copyright information

© Sociedade Brasileira de Matemática 2018

Authors and Affiliations

  1. 1.Faculdade de MatemáticaUniversidade Federal de Uberlândia (UFU)UberlândiaBrazil

Personalised recommendations