Topological type of discriminants of some special families

  1. García Barroso, Evelia R. 1
  2. Hernández Iglesias, M. Fernando 2
  1. 1 Universidad de La Laguna
    info

    Universidad de La Laguna

    San Cristobal de La Laguna, España

    ROR https://ror.org/01r9z8p25

  2. 2 Universidad Nacional Mayor de San Marcos
    info

    Universidad Nacional Mayor de San Marcos

    Lima, Perú

    ROR https://ror.org/006vs7897

Journal:
Periodica Mathematica Hungarica

ISSN: 0031-5303 1588-2829

Year of publication: 2022

Volume: 84

Pages: 321-345

Type: Article

DOI: 10.1007/S10998-021-00410-0 GOOGLE SCHOLAR lock_openOpen access editor

More publications in: Periodica Mathematica Hungarica

Sustainable development goals

Abstract

AbstractWe will describe the topological type of the discriminant curve of the morphism $$(\ell , f)$$ ( ℓ , f ) , where $$\ell $$ ℓ is a smooth curve and f is an irreducible curve (branch) of multiplicity less than five or a branch such that the difference between its Milnor number and Tjurina number is less than 3. We prove that for a branch of these families, the topological type of the discriminant curve is determined by the semigroup, the Zariski invariant and at most two other analytical invariants of the branch.

Bibliographic References

  • V. Bayer, A. Hefez. Algebroid plane curves whose Milnor and Tjurina numbers differ by one or two. Bol. Soc. Brasil. Mat. (N.S.) 32(1), 63–81 (2001)
  • E. Casas-Alvero, Local geometry of planar analytic morphisms. Asian J. Math. 11(3), 373–426 (2007)
  • A. Chenciner. Courbes algébriques planes. Publications Mathématiques de l’Université Paris VII, 1978
  • E. R. García Barroso, J. Gwoździewicz, A discriminant criterion of irreducibility. Kodai Math. J. 35(2), 403–414 (2012)
  • E.R. García Barroso, J. Gwoździewicz, A. Lenarcik, Non-degeneracy of the discriminant. Acta Math. Hungar. 147(1), 220–246. https://doi.org/10.1007/s10474-015-0515-8 (2015)
  • E.R. García Barroso, A. Lenarcik, A. Płoski, Characterization of non-degenerate plane curve singularities. Univ. Iagel. Acta Math. No. 45, 27–36 (2007)
  • A. Hefez. Irreducible Plane Curve Singularities. Sixth Worhshop at Sao Carlos. (2003), 1–120
  • A. Hefez, M.E. Hernandes, Analytic classification of plane branches up to multiplicity 4. J. Symb. Comput. 44, 626–634 (2009)
  • A. Hefez, M.E. Hernandes, M.F. Hernández Iglesias, On Polars of Plane Branches In: Cisneros-Molina J., Tráng Lê D., Oka M., Snoussi J. (eds) Singularities in Geometry, Topology, Foliations and Dynamics. Trends in Mathematics. Birkhäuser (2017), 135–153
  • A. Hefez; M.E. Hernandes; M.F. Hernández Iglesias, Plane branches with Newton non-degenerate polars. Int. J. Math. 29(1), 1850001 (2018)
  • M.F. Hernández Iglesias, Polar de um germe de curva irredutível plana. PhD thesis. Universidade Federal Fluminense, Brasil (2012)
  • M. Merle, Invariants polaires des courbes planes. Invent. Math. 41, 103–111 (1977)
  • M. Oka, Non-Degenerate Complete Intersection Singularity, Actualités Mathématiques. Hermann, Paris, viii+309 pp (1997)
  • B. Teissier, Varietés polaires. I. Invariants polaires des singularités d’hypersurfaces, Invent. Math. 40, 267–292 (1977)
  • O. Zariski, Characterization of plane algebroid curves whose module of differentials has maximum torsion. Proc. Nat. Acad. Sci. 56, 781–786 (1966)
  • O. Zariski, The moduli problem for plane branches, with an appendix by Bernard Teissier. University Lectures Series, Volume 39, AMS 2006, pp. 151