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Geometry of complete cuspidal plane cubics

  • J. M. Miret
  • S. Xambó Descamps
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1389)

Abstract

We show how to compute all fundamental numbers for plane cuspidal cubics. This updates and extends the work of Schubert on this subject. In our approach we need a far more precise description of the first order degenerations (13 in all) than that given by Schubert and this is obtained by proving a number of key geometric relations that are satisfied by cuspidal cubics. Moreover, our procedure does not require using coincidence formulas to derive the basic degeneration relations.

Keywords

Cross Ratio Triple Line Chow Group Fundamental Number Plane Cubics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Casas, E. [1987], A transversality theorem and an enumerative calculus for proper solutions, Preprint, 1987.Google Scholar
  2. Fulton, W. [1984], Intersection Theory, Ergebnisse NF 2, Springer-Verlag, 1984.Google Scholar
  3. Kleiman, S. [1974], The transversality of a general translate, Compositio Math. 38 (1974), 287–297.MathSciNetzbMATHGoogle Scholar
  4. Kleiman, S.; Speiser, R. [1986], Enumerative geometry of cuspidal plane cubics, Proceedings Vancouver Conference in Algebraic Geometry 1984 (eds. Carrell, Geramita and Russell), CMSAMS Conf. Proc. Vol 6, 1986.Google Scholar
  5. Laksov, D.; Speiser, R. [1987], Transversality criteria in any characteristic, Preprint, 1987.Google Scholar
  6. Maillard, S. [1871], Récherche des charactéristiques des systèmes élémentaires de courbes planes du troisième ordre, Thesis, Paris, publ. by Cusset (1871).Google Scholar
  7. Miret, J. M.; Xambó, S. [1987], On Schubert's degenerations of cuspidal plane cubics, Preprint Univ. of Barcelona, 1987.Google Scholar
  8. Rosselló, F.; Xambó, S. [1987], Computing Chow groups, in: Algebraic Geometry Sundance 1986, LN in Math. 1311, 220–234.Google Scholar
  9. Sacchiero, G. [1984], Numeri caratteristici delle cubiche piane cuspidale, Preprint Univ. di Roma II (1984).Google Scholar
  10. Schubert, H. C. H. [1879], Kalkül der abzählenden der Geometrie, Teubner, Leipzig, 1879 (reprinted by Springer-Verlag, 1979).Google Scholar
  11. Zeuthen, H. [1872], Détermination des charactéristiques des systèmes elémentaires des cubiques, CR. Acad. Sc. Paris 74, 521–526.Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • J. M. Miret
    • 1
  • S. Xambó Descamps
    • 1
  1. 1.Dept. Àlgebra i GeometriaUniv. BarcelonaBarcelonaSpain

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