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Enumeration of tropes

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Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas Aims and scope Submit manuscript

Abstract

Singularities of the plane sections of a general surface in three-space are well known, and counted; in particular, all plane sections are reduced. Fixing integers \(d,k\), we give formulas for the degree of the locus of surfaces of degree \(d\) admitting a plane which is tangent along some curve of degree \(k\).

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References

  1. Eklund, D.: Curves on Heisenberg invariant quartic surfaces in projective \(3\)-space. Preprint (2010)

  2. Fulton, W.: Intersection theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, 2nd edn. Springer, Berlin (1998)

    Google Scholar 

  3. Göttsche, L.: A conjectural generating function for numbers of curves on surfaces. Comm. Math. Phys. 196, 523–533 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Kleiman, S., Piene, R., Tyomkin, I.: Enriques diagrams, arbitrarily near points, and Hilbert schemes. Rendiconti Lincei-Matematica e Applicazioni. 22(4), 411–451 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Roberts, S.: Sur l’ordre des conditions de la coexistence des équations algèbriques à plusieurs variables. J. Reine Angew. Math. (Crelle) 67, 266–278 (1867)

    Article  MATH  Google Scholar 

  6. Vainsencher, I.: Enumeration of n-fold tangent hyperplanes to a surface. J. Alg. Geom. 4, 503–526 (1995)

    MathSciNet  MATH  Google Scholar 

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Acknowledgments

Many thanks are due to the referees for kindly offering suggestions and pointing out several corrections.

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Authors and Affiliations

  1. Departamento de Matemática, UFMG, Belo Horizonte, MG, Brazil

    I. Vainsencher

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  1. I. Vainsencher
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Correspondence to I. Vainsencher.

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Dedicated to Professor Heisuke Hironaka on his 80th birthday.

I. Vainsencher is partially supported by CNPQ-Brasil.

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Vainsencher, I. Enumeration of tropes. RACSAM 107, 213–220 (2013). https://doi.org/10.1007/s13398-012-0099-x

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  • DOI: https://doi.org/10.1007/s13398-012-0099-x

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Mathematics Subject Classification (1991)

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