Skip to main content
SpringerLink
Log in

Patchworking singular algebraic curves I

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper we present a general patchworking procedure for the construction of reduced singular curves having prescribed singularities and belonging to a given linear system on algebraic surfaces. It originates in the Viro “gluing” method for the construction of real non-singular algebraic hypersurfaces. The general procedure includes almost all known particular modifications, and goes far beyond. Some applications and examples illustrate the construction.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. Caporaso and J. Harris,Counting plane curves of any genus, Inventiones Mathematicae131 (1998), 345–392.

    Article  MathSciNet  Google Scholar 

  2. X. Chen,Rational curves on K3 surfaces, Journal of Algebraic Geometry8 (1999), 245–278.

    MathSciNet  MATH  Google Scholar 

  3. L. Chiantini and C. Ciliberto,On the Severi varieties on surfaces in ℙ3, Journal of Algebraic Geometry8 (1999), 67–83.

    MathSciNet  MATH  Google Scholar 

  4. R. E. Gompf,A new construction of symplectic manifolds, Annals of Mathematics (2)142 (1995), 527–595.

    Article  MathSciNet  Google Scholar 

  5. G.-M. Greuel and U. Karras,Families of varieties with prescribed singularities, Compositio Mathematica69 (1989), 83–110.

    MathSciNet  MATH  Google Scholar 

  6. G.-M. Greuel and C. Lossen,Equianalytic and equisingular families of curves on surfaces, Manuscripta Mathematica91 (1996), 323–342.

    Article  MathSciNet  Google Scholar 

  7. R. Hartschorne,Algebraic Geometry, Springer-Verlag, Berlin, 1977.

    Book  Google Scholar 

  8. D. Hilbert,Mathematische Probleme, Archiv der Mathematik und Physik3 (1901), 213–237 (German).

    MATH  Google Scholar 

  9. I. Itenberg and E. Shustin,Singular points and limit cycles of planar polynomial vector fields, Duke Mathematical Journal102 (2000), 1–37.

    Article  MathSciNet  Google Scholar 

  10. I. Itenberg and E. Shustin,Combinatorial patchworking of real pseudo-holomorphic curves, Turkish Journal of Mathematics26 (2002), 27–51.

    MathSciNet  MATH  Google Scholar 

  11. T. Keilen and I. Tyomkin,Existence of curves with prescribed topological singularities, Transactions of the American Mathematical Society354 (2002), 1837–1860.

    Article  MathSciNet  Google Scholar 

  12. T. Oda,Convex Bodies and Algebraic Geometry, Springer-Verlag, Berlin, 1988.

    MATH  Google Scholar 

  13. Z. Ran,Enumerative geometry of singular plane curves, Inventiones Mathematicae97 (1989), 447–465.

    Article  MathSciNet  Google Scholar 

  14. E. Shustin,Real plane algebraic curves with prescribed singularities, Topology32 (1993), 845–856.

    Article  MathSciNet  Google Scholar 

  15. E. Shustin,Critical points of real polynomials, subdivisions of Newton polyhedra and topology of real algebraic hypersurfaces, American Mathematical Society Translations (2)173 (1996), 203–223.

    MathSciNet  MATH  Google Scholar 

  16. E. Shustin,Gluing of singular and critical points, Topology37 (1998), 195–217.

    Article  MathSciNet  Google Scholar 

  17. E. Shustin,Lower deformations of isolated hypersurface singularities, Algebra i Analiz10 (1999), 221–249 (English translation: St. Petersburg Mathematical Journal11 (2000), 883–908).

    MATH  Google Scholar 

  18. E. Shustin,Analytic order of singular and critical points, Transactions of the American Mathematical Society356 (2004), 953–985.

    Article  MathSciNet  Google Scholar 

  19. E. Shustin,Patchworking singular algebraic curves, non-Archimedean amoebas and enumerative geometry, A tropical approach to enumerative geometry, Algebra i Analiz17 (2005), 170–214.

    Google Scholar 

  20. E. Shustin and I. Tyomkin,Patchworking singular algebraic curves II, Israel Journal of Mathematics, this volume.

  21. E. Shustin and E. Westenberger,Projective hypersurfaces with many singularities of prescribed types, Journal of the London Mathematical Society (2)70 (2004), 609–624.

    Article  MathSciNet  Google Scholar 

  22. O. Viro,Curves of degree 7, curves of degree 8 and the Ragsdale conjecture, Doklady Akademii Nauk SSSR254 (1980), 1305–1310 (Russian); English translation: Soviet Mathematics Doklady22 (1980), 566–570.

    MathSciNet  MATH  Google Scholar 

  23. O. Ya. Viro,Gluing of algebraic hypersurfaces, smoothing of singularities and construction of curves, Proceedings of the Leningrad International Topological Conference, Leningrad, Aug. 1982, Nauka, Leningrad, 1983, pp. 149–197 (Russian).

    Google Scholar 

  24. O. Viro,Gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7, Lecture Notes in Mathematics1060, Springer-Verlag, Berlin, 1984, pp. 187–200.

    MATH  Google Scholar 

  25. O. Ya. Viro,Real algebraic plane curves: constructions with controlled topology, Leningrad Mathemaical Journal1 (1990), 1059–1134.

    MATH  Google Scholar 

  26. O. Viro,Introduction to Topology of Real Algebraic Varieties, Preprint, appears in the internet at http://www.math.uu.se/~oleg/es/index.html.

  27. O. Viro,Patchworking real algebraic varieties, Preprint, appears in the internet at http://www.math.uu.se/~oleg/pw.ps.

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University Ramat Aviv, 69978, Tel Aviv, Israel

    Eugenii Shustin

  2. Institut de Mathématiques, Algèbres d'opérateurs et répresentations, Université Pierre et Marie Curie, 175 rue du Chevaleret, plateu 7D, F-75013, Paris, France

    Ilya Tyomkin

Authors
  1. Eugenii Shustin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ilya Tyomkin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Both authors were partially supported by the Herman Minkowsky-Minerva Center for Geometry at Tel Aviv University, and by grant no. G-616-15.6/99 from the German-Israeli Foundation for Research and Development. The first author was also supported by the Bessel Research Award from the Alexander von Humboldt Foundation. The second author was also partially supported by the EC-network ‘Algebraic Lie Representations” contract no. ERB-FMRX-CT97-0100.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shustin, E., Tyomkin, I. Patchworking singular algebraic curves I. Isr. J. Math. 151, 125–144 (2006). https://doi.org/10.1007/BF02777358

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02777358

Keywords

Search

Navigation