Skip to main content
SpringerLink
Log in

Patchworking singular algebraic curves II

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

Abstract

In this paper we present various particular versions of the general patch-working procedurefor the construction of reduced algebraic curves with prescribed singularities, on algebraic surfaces. Among the main examples are a deformation of reducible algebraic curves on reducible algebraic surfaces in the presence of non-transverse intersections of a curve with the singular locus of a surface, and a deformation of curves with multiple components. As an application we deduce, a significant asymptotical improvement for the sufficient existence criterion of algebraic curves with arbitrary prescribed singlarities in given linear systems on smooth projective algebraic surfaces.

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. G.-M. Greuel, C. Lossen and E. Shustin,Plane curves of minimal degree with prescribed singularities, Inventiones Mathematicae133 (1998), 539–580.

    Article  MathSciNet  Google Scholar 

  8. R. Hartschorne,Algebraic Geometry, Graduate Texts in Mathematics52, Springer-Verlag, New York, 1977.

    Book  Google Scholar 

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

    MATH  Google Scholar 

  10. 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 

  11. I. Itenberg and E. Shustin,Combinatorial patchworking of real pseudoholomorphic curves, Turkish Journal of Mathematics26 (2002), 27–51.

    MathSciNet  MATH  Google Scholar 

  12. 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 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. 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 

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

    Article  MathSciNet  Google Scholar 

  18. E. Shustin,Lower deformations of isolated hypersurface singularities Algebra i Analiz10 (1999), no. 5, 221–249 (English translation in St. Petersburg Mathematical Journal11 (2000), no. 4, 883–908).

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. E. Shustin,A tropical approach to enumerative geometry, Algebra i Analiz17 (2005), 170–214.

    MathSciNet  Google Scholar 

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

  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 in 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, inProceedings 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 Mathematical Journal1 (1990), 1059–1134.

    MathSciNet  MATH  Google Scholar 

  26. O. Viro,Introduction to topology of real algebraic varieties, Preprint, appears in the internet athttp://www.math.uu.se/≈oleg/es/index.html.

  27. O. Viro,Patchworking, real algebraic varieties, Preprint, appears in the internet athttp://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. Department of Mathematical and Computer Sciences, Weizmann Institute of Science, 76100, Rehovot, Israel

    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.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shustin, E., Tyomkin, I. Patchworking singular algebraic curves II. Isr. J. Math. 151, 145–166 (2006). https://doi.org/10.1007/BF02777359

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02777359

Keywords

Search

Navigation