Inventiones mathematicae

, Volume 98, Issue 3, pp 445–489

Complex algebraic plane curves via their links at infinity

  • Walter D. Neumann


Considering that the study of plane cuves has an over 2000 year history and is the seed from which modern algebraic geometry grew, surprisingly little is known about the topology of affine algebraic plane curves. We topologically classify “regular” algebraic plane curves in complex affine 2-space using “splice diagrams:” certain decorated trees that code Puiseux data at infinity. (The regularity condition — that the curve be a “typical” fiber of its defining polynomial — can conjecturally be avoided.) We also show that the splice diagram determines such algebraic information as the minimal degree of the curve, even in the irregular case. Among other things, this enables algebraic classification of regular algebraic plane curves with given topology.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A] Abhyankar, S.S.: On Expansion techniques in algebraic geometry. Tata Institute of Fundamental Research. Lect. Math. Phys.57, 1977Google Scholar
  2. [A-M1] Abhyankar, S.S., Moh, T.T.: Newton-Puiseux expansion and generalized Tschirnhausen transformation. I. J. Reine Angew. Math.260, 47–83 (1973)Google Scholar
  3. [A-M2] Abhyankar, S.S., Moh, T.T.: Embeddings of the line in the plane. J. Reine Angew. Math.276, 148–166 (1975)Google Scholar
  4. [A-S] Abhyankar, S.S., Singh, B.: Embeddings of certain curves. Am. J. Math.100, 99–175 (1978)Google Scholar
  5. [E-N] Eisenbud, D., Neumann, W.: Three-Dimensional link theory and invariants of plane curve singularities. Ann. Math. Stud.101, Princeton: University Press 1985Google Scholar
  6. [J] Johannson, K.: Homotopy equivalences of 3-manifolds with boundaries. Lect. Notes Math. vol 761. New York Berlin Heidelberg: Springer 1979Google Scholar
  7. [J-S] Jaco, W., Shalen, P.: Seifert fibered spaces in 3-manifolds. Memoirs of the A.M.S.220, 1979.Google Scholar
  8. [M] Milnor, J.: Singular points of complex hypersurfaces. Ann. Math. Stud.61, Princeton: University Press, 1968Google Scholar
  9. [N-R1] Neumann, W.D., Rudolph, L.: Unfoldings in knot theory. Math. Ann.278, 409–439 (1987) and: Corrigendum: Unfoldings in knot theory. ibid.282, 349–351 (1988)Google Scholar
  10. [N-R2] Neumann, W.D., Rudolph, L.: Difference index of vectorfields and the enhanced Milnor number. Topology (to appear)Google Scholar
  11. [R] Rudolph, L.: Isolated critical points of maps from ℝ4 to ℝ2 and a natural splitting of the Milnor number of a classical fibered link. Part I. Comm. Math. Helv.62, 630–645 (1987)Google Scholar
  12. [S] Suzuki, M.: Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l'espace ℂ2. J. Math. Soc. Jpn.26, 241–257 (1974)Google Scholar
  13. [Z-L] Zaidenberg, M.G., Lin, V. Y.: An irreducible, simply connected curve in ℂ2 is equivalent to a quasihomogeneous curve. (English translation) Soviet Math. Dokl.28, 200–204 (1983)Google Scholar
  14. [W] Waldhausen, F.: On irreducible 3-manifolds that are sufficiently large. Ann. Math.87, 56–88 (1968)Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • Walter D. Neumann
    • 1
  1. 1.Department of MathematicsOhio State UniversityColumbusUSA

Personalised recommendations