Multiple Point Formulas for Maps

  • Steven L. Kleiman
Part of the Progress in Mathematics book series (PM, volume 24)


Let f: X → Y be a map. Its set of r-fold points is
$$\begin{array}{*{20}{c}} {{{\text{M}}_{{\text{r}}}}{\text{ = }}\{ {\text{x}}\varepsilon {\text{X|}}} & {{\text{there}} {\text{exist}}} & {{{\text{x}}_{2}},...,{{\text{x}}_{{\text{r}}}}} & {{\text{with}}} & {{\text{f(}}{{\text{x}}_{{\text{i}}}}){\text{ = f}}({\text{x}})\} ;} \\ \end{array}$$
the xi must be distinct from x and from each other, but they may lie “infinitely close” (that is, determine tangent directions along the fiber f-1f(x)). An r-fold-point formula is a polynomial expression in the invariants of f which gives, under appropriate hypotheses, the number of r-fold points, weighted by natural multiplicities, or the class mr of a natural positive cycle supported by Mr. The theory of these formulas will be surveyed here, concentrating on some of the author’s recent work, Kleiman [1981b], [1982]. Aside from a few comments, the setting will be algebraic geometry, although the formulas and their proofs have a universal character.


Algebraic Geometry Double Point Intersection Formula Hilbert Scheme Refined Version 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Arbarello, E., Cornalba, M., Griffiths, P., Harris.: Geometry of Algebraic Curves, to appear.Google Scholar
  2. Canuto, G. [ 1979 ]: Associated Curves and Plücker Formulas in Grassmannians. Inventiones math. 53, 77–90 (1979).MathSciNetGoogle Scholar
  3. Catanese, F. [1979]: On Severi’s Proof of the Double Point Formula. Comm. in Alg. 7 (7), 763–773 (1979).MathSciNetMATHCrossRefGoogle Scholar
  4. Fulton, W. [ 1978 ]: A NOTE ON RESIDUAL INTERSECTIONS AND THE DOUBLE POINT FORMULA. Acta math. 140, 93–101 (1978).MathSciNetGoogle Scholar
  5. Fulton, W., MacPherson [1980]: BIVARIANT THEORIES. Preprint, Brown Univ. Providence (1980).Google Scholar
  6. Fulton, W., Laksov, D. [1977]: Residual intersections and the double-point formula. Real and complex singularities, Oslo 1976, P. Holm, ed., pp. 171–177. Sijthoff & Noordhooff (1977).Google Scholar
  7. Grayson, D. [1979]: COINCIDENCE FORMULAS IN ENUMERATIVE GEOMETRY. Comm. in Alg. 7(16), 1685–1711 (1979).MathSciNetMATHCrossRefGoogle Scholar
  8. Griffiths, P., Harris, J. [1979]: Algebraic Geometry and Local Differential Geometry. Ann. Ec. N.S. 12 (3), 355–452 (1979).MathSciNetMATHGoogle Scholar
  9. Herbert, R. [ 1975 ]: MULTIPLE POINTS OF IMMERSED MANIFOLDS. Thesis, Univ. of Minnesota (1975), Amer. Math. Soc. Memoire No. 250 (Oct. 1981).Google Scholar
  10. Holme, A. [ 1978 ]: Deformation and Stratification of Secant Structure. Algebraic Geometry, Proceedings, Tromϕ, Norway 1977, L. Olson, ed., pp. 60–91. Lecture Notes inMath., 687. Springer (1978).Google Scholar
  11. Holme, A. [ 1979 ]: On the dual of a smooth variety. Algebraic Geometry, Proceedings, Copenhagen 1978, IC. Lonsted, Ed., pp. 144–156. Lecture Notes in Math., 732. Springer (1979).Google Scholar
  12. Johnson, K. [ 1978 ]: Immersion and Embedding of Projective Varieties. Acta math., 140, 49–74 (1978).Google Scholar
  13. Kleiman, S. [1976]: Problem 15. Rigorous foundation of Schubert’s enumerative calculus. Mathematical Developments arising from Hilbert Problems, Proc. Symposia Pure Math., Vol. XXVIII, pp. 445–82. A.M.S. (1976).Google Scholar
  14. Kleiman, S. [1977]: The enumerative theory of singularities. Real and complex singularities, Oslo 1976, P. Holm, ed., pp. 297–396. Sijthoff & Noordhooff (1977).Google Scholar
  15. Kleiman, S. [ 1981a ]: Concerning the dual variety. 18th Scand. Congress Math. Proceedings, Birkhäuser (1981), 386–396.Google Scholar
  16. Kleiman, S. [ 1981b ]: Multiple-point formulas I: Iteration. Acta Math. 147, 13–49 (1981).MathSciNetGoogle Scholar
  17. Kleiman, S. [ 1982 ]: Multiple-point formulas I I: O-cycles. In preparation.Google Scholar
  18. Kleiman, S. [1980]: RELATIVE DUALTIY FOR QUASI-COHERENT SHEAVES. Compositio Math. 41 (1), 39–60 (1980).MathSciNetMATHGoogle Scholar
  19. Laksov, D. [ 1978a ]: RESIDUAL INTERSECTIONS AND TODD’S FORMULA FOR THE DOUBLE LOCUS OF A MORPHISM. Acta Math., 140, 75–92 (1978).MathSciNetGoogle Scholar
  20. Laksov, D. [ 1978b ]: Secant bundles and Todd’s formula for the double points of maps into IPn. Proc. Lond. Math. Soc. 37, 120–142 (1978).MathSciNetGoogle Scholar
  21. Lashof, R., Smale, S. [ 1958 ]: Self-intersections of immersed manifolds. J. Math. Mech. 8, 143–157 (1959).MathSciNetGoogle Scholar
  22. Laudal, O. [ 1978 ]: A Generalized Trisecant Lemma. Algebraic Geometry, Proceedings, Tromsϕ, Norway 1977, L. Olson, ed., pp. 112–149. Lecture Notes in Math., 687. Springer (1978).Google Scholar
  23. LeBarz, P. [ 1978 ]: Géometrie énumérative pour les multisécantes. Variétés analytic compactes, Nice 1977, pp. 116–167. Lecture Notes in Math., 687. Springer (1978).Google Scholar
  24. LeBarz, P. [ 1979a ]: Validité de certaines formules de géometrie énumérative. C.R.A.S. Paris. 289, 755–758 (1979).MathSciNetGoogle Scholar
  25. LeBarz, P. [ 1979b ]: Courbes generales de IP3. Math. Scand. 44, 243–277 (1979).MathSciNetGoogle Scholar
  26. LeBarz, P. [ 1980 ]: Une courbe gauche avec -4 quadrisecantes. Preprint Univ. Nice (1980).Google Scholar
  27. LeBarz, P. [ 1981a ]: Formules pour les multisécantes des surfaces, C.R.A.S. 292, 797–800 (1981).MathSciNetGoogle Scholar
  28. LeBarz, P. [ 1881b ]: Quelques calculs dans Hilbk IPN et la variété des alignements, in preparation.Google Scholar
  29. LeBarz, P. [ 1981c ]: Une Application de la Formule de Fulton-MacPherson. Preprint Univ. Nice (June 1981 ).Google Scholar
  30. Piene, R. [1977]: Numerical characters of a curve in projective n-space. Real and complex singularities, Oslo 1976, P. Holm, ed., pp. 475–496. Sijthoff & Noordhooff (1977).Google Scholar
  31. Piene, R. [ 1978a ]: Polar classes of singular varieties. Ann. S. Ec. N. Sup. 11, 274–276 (1978).MathSciNetGoogle Scholar
  32. Piene, R. [ 1978b ]: Some Formulas for a Surface in IP3. Algebraic Geometry, Proceedings, Tromsϕ, Norway 1977, L. Olson, ed., pp. 196–235. Lecture Notes in Math, 687. Springer (1978).Google Scholar
  33. Piene, R. [1979]: A proof of Noether’s formula for the arithmetic genus of an algebraic surface. Compositio math. (1)38, 113–119 (1979).Google Scholar
  34. Piene, R. [ 1980 ]: CUSPIDAL PROJECTIONS OF SPACE CURVES. Preprint, Univ. of Oslo, Norway, Feb. (1980), Math. Ann. 256, 95–119 (1981).MathSciNetGoogle Scholar
  35. Piene, R., Ronga, F. [ 1979 ]: A GEOMETRIC APPROACH TO THE ARITHMETIC GENUS OF A PROJECTIVE MANIFOLD OF DIMENSION THREE. Preprint July (1979), Topology, 20, 179–190 (1981);MathSciNetCrossRefGoogle Scholar
  36. Roberts, J. [1979]: Some properties of double point schemes. Preprint May (1979), Compositio Math., 41 (1), 61–94 (1980).MathSciNetMATHGoogle Scholar
  37. Roberts, J., Speiser, R., [1980]: Schubert’s enumerative geometry of triangles from a modem viewpoint. Algebraic Geometry, Proceedings, Chicago Circle (1980), A. Libgober and P. Wagreich, ed., pp. 272–281, Lecture Notes in Math, 862. Springer (1981).Google Scholar
  38. Roberts, J., Speiser, R., [1980]: Enumerative geometry of triangles. In preparation.Google Scholar
  39. Ronga, F. [1973]: Le calcul des classes duals aux points doubles d’une application. Compositio math. (2)27, 223–232 (1973).MathSciNetGoogle Scholar
  40. Ronga, F. [1980]: On multiple points of smooth immersions. Preprint March (1980).Google Scholar
  41. Schubert, H. [1879]-: Kalkül der abzänlenden Geometrie. Reprint. Springer (1979).CrossRefGoogle Scholar
  42. Vainsencher, I. [ 1980 ]: COUNTING DIVISORS WITH PRESCRIBED SINGULARITIES. To appear in Proc. A.M.S.Google Scholar
  43. von zu Gathen, J. [ 1980 ]: SEKANTENRAUME VON KURVEN. Inaugural-Dissertation, Univ. Zurich (1980).Google Scholar
  44. Urabe, T. [1980]: Duality of numerical characters of polar loci. Preprint March (? ) (1980).Google Scholar

Copyright information

© Birkhäuser Boston, Inc. 1982

Authors and Affiliations

  • Steven L. Kleiman
    • 1
  1. 1.M.I.T. 2-278CambridgeU.S.A.

Personalised recommendations