Revista Matemática Complutense

, Volume 27, Issue 2, pp 461–499 | Cite as

Analytic theory of curvettes and dicriticals



The geometric theory of pencils in a germ of a smooth complex surface has been studied by several authors and their properties strongly lie on the structure of the dicritical components of their resolution, whereas the strict transform of the general element of the pencil by the resolution can be seen as a union of the so-called curvettes. This theory can be interpreted as the study of ideals (with two generators) in the ring of complex convergent power series in two variables, which is a local regular ring of dimension 2. In this work, we study properties of curvettes and dicriticals in an arbitrary local regular ring of dimension 2 (without restrictions on its characteristic or the one of its residue field). All the results are stated in purely algebraic terms though the ideas come from geometry.


Analytic Contact number Curvette Dicritical divisor 

Mathematics Subject Classification (2000)

Primary 14A05 



Second author is partially supported by MTM2010-21740-C02-02.


  1. 1.
    Abhyankar, S.: On the valuations centered in a local domain. Amer. J. Math. 78, 321–348 (1956)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Abhyankar, S.: Ramification theoretic methods in algebraic geometry. Annals of Mathematics Studies, no. 43. Princeton University Press, Princeton, NJ (1959)Google Scholar
  3. 3.
    Abhyankar, S.: Resolution of singularities of embedded algebraic surfaces. Pure and Applied Mathematics, Vol. 24, Academic Press, New York. Second Enlarged Edition of 1998 Published by Springer-Verlag (1966)Google Scholar
  4. 4.
    Abhyankar, S.: Historical ramblings in algebraic geometry and related algebra. Amer. Math. Mon. 83(6), 409–448 (1976)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Abhyankar, S.: Desingularization of plane curves. In: Singularities, Part 1 (Arcata, Calif., 1981), Proc. Sympos. Pure Math., vol. 40, pp. 1–45. Amer. Math. Soc., Providence, RI (1983)Google Scholar
  6. 6.
    Abhyankar, S.: Algebraic geometry for scientists and engineers, mathematical surveys and monographs, vol. 35. American Mathematical Society, Providence, RI (1990)CrossRefGoogle Scholar
  7. 7.
    Abhyankar, S.: Polynomial expansion. Proc. Amer. Math. Soc. 126(6), 1583–1596 (1998)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Abhyankar, S.: Lectures on algebra. Vol. I. World Scientific Publishing Co., Pte. Ltd., Hackensack, NJ (2006)Google Scholar
  9. 9.
    Abhyankar, S.: Dicritical divisors and Jacobian problem. Indian J. Pure Appl. Math. 41(1), 77–97 (2010)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Abhyankar, S.: Inversion and invariance of characteristic terms: Part I. In: The legacy of Alladi Ramakrishnan in the mathematical sciences, pp. 93–168. Springer, New York (2010).Google Scholar
  11. 11.
    Abhyankar, S.: More about dicriticals. Proc. Amer. Math. Soc. 139(9), 3083–3097 (2011)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Abhyankar, S.: Pillars and towers of quadratic transformations. Proc. Amer. Math. Soc. 139(9), 3067–3082 (2011)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Abhyankar, S.: Quadratic transforms inside their generic incarnations. Proc. Amer. Math. Soc. 140(12), 4111–4126 (2012)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Abhyankar, S.: Dicriticals of pencils and Dedekind’s Gauss lemma. Rev. Mat. Complut. 26(2), 735–752 (2013)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Abhyankar, S.: Generic incarnations of quadratic transforms. Proc. Amer. Math. Soc. 141(12), 4103–4117 (2013)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Abhyankar, S., Artal, E.: Algebraic theory of curvettes and dicriticals. Proc. Amer. Math. Soc. 141(12), 4087–4102 (2013)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Abhyankar, S., Heinzer, W.: Existence of dicritical divisors revisited. Proc. Indian Acad. Sci. Math. Sci. 121(3), 267–290 (2011)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Abhyankar, S., Heinzer, W.: Existence of dicritical divisors. Amer. J. Math. 134(1), 171–192 (2012)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Abhyankar, S., Heinzer, W.: Rees valuations. Proc. Indian Acad. Sci. Math. Sci. 122(4), 525–546 (2012)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Abhyankar, S., Luengo, I.: Algebraic theory of dicritical divisors. Amer. J. Math. 133(6), 1713–1732 (2011)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Abhyankar, S., Luengo, I.: Spiders and multiplicity sequences. Proc. Amer. Math. Soc. 141(12), 4071–4085 (2013)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Artal, E.: Une démonstration géométrique du théorème d’Abhyankar-Moh. J. Reine Angew. Math. 464, 97–108 (1995)MATHMathSciNetGoogle Scholar
  23. 23.
    Lê, D., Weber, C.: A geometrical approach to the Jacobian conjecture for n=2. Kodai Math. J. 17, 374–381 (1994)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Mattei, J.F., Moussu, R.: Holonomie et intégrales premières. Ann. Sci. École Norm. Sup. (4) 13(4), 469–523 (1980)MATHMathSciNetGoogle Scholar
  25. 25.
    Nagata, M.: Local rings. Interscience Tracts in Pure and Applied Mathematics, No. 13. Interscience Publishers a division of John Wiley & Sons, New York-London (1962)Google Scholar
  26. 26.
    Zariski, O., Samuel, P.: Commutative algebra. Vol. 1. Springer-Verlag, New York (1975). With the cooperation of I. S. Cohen, Corrected reprinting of the 1958 edition, Graduate Texts in Mathematics, No. 28Google Scholar
  27. 27.
    Zariski, O., Samuel, P.: Commutative algebra. Vol. II. Springer, New York (1975). (Reprint of the 1960 edition, Graduate Texts in Mathematics, Vol. 29)Google Scholar

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© Universidad Complutense de Madrid 2014

Authors and Affiliations

  1. 1.Mathematics DepartmentPurdue UniversityWest LafayetteUSA
  2. 2.Departamento de Matemáticas-IUMAUniversidad de ZaragozaZaragozaSpain

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