Mathematische Annalen

, Volume 361, Issue 3–4, pp 995–1020 | Cite as

Characterizing normal crossing hypersurfaces

  • Eleonore FaberEmail author


The objective of this article is to give an effective algebraic characterization of normal crossing hypersurfaces in complex manifolds. It is shown that a divisor (=hypersurface) has normal crossings if and only if it is a free divisor, has a radical Jacobian ideal and a smooth normalization. Using K. Saito’s theory of free divisors, also a characterization in terms of logarithmic differential forms and vector fields is found. Finally, we give another description of a normal crossing divisor in terms of the logarithmic residue using recent results of M. Granger and M. Schulze.

Mathematics Subject Classification

32S25 32S10 32A27 14B04 



I thank my advisor Herwig Hauser for introducing me to this topic and for many comments and suggestions. I also thank David Mond, Luis Narváez and Mathias Schulze for several discussions, suggestions and comments. Moreover, ideas for this work originated from discussions with Alexander G. Aleksandrov, Michel Granger, Jan Schepers, Bernard Teissier and Orlando Villamayor, whose help shall be acknowledged.


  1. 1.
    Aleksandrov, A.G.: Euler-homogeneous singularities and logarithmic differential forms. Ann. Glob. Anal. Geom. 4(2), 225–242 (1986)CrossRefzbMATHGoogle Scholar
  2. 2.
    Aleksandrov, A.G.: Nonisolated Saito singularities. Math. USSR Sbornik 65(2), 561–574 (1990)CrossRefzbMATHGoogle Scholar
  3. 3.
    Aleksandrov, A.G.: Logarithmic differential forms, torsion differentials and residue. Complex Var. Theory Appl. 50(7–11), 777–802 (2005)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Aleksandrov, A.G., Tsikh, A.K.: Théorie des résidus de Leray et formes de Barlet sur une intersection complète singulière. C. R. Acad. Sci. Paris Sér. I Math. 333(11), 973–978 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Altman, A., Kleiman, S.: Introduction to Grothendieck duality theory. In: Lecture Notes in Mathematics, vol. 146. Springer, Berlin (1970)Google Scholar
  6. 6.
    Bass, H.: On the ubiquity of Gorenstein rings. Math. Z. 82, 8–28 (1963)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Bodnár, G.: Algorithmic tests for the normal crossing property. In: Automated Deduction in Geometry, Lecture Notes in Computer Science, vol. 2930, pp. 1–20. Springer, Berlin (2004)Google Scholar
  8. 8.
    Buchweitz, R.O., Ebeling, W., von Bothmer, H.G.: Low-dimensional singularities with free divisors as discriminants. J. Algebraic Geom. 18(2), 371–406 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Buchweitz, R.O., Mond, D.: Linear free divisors and quiver representations. In: Singularities and computer algebra. In: London Math. Soc. Lecture Note Ser., vol. 324, pp. 41–77. Cambridge University Press, Cambridge (2006)Google Scholar
  10. 10.
    Calderón Moreno, F.J., Narváez Macarro, L.: The module \({\cal D}\!\!f^s\) for locally quasi-homogeneous free divisors. Comp. Math. 134, 59–64 (2002)Google Scholar
  11. 11.
    Calderón Moreno, F.J., Narváez Macarro, L.: On the logarithmic comparison theorem for integrable logarithmic connections. Proc. Lond. Math. Soc. (3) 98(3), 585–606 (2009)Google Scholar
  12. 12.
    Castro Jiménez, F., Narváez Macarro, L., Mond, D.: Cohomology of the complement of a free divisor. Trans. Am. Math. Soc. 348(8), 3037–3049 (1996)CrossRefzbMATHGoogle Scholar
  13. 13.
    Damon, J.: On the freeness of equisingular deformations of plane curve singularities. Topol. Appl. 118(1–2), 31–43 (2002). Arrangements in Boston: a Conference on Hyperplane Arrangements (1999)Google Scholar
  14. 14.
    De-Concini, C., Procesi, C.: Wonderful models of subspace arrangements. Selecta Mathematica 1(3), 459–494 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Deligne, P.: Théorie de Hodge II. Inst. Hautes Études Sci. Publ. Math. 40, 5–57 (1971)Google Scholar
  16. 16.
    Denham, G., Schenck, H., Schulze, M., Wakefield, M., Walther, U.: Local cohomology of logarithmic forms. Ann. Institut Fourier 63(3), 1177–1203 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Eisenbud, D.: Commutative algebra with a view toward algebraic geometry, graduate texts in mathematics, vol. 150. Springer, New York (1995)Google Scholar
  18. 18.
    Faber, E.: Normal crossings in local analytic geometry. Phd thesis, Universität Wien (2011)Google Scholar
  19. 19.
    Faber, E.: Towards transversality of singular varieties: splayed divisors. Publ. RIMS 49, 393–412 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Faber, E., Hauser, H.: Today’s menu: geometry and resolution of singular algebraic surfaces. Bull. Am. Math. Soc. 47(3), 373–417 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Fulton, W., MacPherson, R.: A compactification of configuration spaces. Ann. Math. 139, 183–225 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Gaffney, T., Hauser, H.: Characterizing singularities of varieties and of mappings. Invent. Math. 81(3), 427–447 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Granger, M., Mond, D., Schulze, M.: Free divisors in prehomogeneous vector spaces. Proc. Lond. Math. Soc. (3) 102(5), 923–950 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Granger, M., Schulze, M.: Normal crossing properties of complex hypersurfaces via logarithmic residues. Compos. Math. (2014, To appear)Google Scholar
  25. 25.
    Hartshorne, R.: Residues and duality. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. In: Lecture Notes in Mathematics, No. 20. Springer, Berlin (1966)Google Scholar
  26. 26.
    Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero. Ann. Math. 79, 109–326 (1964)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Hoşten, S., Smith, G.G.: Monomial ideals. In: Computations in Algebraic Geometry with Macaulay 2, Algorithms Comput. Math., vol. 8, pp. 73–100. Springer, Berlin (2002)Google Scholar
  28. 28.
    de Jong, T., Pfister, G.: Local analytic geometry. In: Advanced Lectures in Mathematics. Vieweg, Braunschweig-Wiesbaden (2000)Google Scholar
  29. 29.
    Lang, S.: Fundamentals of Differential Geometry, Graduate Texts in Mathematics, vol. 191. Springer, New York (1999)CrossRefGoogle Scholar
  30. 30.
    Lejeune-Jalabert, M., Teissier, B.: Clôture intégrale des idéaux et équisingularité with an appendix by Jean–Jacques Risler. Ann. Fac. Sci. Toulouse Math. (6) 17(4), 781–859 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Lipman, J., Teissier, B.: Pseudo-rational local rings and a theorem of Briançon Skoda about the integral closure of ideals. Michigan Math. J. 28(1), 97–116 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Looijenga, E.J.N.: Isolated singular points on complete intersections. In: London Mathematical Society Lecture Note Series, vol. 77. Cambridge University Press, Cambridge (1984)Google Scholar
  33. 33.
    Mather, J.N., Yau, S.S.T.: Classification of isolated hypersurface singularities by their moduli algebras. Invent. Math. 69, 243–251 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Matsumura, H.: Commutative ring theory. In: Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press, Cambridge (1986)Google Scholar
  35. 35.
    Mond, D., Pellikaan, R.: Fitting ideals and multiple points of analytic mappings. In: Algebraic Geometry and Complex Analysis (Pátzcuaro, 1987). Lecture Notes in Mathematics, vol. 1414, pp. 107–161. Springer, Berlin (1989)Google Scholar
  36. 36.
    Mond, D., van Straten, D.: The structure of the discriminant of some space-curve singularities. Q. J. Math. 52(3), 355–365 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Narasimhan, R.: Introduction to the theory of analytic spaces. In: Lecture Notes in Mathematics, vol. 25. Springer, Berlin (1966)Google Scholar
  38. 38.
    Narváez Macarro, L.: Linearity conditions on the Jacobian ideal and logarithmic-meromorphic comparison for free divisors. In: Singularities 1, Contemp. Math., vol. 474, pp. 245–269. American Mathematical Society, Providence (2008)Google Scholar
  39. 39.
    Nobile, A.: Some properties of the Nash blowing-up. Pac. J. Math. 60, 297–305 (1975)zbMATHMathSciNetGoogle Scholar
  40. 40.
    Orlik, P., Terao, H.: Arrangements of hyperplanes. In: Grundlehren der Mathematischen Wissenschaften, vol. 300. Springer, Berlin (1992)Google Scholar
  41. 41.
    Pellikaan, R.: Finite determinacy of funtions with non-isolated singularities. Proc. Lond. Math. Soc. 57, 357–382 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    Piene, R.: Ideals associated to a desingularization. In: Algebraic Geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), Lecture Notes in Mathematics, vol. 732, pp. 503–517. Springer, Berlin (1979)Google Scholar
  43. 43.
    Rossi, H.: Vector fields on analytic spaces. Ann. Math. 2(78), 455–467 (1963)CrossRefGoogle Scholar
  44. 44.
    Saito, K.: Theory of logarithmic differential forms and logarithmic vector fields. J. Fac. Sci. Univ. Tokyo 27(2), 265–291 (1980)zbMATHMathSciNetGoogle Scholar
  45. 45.
    Saito, K.: Primitive forms for a universal unfolding of a function with an isolated critical point. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28(3), 775–792 (1981)zbMATHMathSciNetGoogle Scholar
  46. 46.
    Scheja, G.: Fortsetzungssätze der komplex-analytischen Cohomologie und ihre algebraische Charakterisierung. Math. Ann. 157, 75–94 (1964)CrossRefzbMATHMathSciNetGoogle Scholar
  47. 47.
    Sekiguchi, J.: Three dimensional saito free divisors and singular curves. J. Sib. Fed. Univ. Math. Phys. 1, 33–41 (2008)Google Scholar
  48. 48.
    Simis, A.: Differential idealizers and algebraic free divisors. In: Commutative Algebra, Lect. Notes Pure Appl. Math., vol. 244, pp. 211–226 (2006)Google Scholar
  49. 49.
    Siu, Y.T., Trautmann, G.: Gap-sheaves and extension of coherent analytic subsheaves. In: Lecture Notes in Mathematics, vol. 172. Springer, Berlin (1971)Google Scholar
  50. 50.
    Lê, D.T., Saito K.: The local \(\pi _{1}\) of the complement of a hypersurface with normal crossings in codimension \(1\) is abelian. Ark. Mat. 22(1), 1–24 (1984)Google Scholar
  51. 51.
    Terao, H.: Arrangements of hyperplanes and their freeness I. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27, 293–312 (1980)zbMATHMathSciNetGoogle Scholar
  52. 52.
    Terao, H.: The exponents of a free hypersurface. In: Singularities, Part 2 (Arcata, Calif., 1981). In: Proc. Sympos. Pure Math., vol. 40, pp. 561–566. American Mathematical Society, Providence (1983)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

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