Duke Mathematical Journal

The Alexander polynomial of a plane curve singularity via the ring of functions on it

A. Campillo, F. Delgado, and S. M. Gusein-Zade

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We prove two formulae that express the Alexander polynomial $\Delta\sp C$ of several variables of a plane curve singularity $C$ in terms of the ring $\mathscr {O}\sb C$ of germs of analytic functions on the curve. One of them expresses $\Delta\sp C$ in terms of dimensions of some factors corresponding to a (multi-indexed) filtration on the ring $\mathscr {O}\sb C$. The other one gives the coefficients of the Alexander polynomial $\Delta\sp C$ as Euler characteristics of some explicitly described spaces (complements to arrangements of projective hyperplanes).

Article information

Duke Math. J., Volume 117, Number 1 (2003), 125-156.

First available in Project Euclid: 26 May 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14H20: Singularities, local rings [See also 13Hxx, 14B05]
Secondary: 32Sxx: Singularities [See also 58Kxx]


Campillo, A.; Delgado, F.; Gusein-Zade, S. M. The Alexander polynomial of a plane curve singularity via the ring of functions on it. Duke Math. J. 117 (2003), no. 1, 125--156. doi:10.1215/S0012-7094-03-11712-3. https://projecteuclid.org/euclid.dmj/1085598340

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  • N. A'Campo, La fonction zêta d'une monodromie, Comment. Math. Helv. 50 (1975), 233--248.
  • A. Campillo, F. Delgado, and S. M. Gusein-Zade, Extended semigroup of a plane curve singularity, Proc. Steklov Inst. Math. 221 (1998), 139--156.
  • --. --. --. --., The Alexander polynomial of a plane curve singularity, and the ring of functions on the curve, Russian Math. Surveys 54 (1999), 634--635.
  • --. --. --. --., On generators of the semigroup of a plane curve singularity, J. London Math. Soc. (2) 60 (1999), 420--430.
  • --. --. --. --., On the monodromy of a plane curve singularity and the Poincaré series of its ring of functions, Funct. Anal. Appl. 33 (1999), 56--57.
  • --. --. --. --., Integration with respect to Euler characteristic over a function space, and the Alexander polynomial of a plane curve singularity, Russian Math. Surveys 55 (2000), 1148--1149.
  • --. --. --. --., ``On the monodromy at infinity of a plane curve and the Poincaré series of its coordinate ring (in Russian)'' in Geometry and Topology (in Russian), Itogi Nauki Tekh. Ser. Sovrem. Mat. Prilozh. Temat. Obz. 68, Vseross. Inst. Nauchn. i Tekhn. Inform. (VINITI), Moscow, 1999, 49--54., ; English translation in Pontryagin Conference, 8: Topology (Moscow, 1998), J. Math. Sci. (New York) 105 (2001), 1839--1842.
  • A. Campillo, F. Delgado, and K. Kiyek, Gorenstein property and symmetry for one-dimensional local Cohen-Macaulay rings, Manuscripta Math. 83 (1994), 405--423.
  • F. Delgado de la Mata, The semigroup of values of a curve singularity with several branches, Manuscripta Math. 59 (1987), 347--374.
  • --. --. --. --., ``An arithmetical factorization for the critical point set of some maps from $\C^2$ to $\C^2$'' in Singularities (Lille, 1991), ed. J.-P. Brasselet, London Math. Soc. Lecture Note Ser. 201, Cambridge Univ. Press, Cambridge, 1994, 61--100.
  • W. Ebeling, Poincaré series and monodromy of a two-dimensional quasihomogeneous hypersurface singularity, Manuscripta Math. 107 (2002), 271--282. \CMP1 906 197
  • D. Eisenbud and W. Neumann, Three-Dimensional Link Theory and Invariants of Plane Curve Singularities, Ann. of Math. Stud. 110, Princeton Univ. Press, Princeton, 1985.
  • R. Waldi, Wertehalbgruppe und Singularität einer ebenen algebraischen Kurve, Ph.D. dissertation, University of Regensburg, Regensburg, 1972.
  • M. Yamamoto, Classification of isolated algebraic singularities by their Alexander polynomials, Topology 23 (1984), 277--287.
  • O. Zariski, Le problème des modules pour les branches planes, Hermann, Paris, 1986.