A characterization of quasi-homogeneous complete intersection singularities
Author:
Henrik Vosegaard
Journal:
J. Algebraic Geom. 11 (2002), 581-597
DOI:
https://doi.org/10.1090/S1056-3911-02-00298-9
Published electronically:
March 13, 2002
MathSciNet review:
1894939
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Additional Information
Abstract: It is well-known that quasi-homogeneity is characterized by equality of the Milnor and Tjurina numbers for isolated complex analytic hypersurface singularities and for certain low-dimensional singularities. We prove that this characterization extends to any complex isolated complete intersection singularity of positive dimension.
[BG]BG R.-O. Buchweitz and G.-M. Greuel, The Milnor number and deformations of complex curve singularities, Invent. Math. 58 (1980), 241–281.
[G1]G1 G.-M. Greuel, Der Gauß-Manin-Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten, Math. Ann. 214 (1975), 235–266.
[G2]G2 G.-M. Greuel, Dualität in der lokalen Kohomologie isolierter Singularitäten, Math. Ann. 250 (1980), 157–173.
[G3]G3 G.-M. Greuel, On deformation of curves and a formula of Deligne, Algebraic Geometry, proceedings La Rábida 1981, LNM 961, Springer-Verlag, 1982, pp. 141–168.
[GMP]GMP G.-M. Greuel, B. Martin and G. Pfister, Numerische Charakterisierung Quasihomogener Gorenstein-Kurvensingularitäten, Math. Nachr. 124 (1985), 123-131.
[L]L E. Looijenga, Isolated singular points on complete intersections, London Math. Soc. LNS 77, Cambridge Univ. Press, 1984.
[LS]LS E. Looijenga and J. Steenbrink, Milnor number and Tjurina number of complete intersections, Math. Ann. 271 (1985), 121–124.
[Sa]Sa K. Saito, Quasihomogene isolierte Singularitäten von Hyperflächen, Invent. Math. 14 (1971), 123–142.
[Se]Se J.-P. Serre, Groupes algébriques et corps de classes, Hermann, 1959.
[St]St J.H.M. Steenbrink, Mixed Hodge structures associated with isolated singularities, Proc. Symp. Pure Math., vol. 40, 1983, pp. 513–536.
[SW]SW G. Scheia and H. Wiebe, Über Derivationen in isolierten Singularitäten auf vollständigen Durchschnitten, Math. Ann. 225 (1977), 161-171.
[V1]V1 H. Vosegaard, Generalized Tjurina numbers of an isolated complete intersection singularity, European Singularities Network Preprints Archive 1998 (to appear in Math. Ann.).
[V2]V2 H. Vosegaard, A characterization of quasi-homogeneous purely elliptic complete intersection singularities, Compositio Math. 124 (2000), 111–121.
[W]W J. Wahl, A characterization of quasi-homogeneous Gorenstein surface singularities, Compositio Math. 55 (1985), 269–288.
[BG]BG R.-O. Buchweitz and G.-M. Greuel, The Milnor number and deformations of complex curve singularities, Invent. Math. 58 (1980), 241–281.
[G1]G1 G.-M. Greuel, Der Gauß-Manin-Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten, Math. Ann. 214 (1975), 235–266.
[G2]G2 G.-M. Greuel, Dualität in der lokalen Kohomologie isolierter Singularitäten, Math. Ann. 250 (1980), 157–173.
[G3]G3 G.-M. Greuel, On deformation of curves and a formula of Deligne, Algebraic Geometry, proceedings La Rábida 1981, LNM 961, Springer-Verlag, 1982, pp. 141–168.
[GMP]GMP G.-M. Greuel, B. Martin and G. Pfister, Numerische Charakterisierung Quasihomogener Gorenstein-Kurvensingularitäten, Math. Nachr. 124 (1985), 123-131.
[L]L E. Looijenga, Isolated singular points on complete intersections, London Math. Soc. LNS 77, Cambridge Univ. Press, 1984.
[LS]LS E. Looijenga and J. Steenbrink, Milnor number and Tjurina number of complete intersections, Math. Ann. 271 (1985), 121–124.
[Sa]Sa K. Saito, Quasihomogene isolierte Singularitäten von Hyperflächen, Invent. Math. 14 (1971), 123–142.
[Se]Se J.-P. Serre, Groupes algébriques et corps de classes, Hermann, 1959.
[St]St J.H.M. Steenbrink, Mixed Hodge structures associated with isolated singularities, Proc. Symp. Pure Math., vol. 40, 1983, pp. 513–536.
[SW]SW G. Scheia and H. Wiebe, Über Derivationen in isolierten Singularitäten auf vollständigen Durchschnitten, Math. Ann. 225 (1977), 161-171.
[V1]V1 H. Vosegaard, Generalized Tjurina numbers of an isolated complete intersection singularity, European Singularities Network Preprints Archive 1998 (to appear in Math. Ann.).
[V2]V2 H. Vosegaard, A characterization of quasi-homogeneous purely elliptic complete intersection singularities, Compositio Math. 124 (2000), 111–121.
[W]W J. Wahl, A characterization of quasi-homogeneous Gorenstein surface singularities, Compositio Math. 55 (1985), 269–288.
Additional Information
Henrik Vosegaard
Affiliation:
Department of Mathematics, Aarhus University, Ny Munkegade, DK-8000 Aarhus C, Denmark
Email:
vosegaard@imf.au.dk
Received by editor(s):
July 5, 2000
Received by editor(s) in revised form:
October 10, 2000
Published electronically:
March 13, 2002
