A reduction theorem for the Zariski multiplicity conjecture

Massey, David B.

doi:10.1090/S0002-9939-1989-0949879-0

A reduction theorem for the Zariski multiplicity conjecture
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Proc. Amer. Math. Soc. 106 (1989), 379-383 Request permission

Abstract:

We prove that the Zariski multiplicity conjecture for families of hypersurfaces of dimension ${}\ne 2$ with isolated singularities is equivalent to the conjecture for families of hypersurfaces with line singularities.

References

Gert-Martin Greuel, Constant Milnor number implies constant multiplicity for quasihomogeneous singularities, Manuscripta Math. 56 (1986), no. 2, 159–166. MR 850367, DOI 10.1007/BF01172153
Helmut A. Hamm and Lê Dũng Tráng, Un théorème de Zariski du type de Lefschetz, Ann. Sci. École Norm. Sup. (4) 6 (1973), 317–355. MR 401755
I. N. Iomdin, Complex surfaces with a one-dimensional set of singularities, Sibirsk. Mat. Ž. 15 (1974), 1061–1082, 1181 (Russian). MR 0447621
Lê Dũng Tráng, Calcul du nombre de cycles évanouissants d’une hypersurface complexe, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 4, 261–270 (French, with English summary). MR 330501

Ensembles analytiques complexes avec lieu singulier de dimension un (D’Apres I. N. Iomdine) Seminar on singularities (Paris, 1976/1977)

7

Topologie des singuarlities des hypersurfaces complexes

Lê Dũng Tráng and C. P. Ramanujam, The invariance of Milnor’s number implies the invariance of the topological type, Amer. J. Math. 98 (1976), no. 1, 67–78. MR 399088, DOI 10.2307/2373614

Families of hypersurfaces with one-dimensional singular setes

Donal B. O’Shea, Topologically trivial deformations of isolated quasihomogeneous hypersurface singularities are equimultiple, Proc. Amer. Math. Soc. 101 (1987), no. 2, 260–262. MR 902538, DOI 10.1090/S0002-9939-1987-0902538-0
Oscar Zariski, Some open questions in the theory of singularities, Bull. Amer. Math. Soc. 77 (1971), 481–491. MR 277533, DOI 10.1090/S0002-9904-1971-12729-5