Abstract
To each isolated singularity of a hypersurface of dimensionn, one associates the local fundamental groupG of the moduli space minus the discriminant locus, and a representation σ:G→Aut(H), whereH is then-homology group, with integer coefficients, of the non singular fibre. Although, in general it is very difficult to determine even a presentation ofG, we show that the image of σ can be computed rather easily, by exploiting some relations in a first aproximate presentation ofG, in the case of Brieskorn polynomials namely, polynomials of the type\(x_0^{a_0 } + \cdot \cdot \cdot + x_0^{a_n } \).
In this way we solve an open problem stated by Brieskorn [1] and Pham [8].
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This work was conducted while both authors belonged to the G.N.S.A.G.A. of the C.N.R.1 (Δ,0) is in general a not reduced space and it can be defined in the following way: let σ:(X, x)→(T, t) be the semiuniversal deformation of (ϕ−1;(0),0); then φ is induced by a map τ:(ℂh, 0)→(T,t). If (D, t) denotes the (reduced) discriminant locus of σ, then (Δ, 0) is (τ−1(D),0) with the fibre structure.
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Hefez, A., Lazzeri, F. The intersection matrix of Brieskorn singularities. Invent Math 25, 143–157 (1974). https://doi.org/10.1007/BF01390172
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DOI: https://doi.org/10.1007/BF01390172