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\(A\) and the vanishing topology of discriminants

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Suppose thatf: ℂn, 0→ℂp, 0 is finitely\(A\)-determined withn≧p. We define a “Milnor fiber” for the discriminant off; it is the discriminant of a “stabilization” off. We prove that this “discriminant Milnor fiber” has the homotopy type of a wedge of spheres of dimensionp−1, whose number we denote byµ Δ (f). One of the main theorems of the paper is a “μ=τ” type result: if (n, p) is in the range of nice dimensions in the sense of Mather, then\(\mu _\Delta (f) \geqq A_e \)-codium,with equality iff is weighted homogeneous. Outside the nice dimensions we obtain analogous formulae with correction terms measuring the presence of unstable but topologically stable germs in the stabilization. These results are further extended to nonlinear sections of free divisors.

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Authors and Affiliations

  1. Department of Mathematics, University of North Carolina, 27599, Chapel Hill, NC, USA

    James Damon

  2. Mathematics Institute, University of Warwick, CV4 7AL, Coventry, UK

    David Mond

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Oblatum 15-VIII-1990

Partially supported by a grant from the National Science Foundation and a Fullbright Fellowship

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Damon, J., Mond, D. \(A\) and the vanishing topology of discriminants. Invent Math 106, 217–242 (1991). https://doi.org/10.1007/BF01243911

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