Non-Negative Deformations of Weighted Homogeneous Singularities

Abstract

We consider a weighted homogeneous germ of complex analytic variety (X, 0) ⊂ (ℂn, 0) and a function germ f : (ℂn, 0) → (ℂ, 0). We derive necessary and sufficient conditions for some deformations to have non-negative degree (i.e., for any additional term in the deformation, the weighted degree is not smaller) in terms of an adapted version of the relative Milnor number. We study the cases where (X, 0) is an isolated hypersurface singularity and the invariant is the Bruce-Roberts number of f with respect to (X, 0), and where (X, 0) is an isolated complete intersection or a curve singularity and the invariant is the Milnor number of the germ f: (X, 0) → ℂ. In the last part, we give some formulas for the invariants in terms of the weights and the degrees of the polynomials.

$\mu$-constant deformations of functions on an ICIS

“…A good idea to look for such counterexamples is to see families which we know, at first, that have constant Milnor number. For this, we recall the results of [18] on deformations of weighted homogeneous germs.…”

Image Milnor number and 𝒜 e -codimension for maps between weighted homogeneous irreducible curves

“…It follows from [6] that if (X, 0) is a curve ICIS weighted homogeneous of type (w 1 , · · · , w n ; d 1 , · · · , d n−1 ), then the Milnor number satisfies µ(X, 0) = d 1 · · · d n−1 (d 1 + · · · + d n−1 − w 1 − · · · − w n ) w 1 · · · w n + 1.…”

Equisingularity of map germs from a surface to the plane

“…The result for µ(S) follows now from [15,Corollary 4.2]. The multiplicity m can be computed easily by using Bezout's theorem:…”