The Image Milnor Number And Excellent Unfoldings

Abstract

We show three basic properties of the image Milnor number µI(f) of a germ $f\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ with isolated instability. First, we show the conservation of the image Milnor number, from which one can deduce the upper semi-continuity and the topological invariance for families. Second, we prove the weak Mond’s conjecture, which states that µI(f) = 0 if and only if f is stable. Finally, we show a conjecture by Houston that any family $f_t\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ with $\mu_I(\,f_t)$ constant is excellent in Gaffney’s sense. For technical reasons, in the last two properties, we consider only the corank 1 case.

Whitney equisingular unfoldings of corank 1 germs

“…Furthermore, in the pair of dimensions (n, p), we deduce from [12,Theorem 4.3] that d(f ) is at most the integer part of p p−n . Finally, taking into account these previous remarks, if s(f ) ≥ d(f ) then this maximum is attained because the proof of [8,Lemma 3.3] only relies on the dimension of the multiple point spaces.…”

“…If either (n, p) are nice dimensions or f u 0 has only corank one singularities, then for almost any u in a neighbourhood of u 0 , the mapping f u : X u → C p has only stable singularities (see, for example, [28, Propositions 5.5 and 5.6]). A desirable property of this topological A -invariant is that it is conservative, as it was for the usual image Milnor number (see [8,Theorem 2.6]). The reasoning that proves the conservation of the usual image Milnor number can be applied verbatim for the general version, and is based as well on Theorem 2.11.…”

“…Also, taking into account that the Euler-Poincaré characteristic of every page remains invariant, see [23, Example 1.F], and that the image of the versal unfolding is contractible, it remains to compute χ E * , * 1 to get µ I (f ), and the terms of the sum are arranged to define Houston's alternating Milnor numbers, µ Alt k (f ) (actually he computes them through the limit of the spectral sequence, both ways give the same result). This ideas and the previous proofs inspire the following definition (see also the simplification and developments of it in [8], particularly how to determine d(f ) when s(f ) > d(f ) with [8, Lemma 3.3]). Definition 3.12.…”

“…Corollary 5.8 (see [8,Theorem 3.9]). For f as in Theorem 5.4, the following are equivalent: (i) f is stable, (ii) µ I (f ) = 0, and (iii) µ D (f ) = 0.…”

“…In a recent paper, [8], we solved the second question for families of multigerms f t : (C n , S) → (C n+1 , 0) of corank one. We prove that the family is excellent if the image Milnor number µ I (f t ) is constant on t, which solved a conjecture posed by Houston in [15].…”