Families of ICIS with constant total Milnor number

We show that a family of isolated complete intersection singularities (ICIS) with constant total Milnor number has no coalescence of singularities. This extends a well-known result of Gabriélov, Lazzeri and Lê for hypersurfaces. We use A’Campo’s theorem to see that the Lefschetz number of the generic monodromy of the ICIS is zero when the ICIS is singular. We give a pair applications for families of functions on ICIS which extend also some known results for functions on a smooth variety.

Topological type of discriminants of some special families

We will describe the topological type of the discriminant curve of the morphism $$(\ell , f)$$ ( ℓ , f ) , where $$\ell $$ ℓ is a smooth curve and f is an irreducible curve (branch) of multiplicity less than five or a branch such that the difference between its Milnor number and Tjurina number is less than 3. We prove that for a branch of these families, the topological type of the discriminant curve is determined by the semigroup, the Zariski invariant and at most two other analytical invariants of the branch.

The relative Bruce–Roberts number of a function on a hypersurface

We consider the relative Bruce–Roberts number $\mu _{\textrm {BR}}^{-}(f,\,X)$ of a function on an isolated hypersurface singularity $(X,\,0)$ . We show that $\mu _{\textrm {BR}}^{-}(f,\,X)$ is equal to the sum of the Milnor number of the fibre $\mu (f^{-1}(0)\cap X,\,0)$ plus the difference $\mu (X,\,0)-\tau (X,\,0)$ between the Milnor and the Tjurina numbers of $(X,\,0)$ . As an application, we show that the usual Bruce–Roberts number $\mu _{\textrm {BR}}(f,\,X)$ is equal to $\mu (f)+\mu _{\textrm {BR}}^{-}(f,\,X)$ . We also deduce that the relative logarithmic characteristic variety $LC(X)^{-}$ , obtained from the logarithmic characteristic variety $LC(X)$ by eliminating the component corresponding to the complement of $X$ in the ambient space, is Cohen–Macaulay.

A generalization of Milnor’s formula

The Milnor number $$\mu _f$$ μ f of a holomorphic function $$f :({\mathbb {C}}^n,0) \rightarrow ({\mathbb {C}},0)$$ f : ( C n , 0 ) → ( C , 0 ) with an isolated singularity has several different characterizations as, for example: 1) the number of critical points in a morsification of f, 2) the middle Betti number of its Milnor fiber $$M_f$$ M f , 3) the degree of the differential $${\text {d}}f$$ d f at the origin, and 4) the length of an analytic algebra due to Milnor’s formula $$\mu _f = \dim _{\mathbb {C}}{\mathcal {O}}_n/{\text {Jac}}(f)$$ μ f = dim C O n / Jac ( f ) . Let $$(X,0) \subset ({\mathbb {C}}^n,0)$$ ( X , 0 ) ⊂ ( C n , 0 ) be an arbitrarily singular reduced analytic space, endowed with its canonical Whitney stratification and let $$f :({\mathbb {C}}^n,0) \rightarrow ({\mathbb {C}},0)$$ f : ( C n , 0 ) → ( C , 0 ) be a holomorphic function whose restriction f|(X, 0) has an isolated singularity in the stratified sense. For each stratum $${\mathscr {S}}_\alpha $$ S α let $$\mu _f(\alpha ;X,0)$$ μ f ( α ; X , 0 ) be the number of critical points on $${\mathscr {S}}_\alpha $$ S α in a morsification of f|(X, 0). We show that the numbers $$\mu _f(\alpha ;X,0)$$ μ f ( α ; X , 0 ) generalize the classical Milnor number in all of the four characterizations above. To this end, we describe a homology decomposition of the Milnor fiber $$M_{f|(X,0)}$$ M f | ( X , 0 ) in terms of the $$\mu _f(\alpha ;X,0)$$ μ f ( α ; X , 0 ) and introduce a new homological index which computes these numbers directly as a holomorphic Euler characteristic. We furthermore give an algorithm for this computation when the closure of the stratum is a hypersurface.

Conjectures on Stably Newton Degenerate Singularities

We discuss a problem of Arnold, whether every function is stably equivalent to one which is non-degenerate for its Newton diagram. We argue that the answer is negative. We describe a method to make functions non-degenerate after stabilisation and give examples of singularities where this method does not work. We conjecture that they are in fact stably degenerate, that is not stably equivalent to non-degenerate functions.We review the various non-degeneracy concepts in the literature. For finite characteristic, we conjecture that there are no wild vanishing cycles for non-degenerate singularities. This implies that the simplest example of singularities with finite Milnor number, $$x^p+x^q$$ x p + x q in characteristic p, is not stably equivalent to a non-degenerate function. We argue that irreducible plane curves with an arbitrary number of Puiseux pairs (in characteristic zero) are stably non-degenerate. As the stabilisation involves many variables, it becomes very difficult to determine the Newton diagram in general, but the form of the equations indicates that the defining functions are non-degenerate.

The Image Milnor Number and Excellent Unfoldings

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.