Milnor Number Research Articles

Foliations on $$\mathbb {CP}^2$$ with a unique singular point without invariant algebraic curves

We prove that a foliation on $$\mathbb {CP}^2$$ of degree d with a singular point of type saddle-node with Milnor number $$d^2+d+1$$ does not have invariant algebraic curves. We give a family of this kind of foliations. We also present a family of foliations of degree d with a unique nilpotent singularity without invariant algebraic curves for d odd greater than 1. Finally we prove that the space of foliations on $$\mathbb {CP}^2$$ of degree $$d \ge 2$$ with a unique singular point has dimension at least $$3d+2$$ .

Arnold's problem on monotonicity of the Newton number for surface singularities

According to the Kouchnirenko Theorem, for a generic (meaning non-degenerate in the Kouchnirenko sense) isolated singularity $f$ its Milnor number $\mu (f)$ is equal to the Newton number $\nu (\mathbf{\Gamma}_{+}(f))$ of a combinatorial object associated to $f$, the Newton polyhedron $\mathbf{\Gamma}_+ (f)$. We give a simple condition characterizing, in terms of $\mathbf{\Gamma}_+ (f)$ and $\mathbf{\Gamma}_+ (g)$, the equality $\nu (\mathbf{\Gamma}_{+}(f)) = \nu (\mathbf{\Gamma}_{+}(g))$, for any surface singularities $f$ and $g$ satisfying $\mathbf{\Gamma}_+ (f) \subset \mathbf{\Gamma}_+ (g)$. This is a complete solution to an Arnold problem (No. 1982-16 in his list of problems) in this case.

Hypersurface singularities in arbitrary characteristic

The study of singularities of algebraic or analytic spaces over the field of complex numbers is a traditional subject, but, in contrast, the parallel theory over arbitrary algebraically closed fields is still poor and there are lots of interesting questions to be answered. Our main concern here is the study of the Milnor number μ of an isolated hypersurface singularity which is defined as the codimension of the ideal generated by the partial derivatives of a power series that represents locally the hypersurface. This is an important topological invariant of the singularity over the complex numbers, but its behavior changes dramatically when the base field has positive characteristic, in which case it may be infinite and depends upon the local equation of the hypersurface, not being an intrinsic invariant. In this paper we will study the variation of the Milnor number in terms of the local equations, showing that its minimum value has an intrinsic meaning and give necessary and sufficient conditions for its invariance. We also relate the smoothness of the generic fiber of an isolated hypersurface singularity deformation with the finiteness of this number, connecting it to a Bertini type result. At the end, we will show how, in arbitrary characteristic, one defines the sequence μ⁎ of Milnor numbers of sections of a hypersurface singularity by general linear spaces of increasing codimension.

A Note on Tropical Curves and the Newton Diagrams of Plane Curve Singularities

For an isolated singularity which is Newton non-degenerate and also convenient, the Milnor number can be computed from the complement of its Newton diagram in the first quadrant by using Kouchnirenko's formula. In this paper, we consider tropical curves dual to subdivisions of this complement for a plane curve singularity, and show that there exists a tropical curve by which we can count the Milnor number. Our formula may be regarded as a tropical version of the well-known formula by the real morsification due to A'Campo and Gusein-Zade.

A note on 3-manifolds and complex surface singularities

This article is motivated by the original Casson invariant regarded as an integral lifting of the Rochlin invariant. We aim to defining an integral lifting of the Adams e-invariant of stably framed 3-manifolds, perhaps endowed with some additional structure. We succeed in doing so for manifolds which are links of normal complex Gorenstein smoothable singularities. These manifolds are naturally equipped with a canonical $$\mathrm{SU}(2)$$-frame. To start we notice that the set of homotopy classes of $$\mathrm{SU}(2)$$-frames on the stable tangent bundle of every closed oriented 3-manifold is canonically a $$\mathbb Z$$-torsor. Then we define the $$\widehat{E}$$-invariant for the manifolds in question, an integer that modulo 24 is the Adams e-invariant. The $$\widehat{E}$$-invariant for the canonical frame equals the Milnor number plus 1, so this brings a new viewpoint on the Milnor number of the smoothable Gorenstein surface singularities.

A Lê-Greuel type formula for the image Milnor number

Let $f:(\mathbb{C}^n,0)\rightarrow (\mathbb{C}^{n+1},0)$ be a corank 1 finitely determined map germ. For a generic linear form $p:(\mathbb{C}^{n+1},0)\to(\mathbb{C},0)$ we denote by $g:(\mathbb{C}^{n-1},0)\rightarrow (\mathbb{C}^{n},0)$ the transverse slice of $f$ with respect to $p$. We prove that the sum of the image Milnor numbers $\mu_I(f)+\mu_I(g)$ is equal to the number of critical points of $p|_{X_s}:X_s\to\mathbb{C}$ on all the strata of $X_s$, where $X_s$ is the disentanglement of $f$ (i.e., the image of a stabilisation $f_s$ of $f$).

A Jacobian module for disentanglements and applications to Mond’s conjecture

Given a germ of holomorphic map $f$ from $\mathbb C^n$ to $\mathbb C^{n+1}$, we define a module $M(f)$ whose dimension over $\mathbb C$ is an upper bound for the $\mathscr A$-codimension of $f$, with equality if $f$ is weighted homogeneous. We also define a relative version $M_y(F)$ of the module, for unfoldings $F$ of $f$. The main result is that if $(n,n+1)$ are nice dimensions, then the dimension of $M(f)$ over $\mathbb C$ is an upper bound of the image Milnor number of $f$, with equality if and only if the relative module $M_y(F)$ is Cohen-Macaulay for some stable unfolding $F$. In particular, if $M_y(F)$ is Cohen-Macaulay, then we have Mond's conjecture for $f$. Furthermore, if $f$ is quasi-homogeneous, then Mond's conjecture for $f$ is equivalent to the fact that $M_y(F)$ is Cohen-Macaulay. Finally, we observe that to prove Mond's conjecture, it suffices to prove it in a suitable family of examples.

A note on a question of Dimca and Greuel

In this note, we give a positive answer to a question of Dimca and Greuel about the quotient between the Milnor and Tjurina numbers of an isolated plane curve singularity in the cases of one Puiseux pair and semi-quasi-homogeneous singularities.

Non-degenerate jumps of Milnor numbers of quasihomogeneous singularities

Let $f_0: (\mathbb {C}^n,0)\rightarrow (\mathbb {C},0)$ be a holomorphic function germ having an isolated critical point at $0\in \mathbb {C}^n$ and let $[f_0]$ be the singularity generated by $f_0$, i.e. the equivalence class of $f_0$ with respect t

A remark on the topology of complex polynomial functions

We begin with a survey of the notion of atypical values of R. Thom. We give a theorem on the atypical values of complex polynomials which generalizes a theorem of Broughton (On the topology of polynomial hypersurfaces in singularities, Part 1, 167–178, proceedings of symposia in pure mathematics, vol 40. American Mathematicaol Society, Providence, 1983) and Broughton (Milnor numbers and the topology of polynomial hypersurfaces. Invent Math 92:217–241, 1988). For this purpose we introduce the notion of atypical value from infinity. Our proofs are geometrical. We end with some open problems to understand the difference between tame polynomials at infinity and polynomials without atypical values from infinity.

Milnor and Tjurina numbers for a hypersurface germ with isolated singularity

Assume that f:(Cn,0)→(C,0) is an analytic function germ at the origin with only isolated singularity. Let μ and τ be the corresponding Milnor and Tjurina numbers. We show that μτ≤n. As an application, we give a lower bound for the Tjurina number in terms of n and the multiplicity of f at the origin.

Jumps of Milnor Numbers of Brieskorn\u2013Pham Singularities in Non-degenerate Families

The jump of the Milnor number of an isolated singularity f_{0} is the minimal non-zero difference between the Milnor numbers of f_{0} and one of its deformation (f_{s}). In the case f_{s} are non-degenerate singularities we call the jump non-degenerate. We give a formula (an inductive algorithm using diophantine equations) for the non-degenerate jump of f_{0} in the case f_{0} is a convenient singularity with only one (n-1)-dimensional face of its Newton diagram which equivalently (in our problem) can be replaced by the Brieskorn–Pham singularities.

Milnor Numbers of Deformations of Semi-Quasi-Homogeneous Plane Curve Singularities

The aim of this paper is to show the possible Milnor numbers of deformations of semi-quasi-homogeneous isolated plane curve singularity f. Assuming that f is irreducible, one can write f = sum _{qalpha + pbeta ge pq}c_{alpha beta } x^{alpha }y^{beta } where c_{p0}c_{0q}ne 0, 2le p<q and p, q are coprime. We show that as Milnor numbers of deformations of f one can attain all numbers from mu (f) to mu (f)-r(p-r), where qequiv r (mathrm{mod }p). Moreover, we provide an algorithm which produces the desired deformations.

The Milnor Number of Plane Branches with Tame Semigroups of Values

The Milnor number of an isolated hypersurface singularity, defined as the codimension $$\mu (f)$$ of the ideal generated by the partial derivatives of a power series f that represents locally the hypersurface, is an important topological invariant of the singularity over the complex numbers. However it may loose its significance when the base field is arbitrary. It turns out that if the ground field is of positive characteristic, this number depends upon the equation f representing the hypersurface, hence it is not an invariant of the hypersurface. For a plane branch represented by an irreducible convergent power series f in two indeterminates over the complex numbers, it was shown by Milnor that $$\mu (f)$$ always coincides with the conductor c(f) of the semigroup of values S(f) of the branch. This is not true anymore if the characteristic of the ground field is positive. In this paper we show that, over algebraically closed fields of arbitrary characteristic, this is true, provided that the semigroup S(f) is tame, that is, the characteristic of the field does not divide any of its minimal generators.

On 1-forms on isolated complete intersection curve singularities

We collect some classical results about holomorphic 1-forms of a reduced complex curve singularity. They are used to study the pull-back of holomorphic 1-forms on an isolated complete intersection curve singularity under the normalization morphism. We wonder whether the Milnor number $\mu$ and the Tjurina number $\tau$ of any isolated plane curve singularity satisfy the inequality $3\mu <4\tau$.

A sharp lower bound for the geometric genus and Zariski multiplicity question

It is well known that the geometric genus and multiplicity are two important invariants for isolated singularities. In this paper we give a sharp lower estimate of the geometric genus in terms of the multiplicity for isolated hypersurface singularities. In 1971, Zariski asked whether the multiplicity of an isolated hypersurface singularity depends only on its embedded topological type. This problem remains unsettled. In this paper we answer positively Zariski’s multiplicity question for isolated hypersurface singularity if Milnor number or geometric genus is small.

Kähler differentials on a plane curve and a counterexample to a result of Zariski in positive characteristic

The aim of this article is to develop the theory of Kähler differentials on algebroid irreducible plane curves defined over algebraically closed fields of positive characteristic and to compare it with the theory over C. We emphasize here the study of the properties of the set of values of Kähler differentials, since they were an important invariant for the analytic classification of branches (cf. [8]), hoping that they will play an important role in positive characteristic too. At the end, we give a counterexample in positive characteristic for the result of Zariski that asserts that if a complex algebroid irreducible plane curve has coincident Tjurina and Milnor numbers, then the curve is equivalent to a monomial curve.

On discriminants, Tjurina modifications and the geometry of determinantal singularities

We describe a method for computing discriminants for a large class of families of isolated determinantal singularities – families induced by perturbations of matrices. The approach intrinsically provides a decomposition of the discriminant into two parts and allows the computation of the determinantal and the non-determinantal loci of the family without extra effort; only the latter manifests itself in the Tjurina transform. This knowledge is then applied to the case of Cohen–Macaulay codimension 2 singularities putting several known, but previously unexplained observations into context and explicitly constructing a counterexample to Wahl's conjecture (see [35], section 6) on the relation of Milnor and Tjurina numbers for surface singularities.

The complex gradient inequality with parameter

We prove that given a holomorphic family of holomorphic functions with isolated singularities at the origin and constant Milnor number, it is possible to obtain the gradient inequality with a uniform exponent.

Generic sections of essentially isolated determinantal singularities

We study the essentially isolated determinantal singularities (EIDS), defined by Ebeling and Gusein-Zade [S. M. Guseĭn-Zade and W. Èbeling, On the indices of 1-forms on determinantal singularities, Tr. Mat. Inst. Steklova 267 (2009) 119–131], as a generalization of isolated singularity. We prove in dimension [Formula: see text], a minimality theorem for the Milnor number of a generic hyperplane section of an EIDS, generalizing the previous results by Snoussi in dimension [Formula: see text]. We define strongly generic hyperplane sections of an EIDS and show that they are still EIDS. Using strongly general hyperplanes, we extend a result of Lê concerning the constancy of the Milnor number.