Milnor Number Research Articles

Tjurina number of a local complete intersection curve

In contrast to the Milnor number, there was no known formula relating the Tjurina number of a reducible curve to the Tjurina numbers of its components. In this work we exhibit such a formula relating Tjurina number of a complete intersection algebroid (or analytic) curve over an algebraically closed field of characteristic zero (or over C ) to the Tjurina numbers of its components, involving the intersection indices among the components and numerical analytic invariants extracted from modules of Kähler differentials on unions of branches of the curve.

Bruce–Roberts numbers and quasihomogeneous functions on analytic varieties

Given a germ of an analytic variety X and a germ of a holomorphic function f with a stratified isolated singularity with respect to the logarithmic stratification of X, we show that under certain conditions on the singularity type of the pair (f, X), the following relative analog of the well-known K. Saito’s theorem holds true: equality of the relative Milnor and Tjurina numbers of f with respect to X (also known as Bruce–Roberts numbers) is equivalent to the relative quasihomogeneity of the pair (f, X), i.e. to the existence of a coordinate system such that both f and X are quasihomogeneous with respect to the same positive rational weights.

On the GIT-Stability of Foliations of Degree 3 with a Unique Singular Point

Applying Geometric Invariant Theory (GIT), we study the stability of foliations of degree 3 on P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {P}^{2}$$\\end{document} with a unique singular point of multiplicity 1, 2, or 3 and Milnor number 13. In particular, we characterize those foliations for multiplicity 2 in three cases: stable, strictly semistable, and unstable.

On $\mu$-Zariski pairs of links

The notion of Zariski pairs for projective curves in $\mathbb{P}^{2}$ is known since the pioneer paper of Zariski. In this paper, we introduce a notion of Zariski pair of links in the class of isolated hypersurface singularities. Such a pair is canonically produced from a Zariski (or a weak Zariski) pair of curves $C = \{f(x,y,z) = 0\}$ and $C' = \{g(x, y, z) = 0\}$ of degree $d$ by simply adding a monomial $z^{d+m}$ to $f$ and $g$ so that the corresponding affine hypersurfaces have isolated singularities at the origin. They have a same zeta function and a Milnor number. We give new examples of weak Zariski pairs which have same $\mu^{*}$ sequences and same zeta functions but two functions belong to different connected components of $\mu$-constant strata (Theorem 14). Two link 3-folds are not diffeomorphic and they are distinguished by the first homology, which implies the Jordan forms of their monodromies are different (Theorem 24). We start from weak Zariski pairs of projective curves to construct new Zariski pairs of surfaces which have non-diffeomorphic link 3-folds. We also prove that the hypersurface pair constructed from a Zariski pair of irreducible plane curves with simple singularities give a diffeomorphic links (Theorem 25).

The Bruce–Roberts Numbers of a Function on an ICIS

Abstract We relate a number of invariants of a function germ with isolated singularity over an isolated complete intersection singularity (ICIS), generalizing our previous results, [17, 18], as a consequence we present a new relatively elementary proof of Greuel’s theorem that for a weighted homogeneous ICIS the Milnor number is equal to Tjurina number. With this relation, we are able to prove that the relative logarithmic characteristic variety is Cohen–Macaulay at any point and the logarithmic characteristic variety is Cohen–Macaulay at any point not in $X\times \{0\}$.

Algorithms for computing mixed multiplicities, mixed volumes, and sectional Milnor numbers

Algorithms for computing mixed multiplicities, mixed volumes, and sectional Milnor numbers

On Briançon–Skoda theorem for foliations

We generalize Mattei’s result relative to the Briançon–Skoda theorem for foliations to the family of foliations of the second type. We use this generalization to establish relationships between the Milnor and Tjurina numbers of foliations of second type, inspired by the results obtained by Liu for complex hypersurfaces and we determine a lower bound for the global Tjurina number of an algebraic curve.

The Tjurina Number for Sebastiani–Thom Type Isolated Hypersurface Singularities

In this note, we provide a formula for the Tjurina number of a join of isolated hypersurface singularities in separated variables. From this, we are able to provide a characterization of isolated hypersurface singularities whose difference between the Milnor and Tjurina numbers is less or equal than two arising as the join of isolated hypersurface singularities in separated variables. Also, we are able to provide new upper bounds for the quotient of Milnor and Tjurina numbers of certain join of isolated hypersurface singularities. Finally, we deduce an upper bound for the quotient of Milnor and Tjurina numbers in terms of the singularity index of any isolated hypersurface singularity, not necessarily a join of singularities.

On the Milnor number of non‐isolated singularities of holomorphic foliations and its topological invariance

AbstractWe define the Milnor number of a one‐dimensional holomorphic foliation as the intersection number of two holomorphic sections with respect to a compact connected component of its singular set. Under certain conditions, we prove that the Milnor number of on a three‐dimensional manifold with respect to is invariant by topological equivalences.

K-th Milnor numbers and k-th Tjurina numbers of weighted homogeneous singularities

k-th Milnor numbers and k-th Tjurina numbers of weighted homogeneous singularities

Thom condition and monodromy

We give the definition of the Thom condition and we show that given any germ of complex analytic function f:(X,x)rightarrow ({mathbb {C}},0) on a complex analytic space X, there exists a geometric local monodromy without fixed points, provided that fin {mathfrak {m}}_{X,x}^2, where {mathfrak {m}}_{X,x} is the maximal ideal of {mathcal {O}}_{X,x}. This result generalizes a well-known theorem of the second named author when X is smooth and proves a statement by Tibar in his PhD thesis. It also implies the A’Campo theorem that the Lefschetz number of the monodromy is equal to zero. Moreover, we give an application to the case that X has maximal rectified homotopical depth at x and show that a family of such functions with isolated critical points and constant total Milnor number has no coalescing of singularities.

Minimal exponents of hyperplane sections: a conjecture of Teissier

We prove a conjecture of Teissier asserting that if f has an isolated singularity at P and H is a smooth hypersurface through P , then \widetilde{\alpha}_P(f)\geq \widetilde{\alpha}_P(f\vert_H)+\frac{1}{\theta_P(f)+1} , where \widetilde{\alpha}_P(f) and \widetilde{\alpha}_P(f\vert_H) are the minimal exponents at P of f and f\vert_H , respectively, and \theta_P(f) is an invariant obtained by comparing the integral closures of the powers of the Jacobian ideal of f and of the ideal defining P . The proof builds on the approaches of Loeser (1984) and Elduque–Mustaţă (2021). The new ingredients are a result concerning the behavior of Hodge ideals with respect to finite maps and a result about the behavior of certain Hodge ideals for families of isolated singularities with constant Milnor number. In the opposite direction, we show that for every f , if H is a general hypersurface through P , then \widetilde{\alpha}_P(f)\leq \widetilde{\alpha}_P(f\vert_H)+\frac{1}{{\rm mult}_P(f)} , extending a result of Loeser from the case of isolated singularities.

On Betti numbers of the gluing of germs of formal complex spaces

AbstractIn this paper, we show that the gluing of germs of formal complex spaces is also a formal complex space. Moreover, we study the Betti numbers of the gluing of formal complex spaces and, for instance, we show that the Betti numbers satisfy for all , where is a d‐dimensional germ of formal complex space given by a regular and a singular germ of formal complex spaces with the same dimension. In particular, this proves the Buchsbaum–Eisenbud–Horrocks conjecture for certain gluing of germs of formal complex spaces. Some formulas for the Betti numbers of the gluing of germs of formal complex spaces are given in terms of the invariant multiplicities, Euler obstruction, Milnor number, and polar multiplicities. As an application, some upper bounds for the Betti numbers of the gluing of some germs of complete intersection are provided, using recent progress on the Watanabe's conjecture.

Singularities of mappings on ICIS and applications to Whitney equisingularity

We study germs of analytic maps f:(X,S)→(Cp,0), when X is an icis of dimension n<p. We define an image Milnor number, generalizing Mond's definition, μI(X,f) and give results known for the smooth case such as the conservation of this quantity by deformations. We also use this to characterise the Whitney equisingularity of families of corank one map germs ft:(Cn,S)→(Cn+1,0) with isolated instabilities in terms of the constancy of the μI⁎-sequences of ft and the projections π:D2(ft)→Cn, where D2(ft) is the icis given by double point space of ft in Cn×Cn. The μI⁎-sequence of a map germ consist of the image Milnor number of the map germ and all its successive transverse slices.

On the quotient of Milnor and Tjurina numbers for two‐dimensional isolated hypersurface singularities

AbstractIn this paper we give a complete answer to a question posed by Dimca and Greuel about the quotient of the Milnor and Tjurina numbers of a plane curve singularity. We put this question into a general framework of the study of the difference of Milnor and Tjurina numbers for isolated complete intersection singularities showing its connection with other problems in singularity theory.

Limit spectral distribution for non-degenerate hypersurface singularities

We establish Kyoji Saito's continuous limit distribution for the spectrum of Newton non-degenerate hypersurface singularities. Investigating Saito's notion of dominant value in the case of irreducible plane curve singularities, we find that the log canonical threshold is strictly bounded below by the doubled inverse of the Milnor number. We show that this bound is asymptotically sharp.

On the Milnor and Tjurina Numbers of Zero-Dimensional Singularities

On the Milnor and Tjurina Numbers of Zero-Dimensional Singularities

On the analytic invariants and semiroots of plane branches

The value semigroup of a k-semiroot Ck of a plane branch C allows us to recover part of the value semigroup Γ=〈v0,…,vg〉 of C, that is, it is related to topological invariants of C. In this paper we consider the set of values of differentials Λk of Ck, that is an analytical invariant, and we show how it determine part of the set of values of differentials Λ of C. As a consequence, in a fixed topological class, we relate the Tjurina number τ of C with the Tjurina number of Ck. In particular, we show that τ≤μ−3ng−24μg−1 where ng=gcd(v0,…,vg−1), μ and μg−1 denote the Milnor number of C and Cg−1 respectively. If ng=2, we have that τ=μ−μg−1 for any curve in the topological class determined by Γ that is a generalization of a result obtained by Luengo and Pfister.

TOPOLOGICAL INVARIANTS AND MILNOR FIBRE FOR \(\mathcal{A}\)-FINITE GERMS \(C^2\) to \(C^3\)

This note is the observation that a simple combination of known results shows that the usual analytic invariants of a finitely determined multi-germ \(f : (C^2 , S) → (C^3 , 0) \)—namely, the image Milnor number , the number of cross-caps and triple points, \(C\) and \(T\), and the Milnor number \(μ(Σ)\) of the curve of double points in the target—depend only on the embedded topological type of the image of \(f\). As a consequence, one obtains the topological invariance of the sign-refined Smale invariant for immersions \(j : S^3 \looparrowright S^5\) associated to finitely determined map germs \((C^2 , 0) → (C^3 , 0)\).

Separatrices for Real Analytic Vector Fields in the Plane

Let $X$ be a germ of real analytic vector field at $({\mathbb R}^{2},0)$ with an algebracally isolated singularity. We say that $X$ is a topological generalized curve if there are no topological saddle-nodes in its reduction of singularities. In this case, we prove that if either the order $\nu_{0}(X)$ or the Milnor number $\mu_{0}(X)$ is even, then $X$ has a formal separatrix, that is, a formal invariant curve at $0 \in {\mathbb R}^{2}$. This result is optimal, in the sense that these hypotheses do not assure the existence of a convergent separatrix.