Mutsuo Oka

• Univ. Paris XI
• Tokyo University of Science

Let $f_0$ and $f_1$ be two homogeneous polynomials of degree d in three complex variables $z_1,z_2,z_3$. We show that the Lê–Yomdin surface singularities defined by $g_0:=f_0+z_i^{d+m}$ and $g_1:=f_1+z_i^{d+m}$ have the same abstract topology, the same monodromy zeta-function, the same $\mu ^*$-invariant, but lie in distinct path-connected componen...

It is well known that the diffeomorphism type of the Milnor fibration of a (Newton) nondegenerate polynomial function f is uniquely determined by the Newton boundary of f . In the present paper, we generalize this result to certain degenerate functions, namely we show that the diffeomorphism type of the Milnor fibration of a (possibly degenerate) p...

We give a criterion to test geometric properties such as Whitney equisingularity and Thom’s a f a_f condition for new families of (possibly nonisolated) hypersurface singularities that “behave well” with respect to their Newton diagrams. As an important corollary, we obtain that in such families all members have isomorphic Milnor fibrations.

A Zariski pair of surfaces is a pair of complex polynomial functions in $\mathbb{C}^3$ which is obtained from a classical Zariski pair of projective curves $f_0(z_1,z_2,z_3)=0$ and $f_1(z_1,z_2,z_3)=0$ of degree $d$ in $\mathbb{P}^2$ by adding a same term of the form $z_i^{d+m}$ ($m\geq 1$) to both $f_0$ and $f_1$ so that the corresponding affine s...

It is well known that the diffeomorphism-type of the Milnor fibration of a (Newton) non-degenerate polynomial function $f$ is uniquely determined by the Newton boundary of $f$. In the present paper, we generalize this result to certain degenerate functions, namely we show that the diffeomorphism-type of the Milnor fibration of a (possibly degenerat...

We give a criterion to test geometric properties such as Whitney equisingularity and Thom's $a_f$ condition for new families of (possibly non-isolated) hypersurface singularities that "behave well" with respect to their Newton diagrams. As an important corollary, we obtain that in such families all members have isomorphic Milnor fibrations.

We consider a mixed function of type \(H({\mathbf {z}},\overline{{\mathbf {z}}})=f({\mathbf {z}})\,{\overline{g}}({{\mathbf {z}}})\) where f and g are convenient holomorphic functions which have isolated critical points at the origin, and assume that the intersection \(f=g=0\) is a complete intersection variety with an isolated singularity at the o...

We consider a mixed function of type $H(z,\bar z)=f(z)\bar g(z)$ where $f,g$ are non-degenerate but they are not assumed to be convenient. We assume that $f=0$ and $g=0$ and $f=g=0$ are non-degenerate and locally tame. We will show that $H$ has a tubular Milnor vibration and a spherical Milnor fibration. We show also two vibrations are equivalent.

We consider a mixed function of type $H(\mathbf z,\bar {\mathbf z})=f(\mathbf z)\bar g(\mathbf z)$ where $f$ and $g$ are convenient holomorphic functions which have isolated critical points at the origin and we assume that the intersection $f=g=0$ is a complete intersection variety with an isolated singlarity at theorigin. We assume also that $H$ s...

In this note, we prove the connectivity of the Milnor fiber for a mixed polynomial $f(\mathbf z,\bar{\mathbf z})$, assuming the existence of a sequence of smooth points of $f^{-1}(0)$ converging to the origin. This result gives also a another proof for the connectivity of the Milnor fiber of a non-reduced complex analytic function which is proved b...

We consider zero points of a generalized Lens equation \(L(z,{\bar{z}})={\bar{z}}^m-{p(z)}/{q(z)} \) and also harmonically splitting Lens type equation \(L^{hs}(z,{\bar{z}})=r({\bar{z}})-p(z)/q(z)\) with \(\deg \, q(z)=n,\,\deg \,p(z)\le n\) whose numerator is a mixed polynomials, say \(f(z,{\bar{z}})\), of degree \((n+m; n,m)\). To such a polynomi...

Let $f(\bf z,\bar{\bf z})$ be a strongly mixed homogeneous polynomial of 3 variables $\bf z=(z_1,z_2,z_3)$ of polar degree $q$ with an isolated singularity at the origin. It defines a smooth Riemann surface $C$ in the complex projective space $\mathbb P^2$. The fundamental group of the complement $\mathbb P^2\setminus C$ is cyclic group of order $q...

We consider roots of a generalized Lens polynomial $L(z,\bar z)={\bar z}^m q(z)-p(z)$ and also harmonically splitting Lens type polynomial $L^{hs}(z,\bar z)=r(\bar z)q(z)-p(z)$ and with ${\rm deg}\,q(z)=n$, ${\rm deg}\,r(\bar z)=m$ and ${\rm deg}\,p(z)\le n$. We have shown that there exists a harmonically splitting polynomial $r(\bar z)q(z)-p(z)$ w...

We consider \L ojasiewicz inequalities for a non-degenerate holomorphic function with an isolated singularity at the origin. We give an explicit estimation of the \L ojasiewicz exponent in a slightly weaker form than the assertion in Fukui.For a weighted homogeneous polynomial, we give a better estimation in the form which is conjectured by Brzosto...

We study the fundamental groups of (the complements of) plane complex curves defined by equations of the form f(y) = g(x), where f and g are polynomials with real coefficients and real roots (so-called R-join-type curves). For generic (respectively, semi-generic) such polynomials, the groups in question are already considered in [6] (respectively,...

We investigate the equisingularity question for $1$-parameter deformation families of mixed polynomial functions $f_t(\mathbf{z},\bar{\mathbf{z}})$ from the Newton polygon point of view. We show that if the members $f_t$ of the family satisfy a number of elementary conditions, which can be easily described in terms of the Newton polygon, then the c...

We investigate the equisingularity question for $1$-parameter deformation families of mixed polynomial functions $f_t(\mathbf{z},\bar{\mathbf{z}})$ from the Newton polygon point of view. We show that if the members $f_t$ of the family satisfy a number of elementary conditions, which can be easily described in terms of the Newton polygon, then the c...

We study a simplicial mixed polynomial of cyclic type and its associated weighted homogeneous polynomial. In the present paper, we show that their links are diffeomorphic and their Milnor fibrations are isomorphic.

In an unpublished lecture note, J. Brian\c{c}on observed that if $\{f_t\}$ is a family of isolated complex hypersurface singularities such that the Newton boundary of $f_t$ is independent of $t$ and $f_t$ is non-degenerate, then the corresponding family of hypersurfaces $\{f_t^{-1}(0)\}$ is Whitney equisingular (and hence topologically equisingular...

In an unpublished lecture note, J. Brian\c{c}on observed that if $\{f_t\}$ is a family of isolated complex hypersurface singularities such that the Newton boundary of $f_t$ is independent of $t$ and $f_t$ is non-degenerate, then the corresponding family of hypersurfaces $\{f_t^{-1}(0)\}$ is Whitney equisingular (and hence topologically equisingular...

We consider a certain mixed polynomial which is an extended Lens equation $L_{n,m}=\bar z^m-p(z)/q(z)$ with $\text{degree}\, q=n$, $\text{degree}\, p<n$ whose numerator is a mixed polynomial of degree $(n+m;n,m)$. Then we consider its deformation of type $L_{n,m}+\epsilon/z^m$ to construct a special mixed polynomial of degree $(n+2m;n+m,m)$ with $5...

We compute the fundamental groups π1(double-struck P2 \ C) for all complex curves C of degree 7 defined by an equation of the form Πj=1ℓ(Y - βjZ)νj = c·Πi=1m(X - αiZ)λi, where Σj=1ℓ νj = Σi=1m λi is the degree of the curve, c ∈ double-struck R\{0}, and β1,...,βℓ (respectively α1,...,αm) mutually distinct real numbers.

We consider a canonical $S^1$ action on $S^3$ which is defined by $(\rho,(z_1,z_2))\mapsto (z_1\rho^p,z_2\rho^q)$ for $\rho\in S^1$ and $(z_1,z_2)\in S^3\subset {\mathbb C}^2$. We consider a link consisting of finite orbits of this action, which some of the orbits are reversely oriented. Such a link appears as a link of a certain type of mixed poly...

Convenient mixed functions with strongly non-degenerate Newton boundaries have Milnor fibrations, as the isolatedness of the singularity follows from the convenience. In this paper, we consider the Milnor fibration for non-convenient mixed functions. We also study geometric properties such as Thom's $a_f$ condition, the transversality of the nearby...

The aim of this survey article is to bring together recent advances concerning the fundamental groups of join-type curves. Though the paper is of purely expository nature, we do also announce a new result.

An \emph{$\mathbb{R}$-join-type curve} is a curve in $\mathbb{C}^2$ defined by an equation of the form \begin{equation*} a\cdot\prod_{j=1}^\ell (y-\beta_j)^{\nu_j} = b\cdot\prod_{i=1}^m (x-\alpha_i)^{\lambda_i}, \end{equation*} where the coefficients $a$, $b$, $\alpha_i$ and $\beta_j$ are \emph{real} numbers. For generic values of $a$ and $b$, the...

An ℝ-join-type curve is a plane complex projective curve defined by an equation of the form where is the degree of the curve, c ∈ ℝ\ {0}, and β1, …, βl (respectively, α1, …,αm) mutually distinct real numbers. In this paper, we determine the fundamental group π1(ℙ2\ C) for every ℝ-join-type curve C of degree 6. For higher degrees and non-real coeff...

We study the topology of the moduli space of septics with the set of singularities B 4,4⊕2A 3⊕5A 1. In particular, we construct a new π 1-equivalent weak Zariski pair.

A strongly non-degenerate mixed function has a Milnor open book structures on a sufficiently small sphere. We introduce the notion of {\em a holomorphic-like} mixed function and we will show that a link defined by such a mixed function has a canonical contact structure. Then we will show that this contact structure for a certain holomorphic-like mi...

Let $f(\bf z,\bar{\bf z})$ be a mixed polynomial with strongly non-degenerate face functions. We consider a canonical toric modification $\pi:\,X\to \Bbb C^n$ and a polar modification $\pi_{\Bbb R}:Y\to X$. We will show that the toric modification resolves topologically the singularity of $V$ and the zeta function of the Milnor fibration of $f$ is...

Let (C;O); O = (0; 0) be a plane curve which is dened by f(x;y) 2 U; f(x;y) = 0g where U is an open neighbourhood of O and f(x;y) is an holomorphic function dened on U. The purpose of this survey is to give an elementary proof of an embedded resolution of a curve germ (C;O), rst by ordinary blowing-ups and then by toric modications. This note is pr...

We consider two mixed curve $C,C'\subset {\Bbb C}^2$ which are defined by mixed functions of two variables $\bf z=(z_1,z_2)$. We have shown in \cite{MC}, that they have canonical orientations. If $C$ and $C'$ are smooth and intersect transversely at $P$, the intersection number $I_{top}(C,C';P)$ is topologically defined. We will generalize this def...

We study the geometry of pencil of plane curves span by two smooth curves C, C of degree d. The location of the base locus C ∩ C plays an important role. Two particular cases are studied: the generic case where the base locus consists of d 2 points and another extreme case where the base locus is a single point. In the second case, we will show tha...

We study the geometry of pencil of plane curves span by two smooth curves C, C ′ of degree d such that π1(P 2 −C ∪C ′ ) is abelian. We will show that the fundamental group π1(C 2 − C(⃗τ)) is isomorphic to Z × F (r − 1) where ⃗τ = (τ1,..., τr) and C(⃗τ) = C(τ1) ∪ · · · ∪ C(τr).

Let $f(\bfz,\bar\bfz)$ be a mixed strongly polar homogeneous polynomial of $3$ variables $\bfz=(z_1,z_2, z_3)$. It defines a Riemann surface $V:=\{[\bfz]\in \BP^{2}\,|\,f(\bfz,\bar\bfz)=0 \}$ in the complex projective space $\BP^{2}$. We will show that for an arbitrary given $g\ge 0$, there exists a mixed polar homogeneous polynomial with polar deg...

We give an elementary introduction of the toric modification, using an irreducible plane curve germ. We explain also the relation between the tower of the toric modifications which gives a resolution of the curve and the Puiseux pairs. 2000 Mathematics Subject Classification14H20

Let $f(\bfz,\bar\bfz)$ be a mixed polar homogeneous polynomial of $n$ variables $\bfz=(z_1,..., z_n)$. It defines a projective real algebraic variety $V:=\{[\bfz]\in \BC\BP^{n-1} | f(\bfz,\bar\bfz)=0 \}$ in the projective space $\BC\BP^{n-1}$. The behavior is different from that of the projective hypersurface. The topology is not uniquely determine...

An irreducible non-torus plane sextic with simple singularities is said to be special if its fundamental group factors to a dihedral group. There exist (exactly) ten config-urations of simple singularities that are realizable by such curves. Among them, six are realizable by non-special sextics as well. We conjecture that for each of these six conf...

Let $f_{{\bf a},\{bf b}}({\bf z},\bar{\bf z})=z_1^{a_1+b_1}\bar z_1^{b_1}+...+z_n^{a_n+b_n}\bar z_n^{b_n}$ be a polar weighted homogeneous mixed polynomial with $a_j>0,b_j\ge 0$, $j=1,..., n$ and let $f_{{\bf a}}({\bf z})=z_1^{a_1}+...+z_n^{a_n}$ be the associated weighted homogeneous polynomial. Consider the corresponding link variety $K_{{\bf a},...

Mixed functions are analytic functions in variables $z_1,..., z_n$ and their conjugates $\bar z_1,..., \bar z_n$. We introduce the notion of Newton non-degeneracy for mixed functions and develop a basic tool for the study of mixed hypersurface singularities. We show the existence of a canonical resolution of the singularity, and the existence of th...

The existence of Alexander-equivalent Zariski pairs dealing with irreducible curves of degree 6 was proved by Degtyarev. However, no explicit example of such a pair is available (only the existence is known) in the literature. In this paper, we construct the first concrete example.

In this paper, we compute Alexander polynomials of a torus curve C of type (2, 5), C : f(x, y) = f_2(x, y)^5 + f_5(x, y)^2 = 0, under the assumption that the origin O is the unique inner singularity and f2 = 0 is an irreducible conic. We show that the Alexander polynomial remains the same with that of a generic torus curve as long as C is irreducib...

Polar weighted homogeneous polynomials are the class of special polynomials of real variables $x_i,y_i, i=1,..., n$ with $z_i=x_i+\sqrt{-1} y_i$, which enjoys a "polar action". In many aspects, their behavior looks like that of complex weighted homogeneous polynomials. We study basic properties of hypersurfaces which are defined by polar weighted h...

We analyze irreducible plane sextics whose fundamental group factors to $D_{14}$. We produce explicit equations for all curves and show that, in the simplest case of the set of singularities $3A_6$, the group is $D_{14}\times Z_3$.

We consider a polynomial mapping Φ = (f, g) : C 2 → C 2 where f(x, y), g(x, y) are polynomials with coefficients in C. (Every argument which follows in this note is also true over any algebraically closed field K of characteristic 0). Put J(Φ) = J(f, g) be the Jacobian, which is given by

The moduli space of torus sextics with the configuration of singularities {A2+A5+2E6} has two connected components. We compute the fundamental groups π1(CP2−C) for sextics C in both components and study their differences.

We introduce a notion of tangential Alexander polynomials for plane curves and study the relation with $\theta$^Alexander polynomial. As an application, we use these polynomials to study a non-reduced degeneration $C_t \to D_0+jL$. We show that there exists a certain surjectivity of the fundamental groups and divisibility among their Alexander poly...

Consider a moduli space ${{\cal M}(\Sigma,d)}$ of reduced curves in ${\textbf{CP}^2}$ with a given degree ${d}$ and having a prescribed configuration of singularities ${\Sigma}$. Let ${C,C'\in {\cal M}(\Sigma,d)}$. The pair of curves ${(C,C')}$ is called a weak Zariski pair if the pairs of spaces ${(\textbf{CP}^2,C)}$ and ${(\textbf{CP}^2,C')}$ are...

Let $p$ and $q$ be integers such that $p > q \geq 2$ and $q$ divides $p$ . Let $\varphi (q)$ be the Euler number of $q$ . We exhibit a Zariski $\varphi(q)$ -ple, distinguished by the Alexander polynomial, whose curves are tame torus curves of type $(p,q)$ , with $q$ smooth irreducible components of degree $p$ ,and one single singular point topologi...

The complete list of reducible sextics of torus type with simple singularities is known in our previous paper. In this paper, we give a complete list of existence and non-existence of Zariski partner sextics of non-torus type corresponding to the list of sextics of torus type.

In this paper, we give a brief survey on the fundamental group of the complement of a plane curve and its Alexander polynomial. We also introduce the notion of θ-Alexander polynomials and discuss their basic properties. Résumé (Un état des lieux sur les polynômes d'Alexander des courbes planes) Dans cet article, nous donnons un bre etat des lieux s...

Recently, Oka-Pho proved that the fundamental group of the complement of any plane irreducible tame torus sextic is not abelian. We compute here the fundamental groups of the complements of some plane irreducible sextics which are not of torus type. For all our examples, we obtain that the fundamental group is abelian.

We present a computational method for obtaining generic forms of sextics with a given configuration of local singularities. Using this method, we first complete the classification of topological types of local singularities appearing on reduced sextics. Next we give the list of possible configurations of singularities containing at least one non-si...

Alexander polynomials of sextics with only simple singularities or sextics of torus type with arbitrary singularities are computed. We show that for ieeducible sextics,there are four possibilities: $(t^2-t+1)^j, j=0,1,2,3$.

We give a complete classsification of reduced sextics of torus type with configurations of the singularities and the geometry of the components.

The second author classified configurations of the singularities on tame sextics of torus type. In this paper, we give a complete classification of the singularities on irreducible sextics of torus type, without assuming the tameness of the sextics. We show that there exists 121 configurations and there are 5 pairs and a triple of configurations fo...

We show that the fundamental group of the complement of any irreducible tame torus sextics in ℙ2 is isomorphic to ℤ2 * ℤ3 except one class. The exceptional class has the configuration of the singularities {C 3,9, 3A2} and the fundamental group is bigger than ℤ2 * ℤ3. In fact, the Alexander polynomial is given by (t 2 −t+1)2. For the proof, we first...

On an affine variety $X$ defined by homogeneous polynomials, every line in the tangent cone of $X$ is a subvariety of $X$. However there are many other germs of analytic varieties which are not of cone type but contain ``lines'' passing through the origin. In this paper, we give a method to determine the existence and the ``number'' of such lines o...

Let $\mathcal N$ be the moduli space of sextics with 3 (3,4)-cusps. The quotient moduli space ${\mathcal N}/G$ is one-dimensional and consists of two components, ${\mathcal N}_{torus}/G$ and ${\mathcal N}_{gen}/G$. By quadratic transformations, they are transformed into one-parameter families $C_s$ and $D_s$ of cubic curves respectively. We study t...

We study the topology of the Milnor fibre $F$ of a function $f$ with critical locus a smooth curve $L$ on a surface $X$ , where $X$ has an isolated complete intersection singularity and contains $L$ . We use toric modification to resolve the non-isolated singularity $V=X\cap f^{-1}(0)$ . Then we compute the Euler-Poincar\'e characteristic of $F$ ....

By using tone modifications and a result of Gonzalez-Sprinberg and Lejeune-Jalabert, we answer the following questions completely. On which Brieskorn-Pham surface there exist smooth curves passing through the singular point? If there exist, how “many” and what are the defining equations. © 2000, Department of Mathematics, Tokyo Institute of Technol...

In this paper, we define the notion of the flex curve F ()(f; P) at a nonsingular point P of a plane curve Ca. We construct interesting plane curves using a cyclic covering transform, branched along F ()(f; P). As an application, we show the moduli space of projective curves of degree 12 with 27 cusps has at least three irreducible components. Simu...

We consider the space of smooth affine curves with one place at infinity and a fixed genus. We show that the quotient space by the algebraic automorphism group of ℂ 2 has the structure of an algebraic variety which has finite connected components and each component is isomorphic to a cyclic quotient of a rational variety.

Let C = f(x,y) = 0 be a germ of a reduced plane curve. As examples of the basic invariants of a plane curve, we have the Milnor number, the number of irreducible components, the resolution complexity, the Puiseux pairs of the irreducible components and their intersection multiplicities. In fact, the Puiseux pairs of the irreducible components and t...

We show the existence of toric resolution tower for an irreducible curve singularity which is explicitly described by Tschirnhausen polynomials. We deduce for a smooth affine plane curve from its topology restrictions for its singularity at infinity. For instance, we discribe the singularities at infinity (up to equisingular deformation) for curves...

Let h 1 ( u ),…, h k ( u ) be Laurent polynomials of m -variables and let be a non-degenerate complete intersection variety. Such an intersection variety appears as an exceptional divisor of a resolution of non-degenerate complete intersection varieties with an isolated singularity at the origin (Ok4]).

Let ${\mathbf f}=(f_1,...,f_k) : (\mathbb{C}^{n+k}, O)-(\mathbb{C}^k, O)$ be a germ of an analytic mapping such that $V={z\in \mathbb{C}^{n+k};f_1(z)= \cdots=f_k(z) =0} $ is non-degenerate complete intersection variety with an isolated singularity at the origin. We give a formula for the principal zeta-function of the monodromy of the Milnor fibrat...

In this paper, we study the canonical stratification of the discriminant varieties of Al, Bl and Dl. We prove that this stratification enjoys strong geometric properties including the regularity. © 1989 Department of Mathematics, Tokyo Institute of Technology.

This chapter presents the examples of algebraic surfaces with q = 0 and pg ≤ 1, which are locally hypersurfaces. It discusses a canonical way of the compactification M of Ma through the toroidal embedding theory and three algebraic surfaces M1, M2, M3 with q = pq = 0. M1 and M3 are known as an Enriques surfaces and a Godeaux surface, respectively....