Uniform stable radius, L\^e numbers and topological triviality for line singularities

TL;DR

This paper demonstrates that for families of complex polynomial functions with line singularities, a uniform stable radius ensures the invariance of L extsuperscript{e} numbers and, in higher dimensions, guarantees topological triviality of the family.

Contribution

It establishes the link between uniform stable radius and invariance of L extsuperscript{e} numbers, leading to topological triviality in higher dimensions for line singularities.

Findings

L extsuperscript{e} numbers are independent of parameter t under a uniform stable radius.

Families of weighted homogeneous line singularities have a uniform stable radius when nearby fibers are uniformly non-singular.

In dimensions n ≥ 5, constant L extsuperscript{e} numbers imply topological triviality for line singularity families.

Abstract

Let $\{f_t\}$ be a family of complex polynomial functions with line singularities. We show that if $\{f_t\}$ has a uniform stable radius (for the corresponding Milnor fibrations), then the L\^e numbers of the functions $f_t$ are independent of $t$ for all small $t$. In the case of isolated singularities --- a case for which the only non-zero L\^e number coincides with the Milnor number --- a similar assertion was proved by M. Oka and D. O'Shea. By combining our result with a theorem of J. Fern\'andez de Bobadilla --- which says that families of line singularities in $\mathbb{C}^n$, $n\geq 5$, with constant L\^e numbers are topologically trivial --- it follows that a family of line singularities in $\mathbb{C}^n$, $n\geq 5$, is topologically trivial if it has a uniform stable radius. As an important example, we show that families of weighted homogeneous line singularities have a uniform stable radius if the nearby fibres $f_t^{-1}(\eta)$, $\eta\not=0$, are "uniformly" non-singular with respect to the deformation parameter $t$.

Equations

$$f:(\mathbb{C}\times {\mathbb{C}}^n,\mathbb{C}\times \{0\})\to (\mathbb{C},0),(t,\mathbf{z})\mapsto f(t,\mathbf{z}),$$

$${\lambda }_{f_t,\mathbf{z}}^0(0),\dots ,{\lambda }_{f_t,\mathbf{z}}^{n-1}(0)$$

$${\phi }_t:(U_t,0)\to (\phi (\{t\}\times U_t),0),\mathbf{z}\mapsto {\phi }_t(\mathbf{z}):=\phi (t,\mathbf{z}),$$

$${\phi }_t(V(f_0)\cap U_t)=V(f_t)\cap {\phi }_t(U_t)$$

$$\mathcal{S}_{t}:=\bigl{\{}V(f_{t})\setminus\Sigma f_{t},\Sigma f_{t}\setminus\{\mathbf{0}\},\{\mathbf{0}\}\bigr{\}}$$

$${\lambda }_{f_t,\mathbf{z}}^0(0)\text{mbox}and{\lambda }_{f_t,\mathbf{z}}^1(0)$$

$$(t,\mathbf{z})\mapsto \Phi (t,\mathbf{z}):=(f(t,\mathbf{z}),t),$$

$$U({\mathring{B}}_r):=({\mathring{D}}_{\tau }\times {\mathring{B}}_r)\cap {\Phi }^{-1}(({\mathring{D}}_{\delta (\epsilon )}\setminus \{0\})\times {\mathring{D}}_{\tau }).$$

$${\Phi }_{\mid U(S_{\epsilon })}:U(S_{\epsilon })\to ({\mathring{D}}_{\delta (\epsilon )}\setminus \{0\})\times {\mathring{D}}_{\tau }$$

$${\Phi }_{\mid U({\mathring{B}}_r)}:U({\mathring{B}}_r)\to ({\mathring{D}}_{\delta (\epsilon )}\setminus \{0\})\times {\mathring{D}}_{\tau }$$

$${\Phi }_{\mid U({\mathring{B}}_r)}^{-1}(f(t,\mathbf{z}),t)\text{mbox}and(\{t\}\times S_{\epsilon })\cap U({\mathring{B}}_r)$$

$$\tilde{\chi }(F_{f_t,0})=(-1{)}^{n-1}{\lambda }_{f_t,\mathbf{z}}^0(0)+(-1{)}^{n-2}{\lambda }_{f_t,\mathbf{z}}^1(0)$$

$$\mathbf{z}\in V(f_{t_{\delta }}-{\eta }_{\delta })\cap B_r\mapsto \Vert \mathbf{z}{\Vert }^2=\sum _{1\le i\le n}|z_i{|}^2.$$

$$\bigg{(}\frac{\partial f_{t}}{\partial z_{1}}(\mathbf{z}),\ldots,\frac{\partial f_{t}}{\partial z_{n}}(\mathbf{z})\bigg{)}=\lambda\bar{\mathbf{z}}$$

$$d_{\mathbf{w}}\cdot \underset{=0}{\underbrace{f_{t_{\tau ,r}}({\mathbf{z}}_{\tau ,r})}}\stackrel{\text{mbox}Euler}{=}\sum _{1\le i\le n}{w}_i(z_{\tau ,r}{)}_i\frac{\partial f_{t_{\tau ,r}}}{\partial z_i}({\mathbf{z}}_{\tau ,r})\stackrel{(\text{ref}ld)}{=}\lambda \sum _{1\le i\le n}{w}_i|(z_{\tau ,r}{)}_i{|}^2\ne 0,$$

Full text

By combining our result with a theorem of J. Fernández de Bobadilla — which says that families of line singularities in $\mathbb{C}^{n}$, $n\geq 5$, with constant Lê numbers are topologically trivial — it follows that a family of line singularities in $\mathbb{C}^{n}$, $n\geq 5$, is topologically trivial if it has a uniform stable radius.

As an important example, we show that families of weighted homogeneous line singularities have a uniform stable radius if the nearby fibres $f_{t}^{-1}(\eta)$, $\eta\not=0$, are “uniformly” non-singular with respect to the deformation parameter $t$.

1. Introduction

Let $(t,\mathbf{z}):=(t,z_{1},\ldots,z_{n})$ be linear coordinates for $\mathbb{C}\times\mathbb{C}^{n}$ ($n\geq 2$), and let

[TABLE]

be a polynomial function. As usual, we write $f_{t}(\mathbf{z}):=f(t,\mathbf{z})$, and for any $\eta\in\mathbb{C}$, we denote by $V(f_{t}-\eta)$ the hypersurface in $\mathbb{C}^{n}$ defined by the equation $f_{t}(\mathbf{z})=\eta$. (Note that (1.1) implies $f_{t}(\mathbf{0})=f(t,\mathbf{0})=0$, so that the origin $\mathbf{0}\in\mathbb{C}^{n}$ belongs to the hypersurface $V(f_{t})=f_{t}^{-1}(0)$ for all $t\in\mathbb{C}$.)

The purpose of this paper is to show that if the polynomial function $f$ defines a family $\{f_{t}\}$ of hypersurfaces with line singularities and with a uniform stable radius (for the corresponding Milnor fibrations), then the Lê numbers

[TABLE]

of the polynomial functions $f_{t}$ at $\mathbf{0}$ with respect to the coordinates $\mathbf{z}$ — which do exist in this case — are independent of $t$ for all small $t$ (cf. Theorem 4.1). In the case of hypersurfaces with isolated singularities — a case for which the constancy of the Lê numbers means the constancy of the Milnor number — a similar assertion was proved by M. Oka [13] and D. O’Shea [17].

It is worth to observe that by combining Theorem 4.1 with a theorem of J. Fernández de Bobadilla [4] — which says that a family of hypersurfaces with line singularities in $\mathbb{C}^{n}$, $n\geq 5$, is topologically trivial if it has constant Lê numbers — it follows that a family of hypersurfaces with line singularities in $\mathbb{C}^{n}$, $n\geq 5$, is topologically trivial if it has a uniform stable radius (cf. Corollary 4.2).

It is well known that if $\{f_{t}\}$ is a family of isolated hypersurface singularities such that each $f_{t}$ is weighted homogeneous with respect to a given system of weights, then $\{f_{t}\}$ has a uniform stable radius — a result of M. Oka [13] and D. O’Shea [17]. In Theorem 5.1, we show this still holds true for weighted homogeneous hypersurfaces with line singularities provided that the nearby fibres $V(f_{t}-\eta)$, $\eta\not=0$, are “uniformly” non-singular with respect to the deformation parameter $t$ — that is, non-singular in a small ball the radius of which does not depends on $t$. (Note that this condition always holds true for isolated singularities.) In particular, by Corollary 4.2, such families have constant Lê numbers, and for $n\geq 5$, they are topologically trivial.

Finally, let us observe that by combining Corollary 4.2 with a theorem of M. Oka [15] — which says that a family $\{f_{t}\}$ of non-degenerate functions with constant Newton boundary has a uniform stable radius — we get a new proof of a theorem of J. Damon [3] which says that if $\{f_{t}\}$ is a family of non-degenerate line singularities in $\mathbb{C}^{n}$, $n\geq 5$, with constant Newton boundary, then $\{f_{t}\}$ is topologically trivial.

Notation 1.1**.**

In this paper, we are only interested in the behaviour of functions (or hypersurfaces) near the origin $\mathbf{0}\in\mathbb{C}^{n}$. We denote by $B_{\varepsilon}$ the closed ball centred at $\mathbf{0}\in\mathbb{C}^{n}$ with radius $\varepsilon>0$, and we write $\mathring{B}_{\varepsilon}$ (respectively, $S_{\varepsilon}$) for its interior (respectively, its boundary). As usual, in $\mathbb{C}$, we rather write $D_{\varepsilon}$ and $\mathring{D}_{\varepsilon}$ instead of $B_{\varepsilon}$ and $\mathring{B}_{\varepsilon}$.

2. Uniform stable radius

By [5, Lemme (2.1.4)], we know that for each $t$ there exists a positive number $r_{t}>0$ such that for any pair $(\varepsilon_{t},\varepsilon^{\prime}_{t})$ with $0<\varepsilon^{\prime}_{t}\leq\varepsilon_{t}\leq r_{t}$, there exists $\delta(\varepsilon_{t},\varepsilon^{\prime}_{t})>0$ such that for any non-zero complex number $\eta$ with $0<|\eta|\leq\delta(\varepsilon_{t},\varepsilon^{\prime}_{t})$, the hypersurface $V(f_{t}-\eta)$ is non-singular in $\mathring{B}_{r_{t}}$ and transversely intersects with the sphere $S_{\varepsilon^{\prime\prime}}$ for any $\varepsilon^{\prime\prime}$ with $\varepsilon^{\prime}_{t}\leq\varepsilon^{\prime\prime}\leq\varepsilon_{t}$. Any such a number $r_{t}$ is called a stable radius for the Milnor fibration of $f_{t}$ at $\mathbf{0}$ (cf. [15, §2]).

Definition 2.1** (cf. [15, §3]).**

We say that the family $\{f_{t}\}$ has a uniform stable radius (we also say that $\{f_{t}\}$ is uniformly stable) if there exist $\tau>0$ and $r>0$ such that for any pair $(\varepsilon,\varepsilon^{\prime})$ with $0<\varepsilon^{\prime}\leq\varepsilon\leq r$, there exists $\delta(\varepsilon,\varepsilon^{\prime})>0$ such that for any non-zero complex number $\eta$ with $0<|\eta|\leq\delta(\varepsilon,\varepsilon^{\prime})$, the hypersurface $V(f_{t}-\eta)$ is non-singular in $\mathring{B}_{r}$ and transversely intersects with the sphere $S_{\varepsilon^{\prime\prime}}$ for any $\varepsilon^{\prime\prime}$ with $\varepsilon^{\prime}\leq\varepsilon^{\prime\prime}\leq\varepsilon$ and for any $t$ with $0\leq|t|\leq\tau$. Any such a number $r$ is called a uniform stable radius for $\{f_{t}\}$.

In the special case where the polynomial function $f$ defines a family $\{f_{t}\}$ of isolated hypersurface singularities (i.e., $f_{t}$ has an isolated singularity at $\mathbf{0}$ for all small $t$), then, by [12], we also know that for each $t$ there exists $R_{t}>0$ such that the hypersurface $V(f_{t})$ is non-singular in $\mathring{B}_{R_{t}}\setminus\{\mathbf{0}\}$ and transversely intersects with the sphere $S_{\rho}$ for any $\rho$ with $0<\rho\leq R_{t}$.

Definition 2.2** (cf. [13, §2]).**

Suppose that $f$ defines a family $\{f_{t}\}$ of isolated hypersurface singularities . We say that $\{f_{t}\}$ satisfies condition (A) if there exist $\nu>0$ and $R>0$ such that $V(f_{t})$ is non-singular in $\mathring{B}_{R}\setminus\{\mathbf{0}\}$ and transversely intersects with the sphere $S_{\rho}$ for any $\rho$ with $0<\rho\leq R$ and for any $t$ with $0\leq|t|\leq\nu$.

It is easy to see that a family $\{f_{t}\}$ of isolated hypersurface singularities satisfies condition (A) if and only if it has no vanishing fold and no non-trivial critical arc in the sense of [17]. Also, it is worth to observe that if $\{f_{t}\}$ satisfies condition (A), then it has a uniform stable radius (cf. [13, 17]).

3. The Oka-O’Shea theorem for isolated singularities

Throughout this section we assume that the polynomial function $f$ defines a family $\{f_{t}\}$ of isolated hypersurface singularities. The following theorem is due to M. Oka [13] and D. O’Shea [17].

Theorem 3.1** (Oka-O’Shea).**

Suppose that $f$ defines a family $\{f_{t}\}$ of isolated hypersurface singularities. Under this assumption, if furthermore $\{f_{t}\}$ satisfies condition (A) or if it has a uniform stable radius, then it is $\mu$-constant — that is, the Milnor number $\mu_{f_{t}}(\mathbf{0})$ of $f_{t}$ at $\mathbf{0}$ is independent of $t$ for all small $t$.

Actually, in [13], M. Oka showed that if $\{f_{t}\}$ satisfies condition (A) or if it has a uniform stable radius, then the Milnor fibrations at $\mathbf{0}$ of $f_{0}$ and $f_{t}$ are isomorphic.

In [8], Lê Dũng Tráng and C. P. Ramanujam showed that for $n\not=3$ any family of isolated hypersurface singularities with constant Milnor number is topologically $\mathcal{V}$-equisingular. With the same assumption, J. G. Timourian [19] showed that the family is actually topologically trivial. We recall that a family $\{f_{t}\}$ is topologically $\mathcal{V}$-equisingular (respectively, topologically trivial) if there exist open neighbourhoods $D$ and $U$ of the origins in $\mathbb{C}$ and $\mathbb{C}^{n}$, respectively, together with a continuous map $\varphi\colon(D\times U,D\times\{\mathbf{0}\})\rightarrow(\mathbb{C}^{n},\mathbf{0})$ such that for all sufficiently small $t$, there is an open neighbourhood $U_{t}\subseteq U$ of $\mathbf{0}\in\mathbb{C}^{n}$ such that the map

[TABLE]

is a homeomorphism satisfying the relation

[TABLE]

(respectively, the relation $f_{0}=f_{t}\circ\varphi_{t}$ on $U_{t}$).

Note that, in general, “$\mu$-constant” does not imply condition (A) (cf. [1, 16]).

Finally, observe that the Briançon-Speder famous family shows that condition (A) does not imply the Whitney conditions along the $t$-axis (cf. [2]).

4. Uniformly stable families of line singularities

4.1. Setup and statement of the main result

From now on we suppose that the polynomial function $f$ defines a family $\{f_{t}\}$ of hypersurfaces with line singularities. As in [9, §4], by such a family we mean a family $\{f_{t}\}$ such that for each $t$ small enough, the singular locus $\Sigma f_{t}$ of $f_{t}$ near the origin $\mathbf{0}\in\mathbb{C}^{n}$ is given by the $z_{1}$-axis, and the restriction of $f_{t}$ to the hyperplane $V(z_{1})$ defined by $z_{1}=0$ has an isolated singularity at the origin. Then, by [10, Remark 1.29], the partition of $V(f_{t})$ given by

[TABLE]

is a good stratification for $f_{t}$ at $\mathbf{0}$, and the hyperplane $V(z_{1})$ is a prepolar slice for $f_{t}$ at $\mathbf{0}$ with respect to $\mathcal{S}_{t}$ for all $t$ small enough. In particular, combined with [10, Proposition 1.23], this implies that the Lê numbers

[TABLE]

of $f_{t}$ at $\mathbf{0}$ with respect to the coordinates $\mathbf{z}$ do exist. (For the definitions of good stratifications, prepolarity and Lê numbers, we refer the reader to D. Massey’s book [10].) Note that for line singularities, the only possible non-zero Lê numbers are precisely $\lambda^{0}_{f_{t},\mathbf{z}}(\mathbf{0})$ and $\lambda^{1}_{f_{t},\mathbf{z}}(\mathbf{0})$. All the other Lê numbers $\lambda^{k}_{f_{t},\mathbf{z}}(\mathbf{0})$ for $2\leq k\leq n-1$ are defined and equal to zero (cf. [10]).

Here is our main observation.

Theorem 4.1**.**

Suppose that $f$ defines a family $\{f_{t}\}$ of hypersurfaces with line singularities. Under this assumption, if furthermore $\{f_{t}\}$ has a uniform stable radius, then it is $\lambda_{\mathbf{z}}$-constant — that is, the Lê numbers $\lambda^{0}_{f_{t},\mathbf{z}}(\mathbf{0})$ and $\lambda^{1}_{f_{t},\mathbf{z}}(\mathbf{0})$ are independent of $t$ for all small $t$.

Theorem 4.1 extends to line singularities Oka-O’Shea’s Theorem 3.1 concerning isolated singularities. Indeed, for isolated singularities, the only possible non-zero Lê number is $\lambda^{0}_{f_{t},\mathbf{z}}(\mathbf{0})$ and the latter coincides with the Milnor number $\mu_{f_{t}}(\mathbf{0})$.

Note that if $\{f_{t}\}$ is a $\lambda_{\mathbf{z}}$-constant family of line singularities in $\mathbb{C}^{n}$ with $n\geq 5$, then, by a theorem of D. Massey [9, Theorem (5.2)], the diffeomorphism type of the Milnor fibration of $f_{t}$ at $\mathbf{0}$ is independent of $t$ for all small $t$. Under the same assumption, in [4, Theorem 42], J. Fernández de Bobadilla showed that $\{f_{t}\}$ is actually topologically trivial. Combining Fernández de Bobadilla’s result with our Theorem 4.1 gives the following corollary.

Corollary 4.2**.**

Suppose that $f$ defines a family $\{f_{t}\}$ of hypersurfaces with line singularities in $\mathbb{C}^{n}$ with $n\geq 5$. Under this assumption, if furthermore $\{f_{t}\}$ has a uniform stable radius, then it is topologically trivial.

4.2. Application to families of non-degenerate line singularities with constant Newton boundary

In [15, Corollary 1], M. Oka showed that if $\{f_{t}\}$ is a family of hypersurface singularities — not necessary line singularities — such that for all small $t$ the polynomial function $f_{t}$ is non-degenerate and the Newton boundary of $f_{t}$ at $\mathbf{0}$ with respect to the coordinates $\mathbf{z}$ is independent of $t$, then $\{f_{t}\}$ has a uniform stable radius. (For the definitions of non-degeneracy and Newton boundary, see [6, 14].) Combined with Oka’s result, Corollary 4.2 provides a new proof of the following theorem of J. Damon [3]. (Actually, the theorem of Damon given in [3] is much more general than the special case stated below.)

Theorem 4.3** (Damon).**

Suppose that $f$ defines a family $\{f_{t}\}$ of hypersurfaces with line singularities in $\mathbb{C}^{n}$ with $n\geq 5$. Under this assumption, if furthermore for any sufficiently small $t$ the polynomial function $f_{t}$ is non-degenerate and the Newton boundary of $f_{t}$ at $\mathbf{0}$ with respect to the coordinates $\mathbf{z}$ is independent of $t$, then the family $\{f_{t}\}$ is topologically trivial.

4.3. Proof of Theorem 4.1

Consider the map $\Phi\colon\mathbb{C}\times\mathbb{C}^{n}\to\mathbb{C}^{2}$ defined by

[TABLE]

and pick positive numbers $\tau$ and $r$ which satisfy the condition of Definition 2.1. Then, in particular, the following property ($\mathcal{P}$) holds true:

1. ($\mathcal{P}$)

for any $\varepsilon$ with $0<\varepsilon<r$, there exists $\delta(\varepsilon)>0$ such that for any $t$ with $0\leq|t|\leq\tau$ and for any $\eta$ with $0<|\eta|\leq\delta(\varepsilon)$, the hypersurface $V(f_{t}-\eta)$ is non-singular in $\mathring{B}_{r}$ and transversely intersects with the sphere $S_{\varepsilon}$.

This property implies that the critical set $\Sigma\Phi$ of $\Phi$ does not intersect with the set

[TABLE]

Indeed, suppose there is a point $(t_{0},\mathbf{z}_{0})\in\Sigma\Phi\cap U(\mathring{B}_{r})$. Then $\mathbf{z}_{0}\in\Sigma(f_{t_{0}}-f_{t_{0}}(\mathbf{z}_{0}))$. But this is not possible, since by ($\mathcal{P}$), the hypersurface $V(f_{t_{0}}-f_{t_{0}}(\mathbf{z}_{0}))$ is smooth. (We recall that a complex variety can never be a smooth manifold throughout a neighbourhood of a critical point (cf. [12, §2]).)

It also follows from Property ($\mathcal{P}$) that the map

[TABLE]

(restriction of $\Phi$ to $U(S_{\varepsilon}):=(\mathring{D}_{\tau}\times S_{\varepsilon})\cap\Phi^{-1}((\mathring{D}_{\delta(\varepsilon)}\setminus\{\mathbf{0}\})\times\mathring{D}_{\tau})$) is a submersion. Indeed, as $\Sigma\Phi\cap U(\mathring{B}_{r})=\emptyset$ and $U(\mathring{B}_{r})$ is an open subset of $\mathbb{C}\times\mathbb{C}^{n}$, the map

[TABLE]

is a submersion. Thus, to show that $\Phi_{\mid U(S_{\varepsilon})}$ is a submersion, it suffices to observe that the inclusion $U(S_{\varepsilon})\hookrightarrow U(\mathring{B}_{r})$ is transverse to the submanifold $\Phi_{\mid U(\mathring{B}_{r})}^{-1}(f(t,\mathbf{z}),t)$ for any point $(t,\mathbf{z})\in U(S_{\varepsilon})$ — or equivalently that the submanifolds

[TABLE]

are transverse to each other. And this is exactly the content of ($\mathcal{P}$).

Now, as $\Phi_{\mid U(S_{\varepsilon})}$ is also a proper map, a result of D. Massey and D. Siersma (cf. [11, Proposition 1.10]) shows that the Milnor number of a generic hyperplane slice of $f_{t}$ at a point on $\Sigma f_{t}$ sufficiently close to the origin — which coincides with the Lê number $\lambda^{1}_{f_{t},\mathbf{z}}(\mathbf{0})$ for line singularities (cf. [7, 9]) — is independent of $t$ for all small $t$.

Finally, since the family $\{f_{t}\}$ has a uniform stable radius — the full strength of this assumption is used here — it follows from a result of M. Oka (cf. [15, Lemma 2]) that the diffeomorphism type of the Milnor fibration of $f_{t}$ at the origin is independent of $t$ for all small $t$. In particular, the reduced Euler characteristic $\tilde{\chi}(F_{f_{t},\mathbf{0}})$ of the Milnor fibre $F_{f_{t},\mathbf{0}}$ of $f_{t}$ at $\mathbf{0}$, which is given by

[TABLE]

(cf. [10, Theorem 3.3]), is independent of $t$ for all small $t$. The constancy of $\lambda^{0}_{f_{t},\mathbf{z}}(\mathbf{0})$ now follows from that of $\lambda^{1}_{f_{t},\mathbf{z}}(\mathbf{0})$.

5. Uniform stable radius and weighted homogeneous

line singularities

By a result of M. Oka [13] and D. O’Shea [17], we know that if $\{f_{t}\}$ is a family of isolated hypersurface singularities such that each $f_{t}$ is weighted homogeneous with respect to a given system of weights, then $\{f_{t}\}$ satisfies condition (A), and hence, is uniformly stable. Our next observation says this still holds true for weighted homogeneous line singularities provided that the nearby fibres $V(f_{t}-\eta)$, $\eta\not=0$, of the functions $f_{t}$ are “uniformly” non-singular with respect to the deformation parameter $t$ — that is, non-singular in a small ball the radius of which does not depends on $t$. (We recall that by [5] the nearby fibres are “individually” non-singular — that is, non-singular in a small ball the radius of which depends on $t$.)

Theorem 5.1**.**

Suppose that $f$ defines a family $\{f_{t}\}$ of hypersurfaces with line singularities such that each $f_{t}$ is weighted homogeneous with respect to a given system of weights $\mathbf{w}=(w_{1},\ldots,w_{n})$ on the variables $(z_{1},\ldots,z_{n})$, with $w_{i}\in\mathbb{N}\setminus\{0\}$. Also, assume that the nearby fibres $V(f_{t}-\eta)$, $\eta\not=0$, of the functions $f_{t}$ are uniformly non-singular with respect to the deformation parameter $t$ — that is, there exist positive numbers $\tau,r,\delta$ such that for any $0<|\eta|\leq\delta$ and any $0\leq|t|\leq\tau$, the hypersurface $V(f_{t}-\eta)$ is non-singular in $\mathring{B}_{r}$. Under these assumptions, the family $\{f_{t}\}$ has a uniform stable radius. (In particular, $\{f_{t}\}$ is $\lambda_{\mathbf{z}}$-constant, and for $n\geq 5$, it is topologically trivial.)

Proof.

It is based on similar arguments than those used in [13] and [17]. Suppose that the family $\{f_{t}\}$ does not have a uniform stable radius. Then, as the nearby fibres of the functions $f_{t}$ are uniformly non-singular with respect to the deformation parameter $t$, for all $\tau>0$ and all $r>0$ small enough, there exist $0<\varepsilon^{\prime}\leq\varepsilon\leq r$ such that for all sufficiently small $\delta>0$ there exist $\eta_{\delta}$, $\varepsilon_{\delta}$ and $t_{\delta}$, with $0<|\eta_{\delta}|\leq\delta$, $\varepsilon^{\prime}\leq\varepsilon_{\delta}\leq\varepsilon$ and $|t_{\delta}|\leq\tau$, such that $V(f_{t_{\delta}}-\eta_{\delta})$ is non-singular in $\mathring{B}_{r}$ and does not transversely intersect with the sphere $S_{\varepsilon_{\delta}}$. It follows that there is a point $\mathbf{z}_{\delta}\in V(f_{t_{\delta}}-\eta_{\delta})\cap S_{\varepsilon_{\delta}}$ which is a critical point of the restriction to $V(f_{t_{\delta}}-\eta_{\delta})\cap B_{r}$ of the squared distance function:

[TABLE]

In other words, the point $(t_{\delta},\mathbf{z}_{\delta})$ lies in the intersection of $D_{\tau}\times(B_{\varepsilon}\setminus\mathring{B}_{\varepsilon^{\prime}})$ with the real algebraic set $C$ consisting of the points $(t,\mathbf{z})$ such that

[TABLE]

for some $\lambda\in\mathbb{C}\setminus\{0\}$, where $\bar{\mathbf{z}}:=(\bar{z}_{1},\ldots,\bar{z}_{n})$ and $\bar{z}_{i}$ denotes the complex conjugate of $z_{i}$ (see e.g. [18, Lemma 1]). Let $C_{\tau,r}:=C\cap(D_{\tau}\times(B_{\varepsilon}\setminus\mathring{B}_{\varepsilon^{\prime}}))$. Take $\delta:=\delta(m):=1/m$ (where $m\in\mathbb{N}\setminus\{0\}$ is sufficiently large), and consider the corresponding sequence of points $(t_{\delta(m)},\mathbf{z}_{\delta(m)})$ in $C_{\tau,r}$. As $C_{\tau,r}$ is compact, taking a subsequence if necessary, we may assume that $(t_{\delta(m)},\mathbf{z}_{\delta(m)})$ converges to a point $(t_{\tau,r},\mathbf{z}_{\tau,r})\in C_{\tau,r}$, and hence $\eta_{\delta(m)}:=f(t_{\delta(m)},\mathbf{z}_{\delta(m)})$ tends to $f(t_{\tau,r},\mathbf{z}_{\tau,r})$ as $m\to\infty$. Since $0<|\eta_{\delta(m)}|\leq\delta(m)=1/m\to 0$ as $m\to\infty$, we have $f(t_{\tau,r},\mathbf{z}_{\tau,r})=0$. Thus $(t_{\tau,r},\mathbf{z}_{\tau,r})\in V(f)\cap C_{\tau,r}$.

Now, since $f_{t_{\tau,r}}$ is weighted homogeneous with respect to the weights $\mathbf{w}=(w_{1},\ldots,w_{n})$, the Euler identity implies the following contradiction:

[TABLE]

where $d_{\mathbf{w}}$ is the weighted degree of $f_{t_{\tau,r}}$ with respect to the weights $\mathbf{w}$ and $(z_{\tau,r})_{i}$ is the $i$th component of $\mathbf{z}_{\tau,r}$. ∎

Remark 5.2*.*

Actually, the proof shows that if $f$ defines a family $\{f_{t}\}$ of hypersurfaces — not necessarily with line singularities — such that each $f_{t}$ is weighted homogeneous with respect to a given system of weights $\mathbf{w}$, and if furthermore, the nearby fibres $V(f_{t}-\eta)$, $\eta\not=0$, of the functions $f_{t}$ are uniformly non-singular with respect to the deformation parameter $t$, then the family $\{f_{t}\}$ has a uniform stable radius.