$\mu ^*$-ZARISKI PAIRS OF SURFACE SINGULARITIES

Abstract

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 components of the $\mu ^*$-constant stratum if their projective tangent cones (defined by $f_0$ and $f_1$, respectively) make a Zariski pair of curves in $\mathbb {P}^2$, the singularities of which are Newton non-degenerate. In this case, we say that $V(g_0):=g_0^{-1}(0)$ and $V(g_1):=g_1^{-1}(0)$ make a $\mu ^*$-Zariski pair of surface singularities. Being such a pair is a necessary condition for the germs $V(g_0)$ and $V(g_1)$ to have distinct embedded topologies.

1 Introduction and statement of the result

Let $g_0$ and $g_1$ be two polynomials in three complex variables $z_1,z_2,z_3$ . We assume that they vanish at the origin $\mathbf {0}\in \mathbb {C}^3$ and that the corresponding germs of surfaces, $V(g_0):=g_0^{-1}(0)$ and $V(g_1):=g_1^{-1}(0)$ , have an isolated singularity at $\mathbf {0}$ . It is well known that if $V(g_0)$ and $V(g_1)$ have the same embedded topology (i.e., if the pairs $(\mathbb {C}^3,V(g_0))$ and $(\mathbb {C}^3,V(g_1))$ are homeomorphic in a neighborhood of the origin, or equivalently, by [28], if the pairs $(\mathbb {S}_\varepsilon ^5,K_{g_0})$ and $(\mathbb {S}_\varepsilon ^5,K_{g_1})$ are diffeomorphic for any $\varepsilon $ small enough), then they have the same Milnor number (see [18], [23], [33]). Here, $K_{g_l}$ denotes the link of $g_l$ ( $l\in \{0,1\}$ ), that is, $K_{g_l}:=\mathbb {S}^5_\varepsilon \cap V(g_l)$ for $\varepsilon $ small enough, where $\mathbb {S}_\varepsilon ^5$ is the sphere with radius $\varepsilon $ centered at $\mathbf {0}\in \mathbb {C}^3$ . (Note that the diffeomorphism type of the embedded link $(\mathbb {S}_\varepsilon ^5,K_{g_l})$ is independent of $\varepsilon $ , provided that $\varepsilon $ is small enough.) On the other hand, it is quite possible for two isolated surface singularities $V(g_0)$ and $V(g_1)$ to have the same Milnor number and non-diffeomorphic embedded links. In [3], [4], using Luengo’s theory of superisolated singularities [20], Artal-Bartolo even showed that the embedded topology of the link of a superisolated surface singularity is not determined by the topology of the abstract link and the characteristic polynomial of the monodromy. However, in practice, given $g_0$ and $g_1$ with the same characteristic polynomial (or equivalently, the same monodromy zeta-function), the same abstract topology, and even with the same Teissier $\mu ^*$ -invariant, it is extremely difficult to determine whether $(\mathbb {S}_\varepsilon ^5,K_{g_0})$ and $(\mathbb {S}_\varepsilon ^5,K_{g_1})$ are diffeomorphic or not. The goal of this paper is to investigate a special class of Lê–Yomdin surface singularities which are “likely to systematically produce” pairs of germs sharing all these invariants but having non-diffeomorphic embedded links. Such pairs are called $\mu ^*$ -Zariski pairs of surface singularities and are defined as follows.

Consider a classical Zariski pair of (reduced) projective curves $C_0=\{f_0=0\}$ and $C_1=\{f_1=0\}$ of degree d in the complex projective plane $\mathbb {P}^2$ , that is, there are regular neighborhoods $N_0$ and $N_1$ of $C_0$ and $C_1$ , respectively, such that $(N_0,C_0)$ and $(N_1,C_1)$ are homeomorphic, while $(\mathbb {P}^2,C_0)$ and $(\mathbb {P}^2,C_1)$ are not. The first example of such a pair was found by Zariski [36] in the early 1930s, and their systematic study was initiated by Artal-Bartolo [5] in the mid-1990s (for a detailed survey on this topic, see [6], [25]). By a linear change of the coordinates $z_1,z_2,z_3$ , we may assume that the singularities of the curves $C_0$ and $C_1$ are not located on the coordinate lines $z_i=0$ ( $1\leq i\leq 3$ ) and that their defining polynomials $f_0$ and $f_1$ are convenient ¹ and Newton non-degenerate on any face $\Delta $ of their (common) Newton diagram if $\Delta $ is not top-dimensional. The fact that the singularities of the curves do not sit on the coordinate lines implies that for any integers $m\geq 1$ and $1\leq i\leq 3$ , the polynomials

$$ \begin{align*} g_0:=f_0+z_i^{d+m} \quad\mbox{and}\quad g_1:=f_1+z_i^{d+m} \end{align*} $$

define an isolated surface singularity at $\mathbf {0}$ (see [21, Th. 2]). Such singularities are called m-Lê–Yomdin singularities and were first investigated by Yomdin and Lê in [19], [13], respectively. The monodromy zeta-function (or the characteristic polynomial) of such a singularity was computed by Siersma [29], [30], Stevens [31], and Gusein-Zade, Luengo, and Melle-Hernández [11] (see also [26]). (The Milnor number was already known from [21].) In [7], Artal-Bartolo, Cogolludo-Agustín, and Martín-Morales gave a characterization for the abstract link of a Lê–Yomdin singularity to be a rational homology sphere.

In the special case where $m=1$ , a $1$ -Lê–Yomdin singularity is called a superisolated singularity. Superisolated singularities were introduced by Luengo [20] to answer important questions and conjectures. For example, in [20], Luengo gave examples of superisolated surface singularities for which the $\mu $ -constant stratum in the miniversal deformation is not smooth.

Now, let us make precise the notion of Zariski pair of surface singularities. Let $g_0=f_0+z_i^{d+m}$ and $g_1=f_1+z_i^{d+m}$ be two Lê–Yomdin surface singularities obtained from a Zariski pair of curves $f_0$ and $f_1$ as above.

• • We say that $(V(g_0),V(g_1))$ is a weak $\zeta $ -Zariski pair of surface singularities if $g_0$ and $g_1$ have the same monodromy zeta-function (in particular, the same Milnor number).

• • A weak $\zeta $ -Zariski pair for which the germs $V(g_0)$ and $V(g_1)$ (or equivalently, the links $K_{g_0}$ and $K_{g_1}$ ) have the same abstract topology is called a $\zeta $ -Zariski pair (without the adjective “weak”).

• • A (weak) $\zeta $ -Zariski pair is said to be a (weak) $\mu ^*$ -Zariski pair if $g_0$ and $g_1$ have the same $\mu ^*$ -invariant while belonging to distinct path-connected components of the $\mu ^*$ -constant stratum.

• • A (weak) $\mu ^*$ -Zariski pair is called a (weak) $\mu $ -Zariski pair if furthermore $g_0$ and $g_1$ lie in different path-connected components of the $\mu $ -constant stratum.

• • Finally, a (weak) $\zeta $ -Zariski pair is called a (weak) Zariski pair if the germs $V(g_0)$ and $V(g_1)$ (or equivalently, $K_{g_0}$ and $K_{g_1}$ ) have distinct embedded topologies.

Note that a (weak) Zariski pair of surface singularities $V(g_0)$ and $V(g_1)$ sharing the same $\mu ^*$ -invariant is always a (weak) $\mu $ -Zariski pair, and hence a (weak) $\mu ^*$ -Zariski pair. That is, being a (weak) $\mu ^*$ -Zariski pair is a necessary condition for being a (weak) Zariski pair. Indeed, by [10, Th. 5.3], if $g_0$ and $g_1$ lie in the same path-connected component of the $\mu ^*$ -constant stratum, then they can always be joined by a piecewise complex-analytic path (defined in the relevant natural way), and by a well-known theorem of Teissier [32, théorème 3.9], this in turn implies that the diffeomorphism type of the pairs $(\mathbb {S}_\varepsilon ^5,K_{g_0})$ and $(\mathbb {S}_\varepsilon ^5,K_{g_1})$ is identical.

In [20], Luengo proved that for superisolated singularities (i.e., for $m=1$ ), the abstract links $K_{g_0}$ and $K_{g_1}$ are homeomorphic. The second-named author showed a similar property for $m\geq 1$ if the singularities of the corresponding curves $C_0$ and $C_1$ are Newton non-degenerate (see [27, Th. 24 and Rem. 25]). In [3, théorème 4.4] and [4, théorème 1.6, §1.7, and corollaire 5.6.6], Artal-Bartolo proved that if $m=1$ , then $V(g_0)$ and $V(g_1)$ also share the same characteristic polynomial of the monodromy, and if furthermore the Alexander polynomials of the curves $C_0$ and $C_1$ do not coincide, then $V(g_0)$ and $V(g_1)$ do not have the same embedded topology. In particular, combined with Luengo’s result, this shows that, in this latter case, $(V(g_0),V(g_1))$ is a Zariski pair of surface singularities.

In this paper, we prove the following theorem.

Theorem 1.1. If the singularities of the curves $C_0$ and $C_1$ are Newton non-degenerate in some suitable local coordinates, ² then the pair made up of the m-Lê–Yomdin singularities $V(g_0)$ and $V(g_1)$ is a $\mu ^*$ -Zariski pair of surface singularities.

Again, we emphasize that being a $\mu ^*$ -Zariski pair is a necessary condition for being a Zariski pair of surface singularities. We also highlight that in the above theorem, the Alexander polynomials of the curves $C_0$ and $C_1$ may coincide.

We expect that with the assumption of the theorem, $(V(g_0),V(g_1))$ is a $\mu $ -Zariski pair, and in fact, a Zariski pair of surface singularities. As mentioned above, in the special case of superisolated singularities (i.e., $m=1$ ), and provided that the curves have distinct Alexander polynomials (but not necessarily Newton non-degenerate singularities), this is already proved by combining Artal-Bartolo’s [3], [4] and Luengo’s [20] results.

2 Proof of Theorem 1.1

First, we show that $(V(g_0),V(g_1))$ is a $\zeta $ -Zariski pair of surface singularities, and then we prove that it is in fact a $\mu ^*$ -Zariski pair. To simplify, we assume that $i=1$ , that is, $g_l=f_l+z_1^{d+m}$ ( $l\in \{0,1\}$ ).

To compute the monodromy zeta-function $\zeta _{g_l,\mathbf {0}}(t)$ of $g_l$ , we use the classical formula of Siersma (see [29, Main theorem, p. 183] and [30, Th. 3.4 and Rem. 3.6]), Stevens (see [31, p. 140]), and Gusein-Zade, Luengo, and Melle-Hernández (see [11, p. 250]) (see also [26, Lem. 3.2 and Th. 3.7]). More precisely, the ordinary point blowing up at $\mathbf {0}\in \mathbb {C}^3$ , denoted by $\pi \colon X\to \mathbb {C}^3$ , being a biholomorphism over $\mathbb {C}^3\setminus V(g_l)$ , the tubular Milnor fibration of $g_l$ at $\mathbf {0}$ can be lifted to X, so that the pullback $\pi ^* g_l\equiv g_l\circ \pi $ is a locally trivial fibration which is isomorphic to it. Let $U_1:=\mathbb {P}^2\setminus \{z_1=0\}$ be the standard affine chart of $\mathbb {P}^2$ with coordinates $(z_2/z_1,z_3/z_1)$ . In the corresponding chart $X\cap (\mathbb {C}^3\times U_1)$ of X, with coordinates $\mathbf {y}\equiv (y_1,y_2,y_3):=(z_1,z_2/z_1,z_3/z_1)$ , the pullback $\pi ^*g_l$ is written as

$$ \begin{align*} \pi^*g_l (\mathbf{y}) = y_1^d(f_l(1,y_2,y_3)+y_1^m). \end{align*} $$

The first factor, $y_1^d$ , corresponds to the exceptional divisor $E\simeq \mathbb {P}^2$ , while the second one represents the strict transform $\tilde V(g_l)$ of $V(g_l)$ . Outside of the exceptional divisor, $\tilde V(g_l)$ has no singularities. On the exceptional divisor, it has a finite number of isolated singularities, which are given by the singular points $\mathbf {p}\in \Sigma (C_l)$ of the reduced curve $C_l$ . Then the formula for the zeta-function mentioned above is written as

(2.1) $$ \begin{align} \zeta_{g_l,\mathbf{0}}(t)=\zeta_d(t)\times (1-t^{d})^{\mu^{\scriptscriptstyle{\text{tot}}}(C_l)} \times \prod_{\substack{\mathbf{p}\in\Sigma(C_l)}} \zeta_{\pi^*g_l,\mathbf{p}}(t), \end{align} $$

where $\zeta _d(t)$ is the zeta-function of a Newton non-degenerate homogeneous polynomial of degree d (i.e., $\zeta _d(t)=(1-t^{d})^{-d^2+3d-3}$ ), $\Sigma (C_l)$ is the set of singular points of $C_l$ , and $\mu ^{\scriptscriptstyle{\text{tot}}}(C_l)$ is the total Milnor number of $C_l$ (i.e., the sum of the local Milnor numbers at the singular points of $C_l$ ).

By our assumption, there exist local coordinates $\mathbf {x}=(x_1,x_2,x_3)$ and $\mathbf {u}=(u_1,u_2,u_3)$ near $\mathbf {p}_0\in \Sigma (C_0)$ and $\mathbf {p}_1\in \Sigma (C_1)$ , respectively, where $x_1=u_1=y_1$ and $(x_2,x_3)$ and $(u_2,u_3)$ are analytic coordinate changes of $(y_2,y_3)$ , ³ such that

$$ \begin{align*} \pi^* g_0 (\mathbf{x}) = x_1^d(h_0(x_2,x_3)+x_1^m) \quad\mbox{and}\quad \pi^* g_1 (\mathbf{u}) = u_1^d(h_1(u_2,u_3)+u_1^m), \end{align*} $$

where $h_0$ and $h_1$ are Newton non-degenerate. Moreover, if the singularities $(C_1,\mathbf {p}_1)$ and $(C_0,\mathbf {p}_0)$ are topologically equivalent, then we may assume that the Newton diagrams, $\Gamma (h_0)$ and $\Gamma (h_1)$ , of $h_0$ and $h_1$ coincide. It follows that $\pi ^* g_0$ and $\pi ^* g_1$ are Newton non-degenerate with the same Newton diagram, and hence, by Varchenko’s formula (see [34, Th. (4.1)]), we have

$$ \begin{align*} \zeta_{\pi^*\!g_0,\mathbf{p}_0}(t)=\zeta_{\pi^*\!g_1,\mathbf{p}_1}(t). \end{align*} $$

Since $(C_0,C_1)$ is a Zariski pair of projective curves, the total Milnor numbers $\mu ^{\scriptscriptstyle{\text{tot}}}(C_0)$ and $\mu ^{\scriptscriptstyle{\text{tot}}}(C_1)$ coincide, and the equality $\zeta _{g_0,\mathbf {0}}(t)=\zeta _{g_1,\mathbf {0}}(t)$ follows immediately from (2.1).

To conclude that $(V(g_0),V(g_1))$ is a $\zeta $ -Zariski pair, it remains to observe that the links $K_{g_0}$ and $K_{g_1}$ have the same abstract topology; this is proved in [27, Th. 24 and Rem. 25].

Now, let us show that $(V(g_0),V(g_1))$ is a $\mu ^*$ -Zariski pair of surface singularities. For that, we must first show that $g_0$ and $g_1$ have the same $\mu ^*$ -invariant at $\mathbf {0}$ . We recall that the $\mu ^*$ -invariant of $g_l$ at $\mathbf {0}$ , introduced by Teissier in [32], is the triple

$$ \begin{align*} \mu^*_{\mathbf{0}}(g_l):=(\mu_{\mathbf{0}}(g_l),\mu_{\mathbf{0}}({g_l}\vert_{H}),\mbox{mult}_{\mathbf{0}}(g_l)-1), \end{align*} $$

where $\mu _{\mathbf {0}}(g_l)$ is the Milnor number of $g_l$ at $\mathbf {0}$ , $\mu _{\mathbf {0}}({g_l}\vert _{H})$ is the Milnor number at $\mathbf {0}$ of the restriction of $g_l$ to a generic plane H of $\mathbb {C}^3$ through the origin (this number is usually denoted by $\mu _{\mathbf {0}}^{(2)}({g_l})$ ), and $\mbox {mult}_{\mathbf {0}}(g_l)$ is the multiplicity of $g_l$ at $\mathbf {0}$ .

By [21, Th. 2], for any $l\in \{0,1\}$ , the Milnor number $\mu _{\mathbf {0}}(g_l)$ is given by

$$ \begin{align*} \mu_{\mathbf{0}}(g_l)=(d-1)^3+m \mu^{\scriptscriptstyle{\text{tot}}}, \end{align*} $$

where $\mu ^{\scriptscriptstyle{\text{tot}}}$ is the (common) total Milnor number of $C_0$ and $C_1$ .

For a generic plane H of $\mathbb {C}^3$ through the origin, the restriction $f_l\vert _{H}$ is a homogeneous polynomial of degree d with an isolated singularity at $\mathbf {0}$ , so that its Milnor number at $\mathbf {0}$ is $\mu _{\mathbf {0}}(f_l\vert _{H})=(d-1)^2$ . Since $f_l\vert _{H}$ is Newton non-degenerate and the term $z_1^{d+m}$ is above the Newton diagram $\Gamma (g_l\vert _{H})=\Gamma (f_l\vert _{H})$ , the restriction $g_l\vert _{H}$ is Newton non-degenerate too. Thus, its Milnor number at $\mathbf {0}$ is determined by $\Gamma (g_l\vert _{H})$ , and hence we have

$$ \begin{align*} \mu_{\mathbf{0}}^{(2)}({g_l}):=\mu_{\mathbf{0}}(g_l\vert_{H})=\mu_{\mathbf{0}}(f_l\vert_{H})=(d-1)^2. \end{align*} $$

Lastly, since the multiplicities of $g_0$ and $g_1$ at $\mathbf {0}$ are equal to d, it follows that $g_0$ and $g_1$ have the same $\mu ^*$ -invariant at $\mathbf {0}$ , namely, for any $l\in \{0,1\}$ , we have

$$ \begin{align*} \mu_{\mathbf{0}}^*(g_l)=((d-1)^3+m \mu^{\scriptscriptstyle{\text{tot}}},(d-1)^2,d-1). \end{align*} $$

Finally, and this is the heart of the proof, we must now show that $g_0$ and $g_1$ lie in different path-connected components of the $\mu ^*$ -constant stratum. To this end, we argue by contradiction. Suppose that $g_0$ and $g_1$ belong to the same component. Then, by [10, Th. 5.3], there exists a $\mu ^*$ -constant piecewise complex-analytic family $\{g_s\}_{0\leq s\leq 1}$ connecting $g_0$ and $g_1$ . In particular, the multiplicity $\mbox {mult}_{\mathbf {0}}(g_s)$ of $g_s$ at $\mathbf {0}$ is independent of $s\in [0,1]$ , and the initial polynomial $\mbox {in}(g_s)$ of $g_s$ (i.e., the sum of the monomials of $g_s$ of lowest degree) has degree d.

Lemma 2.1. For each $s\in [0,1]$ , the homogeneous polynomial ${\mathrm{in}}(g_s)$ is reduced, so that the projective curve $C_{s}\subseteq \mathbb {P}^2$ defined by ${\mathrm{in}}(g_s)$ has only isolated singularities.

Proof. We argue by contradiction. Suppose there exists $s_0\in [0,1]$ such that $\mbox {in}(g_{s_0})$ is not reduced (i.e., $C_{s_0}$ has non-isolated singularities). Then, for a generic linear plane H of $\mathbb {C}^3$ , there are coordinates $(x,y)$ for H and linear forms $\ell _1(x,y),\ldots ,\ell _q(x,y)$ such that

$$ \begin{align*} \mbox{in}(g_{s_0})\vert_H(x,y)=\ell_1(x,y)^{p_1}\cdots\ell_q(x,y)^{p_q} \end{align*} $$

with $p_1\geq \cdots \geq p_q$ and $p_1\geq 2$ . By a linear change of coordinates, we may assume that $\ell _1(x,y)\equiv x$ , so that

$$ \begin{align*} \mbox{in}(g_{s_0})\vert_H(x,y)=x^{p_1}h(x,y), \end{align*} $$

where h is a homogeneous polynomial of degree $d-p_1$ (in particular, $\mbox {in}(g_{s_0})\vert _H$ is not convenient with respect to the coordinates $(x,y)$ ). By adding monomials of the form $x^{\alpha }$ and $y^{\beta }$ for $\alpha ,\, \beta $ large enough, we may also assume that $g_{s_0}\vert _H$ is convenient. Now, since the integral point $(1,d-1)$ is not on the Newton diagram $\Gamma (\mbox {in}(g_{s_0})\vert _H)$ of $\mbox {in}(g_{s_0})\vert _H$ with respect to the coordinates $(x,y)$ , it follows ⁴ that

$$ \begin{align*} \nu(\Gamma_{\!-}(g_{s_0}\vert_H))>\nu(\Gamma_{\!-}(g_{0}\vert_H)) \end{align*} $$

(see Figure 1, where $\Gamma _{\!+}(\mbox {in}(g_{s_0})\vert _H)$ is the Newton polyhedron of $\mbox {in}(g_{s_0})\vert _H$ in the coordinates $(x,y)$ ). Here, $\nu (\cdot )$ denotes the Newton number (see [14] for the definition) and $\Gamma _{\!-}(g_{s_0}\vert _H)$ stands for the cone over $\Gamma (g_{s_0}\vert _H)$ with the origin as vertex. (Again, $\Gamma (g_{s_0}\vert _H)$ denotes the Newton diagram of $g_{s_0}\vert _H$ with respect to the coordinates $(x,y)$ .) The polyhedron $\Gamma _{\!-}(g_{0}\vert _H)$ is defined similarly. Since

$$ \begin{align*} \mu_{\mathbf{0}}(g_{s_0}\vert_H)\geq \nu(\Gamma_{\!-}(g_{s_0}\vert_H)) \end{align*} $$

(see [14, théorème 1.10]), altogether we have

$$ \begin{align*} \mu^{(2)}_{\mathbf{0}}(g_{s_0})=\mu_{\mathbf{0}}(g_{s_0}\vert_H)\geq \nu(\Gamma_{\!-}(g_{s_0}\vert_H))>\nu(\Gamma_{\!-}(g_{0}\vert_H))=(d-1)^2=\mu^{(2)}_{\mathbf{0}}(g_{0}), \end{align*} $$

which is a contradiction to the $\mu ^*$ -constancy.

Lemma 2.2. The zeta-function $\zeta _{g_s,\mathbf {0}}(t)$ is independent of $s\in [0,1]$ .

Figure 1 Newton diagrams.

Proof. It is well known that in a $\mu ^*$ -constant piecewise complex-analytic family $\{g_s\}$ , the diffeomorphism type of the embedded link $(\mathbb {S}_\varepsilon ^5,K_{g_s})$ is independent of s (see [32, théorème 3.9 and remarque 3.12]). Alternatively, we may use [27, Lem. 12], which asserts that in a $\mu $ -constant (a fortiori in a $\mu ^*$ -constant) piecewise complex-analytic family $\{g_s\}$ , the zeta-function $\zeta _{g_s,\mathbf {0}}(t)$ is independent of s.

Now, by the A’Campo formula (see [1, théorème 3]), we know that the zeta-function $\zeta _{g_s,\mathbf {0}}(t)$ is uniquely written as

(2.2) $$ \begin{align} \zeta_{g_s,\mathbf{0}}(t)=\prod_{i=1}^{\ell} (1-t^{d_i})^{\nu_i}, \end{align} $$

where $d_1,\ldots ,d_{\ell }$ are mutually disjoint and $\nu _1,\ldots ,\nu _{\ell }$ are nonzero integers. The smallest integer $d_{i_0}$ among $d_1,\ldots ,d_\ell $ is called the zeta-multiplicity of $g_s$ and is denoted by $m_\zeta (g_s)$ . We define the zeta-multiplicity factor of $\zeta _{g_s,\mathbf {0}}(t)$ as the factor $(1-t^{d_{i_0}})^{\nu _{i_0}}$ of (2.2) corresponding to the zeta-multiplicity $d_{i_0}\equiv m_\zeta (g_s)$ . Note that, by Lemma 2.2, the zeta-multiplicity of $g_s$ and the zeta-multiplicity factor of $\zeta _{g_s,\mathbf {0}}(t)$ are independent of s. Moreover, by [27, Prop. 11], we know that $m_\zeta (g_s)\geq \mbox {mult}_{\mathbf {0}}(g_s)=d$ , and the formula (2.1) shows that for $s=0$ we have $m_{\zeta }(g_0)\leq d$ . So, altogether, $m_{\zeta }(g_s)=d$ for any $s\in [0,1]$ .

Lemma 2.3. For any $s\in [0,1]$ , the zeta-multiplicity factor of $\zeta _{g_s,\mathbf {0}}(t)$ is given by

$$ \begin{align*} (1-t^d)^{-d^2+3d-3+\mu^{\scriptscriptstyle{\mathrm{tot}}}(C_{s})}, \end{align*} $$

and since the latter is independent of s, so is the total Milnor number $\mu ^{\scriptscriptstyle{\mathrm{tot}}}(C_{s})$ .

Proof. Here, to compute $\zeta _{g_s,\mathbf {0}}(t)$ , we apply a method developed by the second-named author in [24]. This method, inspired by an approach of Clemens [8], was used in [24, Chap. I, Proof of Th. 5.2] to generalize the classical zeta-function formula of A’Campo [1]. Roughly, the idea is to decompose the lifted Milnor fibration $\pi ^*g_s$ (which is isomorphic to the original Milnor fibration of $g_s$ at $\mathbf {0}$ ) into its restrictions along “controlled” tubular neighborhoods of the strata in a canonical regular stratification of $\pi ^{-1}(V(g_s))$ . Then, by the multiplicativeness of the zeta-function, it suffices to compute the zeta-functions of the induced restricted fibrations. More precisely, let $\mathbf {p}_1,\ldots ,\mathbf {p}_{k_0}$ be the points of the singular set $\Sigma (C_s)$ of $C_s$ , and for each $\mathbf {p}_k$ , let $B_\varepsilon (\mathbf {p}_k)$ be a small ball centered at $\mathbf {p}_k$ . Put

$$ \begin{align*} B:=\bigcup_{k=1}^{k_0}B_\varepsilon(\mathbf{p}_k), \end{align*} $$

and consider tubular neighborhoods $N(C_s)$ and $N(E)$ of $C_s\setminus B$ and $E\setminus (N(C_s)\cup B)$ , respectively. As in [24, Chap. I, p. 56], we assume that the triple

(2.3) $$ \begin{align} \{B,N(C_s),N(E)\}, \end{align} $$

together with its natural associated projections and distance functions, makes a family of “control data” in the sense of Mather [22, §7]. Consider the restrictions of $\hat g_s:=\pi ^*g_s$ to $N(E)$ , $N(C_s)$ and the balls $B_\varepsilon (\mathbf {p}_k)$ , respectively. The relations (5.2.4) and (5.2.5), together with Lemmas (5.3) and (5.4), of [24, Chap. I] say that

(2.4) $$ \begin{align} \zeta_{g_s,\mathbf{0}}(t)\equiv \zeta_{\hat g_s}(t) = \zeta_{\hat g_s\vert_{N(E)}}(t)\cdot \zeta_{\hat g_s\vert_{N(C_s)}}(t)\cdot \prod_{k=1}^{k_0} \zeta_{\hat g_s\vert_{B_\varepsilon(\mathbf{p}_k)}}(t). \end{align} $$

Thus, it suffices to compute each piece $\zeta _{\hat g_s\vert _{N(E)}}(t)$ , $\zeta _{\hat g_s\vert _{N(C_s)}}(t)$ , and $\zeta _{\hat g_s\vert _{B_\varepsilon (\mathbf {p}_k)}}(t)$ separately.

We start with the calculation of the zeta-function $\zeta _{\hat g_s\vert _{N(E)}}(t)$ of the fibration $\hat g_s\vert _{N(E)}$ . For admissible coordinates $\mathbf {x}=(x_1,x_2,x_3)$ in a neighborhood $U_{\mathbf {p}}$ of a point $\mathbf {p}\in E':=E\setminus (N(C_s)\cup B)$ , we may assume that the projection

$$ \begin{align*} p\colon U_{\mathbf{p}}\cap N(E)\to E' \end{align*} $$

associated with the family of control data (2.3) is given by $\mathbf {x}\mapsto (0,x_2,x_3)$ , so that $E'$ is defined by $x_1=0$ and the restriction of $\hat g_s$ to $p^{-1}(\mathbf {p})$ is given by $x_1^d$ . Then, by the relation (5.2.5) of [24, Chap. I], the normal zeta-function $\zeta _{E'}^\bot (t)$ of $\hat g_s$ along $E'$ (see [24, Chap. I, p. 59] for the definition) is given by

$$ \begin{align*} \zeta_{E'}^\bot(t)=(1-t^d)^{-1}. \end{align*} $$

Thus, by [24, Chap. I, Lems. (5.3) and (5.4)], we get

$$ \begin{align*} \zeta_{\hat g_s\vert_{N(E)}}(t) & =(\zeta_{E'}^\bot(t))^{\chi(E\setminus \tilde V(g_s))}=(\zeta_{E'}^\bot(t))^{\chi(\mathbb{P}^2\setminus C_s)}=(\zeta_{E'}^\bot(t))^{\chi(\mathbb{P}^2)-\chi(C_s)}\\ &=(1-t^d)^{-\chi(\mathbb{P}^2)+\chi(C_s)}=(1-t^d)^{-3+\chi(C_s)} =(1-t^d)^{-3+3d-d^2+\mu^{\scriptscriptstyle{\text{tot}}}(C_s)}. \end{align*} $$

Here, $\chi (\cdot )$ denotes the Euler–Poincaré characteristic, and we recall that for a reduced curve $C_s$ of degree d, we have $\chi (C_s)=3d-d^2+\mu ^{\scriptscriptstyle{\text{tot}}}(C_s)$ (see, e.g., [35, Cor. 7.1.4]).

Next, we look at the zeta-function $\zeta _{\hat g_s\vert _{N(C_s)}}(t)$ . This time, for admissible coordinates $\mathbf {x}=(x_1,x_2,x_3)$ in a neighborhood $U_{\mathbf {p}}$ of a point $\mathbf {p}\in C^{\prime }_s:=C_s\setminus B$ , we may assume that the projection

$$ \begin{align*} p'\colon U_{\mathbf{p}}\cap N(C_s)\to C^{\prime}_s \end{align*} $$

associated with the family of control data (2.3) is given by $\mathbf {x}\mapsto (0,x_2,0)$ , so that $C^{\prime }_s$ is defined by $x_1=x_3=0$ and the restriction of $\hat g_s$ to $p^{\prime -1}(\mathbf {p})$ is given by $x_1^dx_3$ . Then, by the relation (5.2.5) of [24, Chap. I], the normal zeta-function of $\hat g_s$ along $C^{\prime }_s$ is given by $\zeta _{C^{\prime }_s}^\bot (t)=1$ , and hence, by [24, Chap. I, Lems. (5.3) and (5.4)] again, we get

$$ \begin{align*} \zeta_{\hat g_s\vert_{N(C_s)}}(t)=1. \end{align*} $$

As for the zeta-function $\zeta _{\hat g_s\vert _{B_\varepsilon (\mathbf {p}_k)}}(t)$ , since the zeta-multiplicity of $g_s$ is d and the (usual) multiplicity of $\hat g_s$ at $\mathbf {p}_{k}$ is greater than or equal to $d+1$ , it follows from [27, Prop. 11] that $\zeta _{\hat g_s\vert _{B_\varepsilon (\mathbf {p}_k)}}(t)$ does not contribute to the zeta-multiplicity factor of $\zeta _{\hat g_s}(t)$ .

So, altogether, the unique contribution to the zeta-multiplicity factor of $\zeta _{\hat g_s}(t)$ comes from the zeta-function $\zeta _{\hat g_s\vert _{N(E)}}(t)$ and is given by $(1-t^d)^{-3+3d-d^2+\mu ^{\scriptscriptstyle{\text{tot}}}(C_s)}$ .

We can now easily complete the proof of Theorem 1.1 thanks to two theorems of Lê. Indeed, we first observe that if there exists $s_0\in [0,1]$ such that the family $\{\mbox {in}(g_s)\}$ has a bifurcation of the singularities in a small ball B centered at a singular point $\mathbf {p}_0$ of $C_{s_0}$ , ⁵ then, by [17, théorème B] (see also [12], [15]), for $s\not =s_0$ near $s_0$ , we have

$$ \begin{align*} \sum_{\mathbf{p}\in B\cap \Sigma(C_s)}\mu_{\mathbf{p}}(\mbox{in}(g_s)) < \mu_{\mathbf{p}_0}(\mbox{in}(g_{s_0})), \end{align*} $$

and hence $\mu ^{\scriptscriptstyle{\text{tot}}}(C_{s})<\mu ^{\scriptscriptstyle{\text{tot}}}(C_{s_0})$ , which contradicts Lemma 2.3. Therefore, there is no such an $s_0$ . But in this case it follows from [16] and the discussion in [9, pp. 17–18, 121] that the topological type of the pair $(\mathbb {P}^2,C_s)$ is independent of s, so that $(C_0,C_1)$ is not a Zariski pair—again a contradiction.

Figure 2 Bifurcation of singularities.