Pencils and critical loci on normal surfaces

Abstract

We study linear pencils of curves on normal surface singularities. Using the minimal good resolution of the pencil, we describe the topological type of generic elements of the pencil and characterize the behaviour of special elements. Furthermore, we show that the critical locus associated to the pencil is linked to the special elements. This gives a decomposition of the critical locus through the minimal good resolution and as a consequence, some information on the topological type of the critical locus.

1 Introduction

Let (Z, z) be a complex analytic normal surface, and let $\pi : (Z,z)\rightarrow ({{\mathbb {C}}}^2,0)$ be a finite complex analytic morphism germ. We choose coordinates (u, v) in $({{\mathbb {C}}}^2, 0)$ and denote $f:=u\circ \pi$ and $g:=v\circ \pi$. We consider the meromorphic function $h:=f/g$ defined in a punctured neighbourhood V of z in Z. It can be seen as a map $h: V\rightarrow {{\mathbb {C}}}{{\mathbb {P}}}^1$ defined by $h(x):=(f(x):g(x))$. For $w = (w_1:w_2)\in {{\mathbb {C}}}{{\mathbb {P}}}^1$, the closure of $h^{-1}(w)$ defines the curve $w_2f-w_1g=0$ on the surface (Z, z). The set $\Lambda := \{ w_2f-w_1g \; | \; (w_1:w_2) \in {{\mathbb {C}}}{{\mathbb {P}}}^1\}$ is the pencil defined by f and g. Let denote $\phi _w$ the element of the pencil $\Lambda$ equal to $w_2f-w_1g$. Its (non reduced) zero locus, denoted by $\Phi _w$, is called the fiber defined by $\phi _w$. Assume $(Z,z)\subset ({{\mathbb {C}}}^n,0)$, then the topological type of $\phi _w$ is the homeomorphism class of the pair $(B_\varepsilon \cap Z, B_\varepsilon \cap \phi _w^{-1}(0))$ where $B_\varepsilon$ is the ball of ${{\mathbb {C}}}^n$ centered at z of radius $\varepsilon$ small enough and the components of $\phi _w^{-1}(0)$ are pondered by the multiplicity of the irreducible components of $\phi _w$. If $\phi _w$ and $\phi _{w'}$ have the same topological type, we also say that $\Phi _w$ and $\Phi _{w'}$ are topologically equivalent.

Such linear families of curves have been studied independently and through different approaches for (Z, z) equal to $({{\mathbb {C}}}^2,0)$ in [8, 11, 16]. In the general case (it means (Z, z) is the germ of a normal complex analytic surface which is not smooth anymore), Lê and Bondil give in [3] a definition of general elements of the pencil which are characterized by the minimality of their Milnor number. In [2] Bondil gives an algebraic $\mu$-constant theorem for linear families of plane curves. Other results have been obtained in the case where $\pi$ is the restriction to (Z, z) of a linear projection of $({{\mathbb {C}}}^n, 0)$ onto $({{\mathbb {C}}}^2,0)$ (see [1, 4, 18]). At last, the topology of the morphism $\pi$ has been studied in [13, 14]. In [13], the authors define rational quotients which are topological invariants of $(\pi , u,v)$ and give different ways to compute them. In [14], F. Michel presents another proof of the topological invariance of this set of rational numbers and besides, she gives a decomposition of the critical locus of $\pi$ in bunches linked to the set of invariants.

Let $\rho : (X,E)\rightarrow (Z,z)$ be a good resolution of the singularity (Z, z). It is a resolution of the singularity (Z, z) such that the exceptional divisor is a union of smooth projective curves with normal crossings. In particular three irreducible components of the exceptional divisor do not meet at the same point. The lifting $h\circ \rho$ is a meromorphic function defined in a suitable neighbourhood of E in X but in a finite set of points.

A good resolution $\rho$ of the pencil $\Lambda$ is a good resolution of the singularity (Z, z) in which $h\circ \rho$ is a morphism. A good resolution of the pencil $\Lambda$ is said to be minimal if and only if by the contraction of any rational component of self-intersection -1 of the exceptional divisor we do not obtain a good resolution of $\Lambda$ anymore. We will see in Sect. 2 that there exists a unique minimal good resolution of $\Lambda$.

Let $\rho : (Y,E)\rightarrow (Z,z)$ be the minimal good resolution of the pencil $\Lambda$.

Definition

An irreducible component $E_\alpha$ of E is called dicritical if the restriction of $\widehat{h}= h\circ \rho$ to $E_\alpha$ is not constant. Let denote ${{\mathcal {D}}}$ the union of the dicritical components.

Definition

We say that a subset $\Delta$ of E is a special zone if it is, either the closure of a connected component of $\overline{E\setminus {{\mathcal {D}}}}$ or a critical point of the restriction of $\widehat{h}$ to ${{\mathcal {D}}}$. In the last case, P can be either a smooth point of ${{{\mathcal {D}}}}$ or a singular point of ${{{\mathcal {D}}}}$. Let $SZ(\Lambda )$ denote the (finite) set of special zones.

Notice that, if $\Delta$ is a special zone then $\widehat{h}_{|_{\Delta }}$ is constant.

Definition

The set of special values of $\Lambda$ is constituted of the values $\widehat{h} (\Delta )$ for $\Delta \in SZ(\Lambda )$. A fiber associated to a special value is called a special fiber of $\Lambda$.

The other values of ${{\mathbb {C}}}{{\mathbb {P}}}^1$ are called generic values for the pencil $\Lambda$. A fiber associated to a generic value is called a generic fiber of $\Lambda$.

Notice that, a change of the set of generators (f, g) of the pencil $\Lambda$ is reflected by a linear change of coordinates in ${{\mathbb {C}}}{{\mathbb {P}}}^1$ (a projectivity). Therefore, neither the definition of a good resolution of the pencil nor the set of special zones depend on the pair of functions of $\Lambda$ chosen. Obviously, the concrete set of special values depends on the pair of functions (f, g).

We prove the following results.

Theorem 1

Let $w,w'$ be generic values for the pencil $\Lambda$, then the fibers $\Phi _w$ and $\Phi _{w'}$ have the same topological type.

Moreover, if $e\in {{\mathbb {C}}}{{\mathbb {P}}}^1$ is a special value for the pencil $\Lambda$, then the fibers $\Phi _w$ and $\Phi _e$ do not have the same topological type.

The above Definitions and Theorem generalize some of the results contained in [11] (see e.g. Theorem 4.1) where the authors study pencils on ${{\mathbb {C}}}^2$. Moreover, we prove the following characterization of the special fibers in terms of the minimal resolution, which extends to the case of normal surfaces the second item of Theorems 1, 2, 3 in [8] (there for pencils defined on ${{\mathbb {C}}}^2$).

Theorem 2

Let $\rho : (Y,E)\rightarrow (Z,z)$ be the minimal good resolution of the pencil $\Lambda$, $\Delta \in SZ(\Lambda )$, and let $e\in {{\mathbb {C}}}{{\mathbb {P}}}^1$. Then, the strict transform of $\Phi _e$ by $\rho$ intersects $\Delta$ if and only if $\Phi _e$ is special and $\widehat{h}(\Delta )=e$.

In a second part we are interested in understanding the behaviour of the critical locus of the map $\pi$. We denote by $I_z(\ ,\ )$ the local intersection multiplicity at z (see Sect. 2.1). The following result generalizes the third item of Theorems 1, 2, 3 of [8].

Theorem 3

Let $\rho : (Y,E)\rightarrow (Z,z)$ be the minimal good resolution of $\Lambda$. For each element $\Delta \in SZ(\Lambda )$ there exists an irreducible component of the critical locus $C(\pi )$ of $\pi$ such that its strict transform by $\rho$ intersects $\Delta$.

Moreover, for each branch $\Gamma$ of $C(\pi )$ there exists $\Delta \in SZ(\Lambda )$ such that the strict transform of $\Gamma$ by $\rho$ intersects $\Delta$ and the value $e=\widehat{h}(\Delta )$ is the unique one that satisfies $I_z(\phi _{e},\Gamma )>I_{z}(\phi _w,\Gamma )$ for all $w\ne e$.

A consequence of these results is Theorem 4:

Theorem 4

Let $\Phi _e$ be a fiber of $\Lambda$. Then the three following properties are equivalent:

• 1. $\Phi _{e}$ is a special fiber of $\Lambda$.

• 2. $I_z(\phi _e, C(\pi ))>\displaystyle {\min }_{\phi \in \Lambda }I_z( \phi , C(\pi ))$.

• 3. $\mu (\phi _{e})> \displaystyle {\min }_{\phi \in \Lambda }\mu (\phi )$.

Remark

Let (f, g) be a pair of linear forms in such a way that $\pi$ is a generic plane projection of (Z, z) in the sense of Teissier (see [20]). Then $\Lambda$ is a pencil of hyperplane sections of (Z, z) and the generic members of the pencil are exactly the generic hyperplane sections among them. So, the pencil is resolved by the normalized blow-up $\psi : (X,E)\rightarrow (Z,z)$ of the maximal ideal and the minimal good resolution of $\Lambda$ is just the minimal good resolution of (Z, z) which factors through $\psi$. In this case, the branches of the critical locus $C(\pi )$ (i.e. the polar curve) appear as curvettas in the Nash modification $N: (X',E') \rightarrow (Z,z)$ of the surface. Thus, Theorem 3 is strongly related with the configuration of the irreducible components of E and $E'$. In a more general context, if g is a generic linear form with respect to f, then the localization of the branches of the polar curve is related with the relative Nash transform of f in the same way.

The organization of the paper is as follows. In Sect. 2, once we have set some preliminary results, we construct and study the minimal good resolution of $\Lambda$. In Sect. 3, we prove Theorems 1 and 2 and in Sect. 4 we prove Theorems 3 and  4. To finish, in Sect. 5, we present some examples.

2 Preliminary results and notations

Let (Z, z) be a normal surface singularity and let $\rho : (X,E)\rightarrow (Z,z)$ be a good resolution of it. We denote $\{ E_{\alpha }, {\alpha \in G(\rho )}\}$ the set of irreducible components of the exceptional divisor E. For $\alpha \in G(\rho )$ and for each holomorphic function $f: (Z,z)\rightarrow ({{\mathbb {C}}},0)$ let $\nu _{\alpha }(f)$ denote the vanishing order of $\overline{f} = f\circ \rho : X\rightarrow {{\mathbb {C}}}$ along the irreducible exceptional curve $E_{\alpha }$ ($\nu _{\alpha }$ is just the divisorial valuation defined by $E_{\alpha }$). The divisor $(\overline{f})$ defined by $\overline{f} = f\circ \rho$ on X could be written as

$$\begin{aligned} (\overline{f}) = (\widetilde{f}) + \sum _{\alpha \in G(\rho )}\nu _{\alpha }(f) E_{\alpha } \; , \end{aligned}$$

where the local part $(\widetilde{f})$ is the strict transform of the germ $\{f=0\}$. For each $\beta \in G(\rho )$ one has the known Mumford formula (see [15]):

$$\begin{aligned} (\overline{f})\cdot E_{\beta } = (\widetilde{f})\cdot E_{\beta } + \sum _{\alpha } \nu _{\alpha }(f) (E_{\alpha }\cdot E_{\beta }) = 0\; . \end{aligned}$$

(1)

(Here “$\cdot$” stands for the intersection form on the smooth surface X). Notice that the intersection matrix $(E_{\alpha }\cdot E_{\beta })$ is negative definite and so $\{\nu _\alpha (f)\}$ is the unique solution of the linear system defined by Eq. (1).

2.1 Intersection multiplicity

Let $C\subset (Z,z)$ be an irreducible germ of curve in (Z, z) and let $f\in {{\mathcal {O}}}_{Z,z}$ be a function. Let $\varphi : ({{\mathbb {C}}},0)\rightarrow (C,z)$ be a parametrization (uniformization) of (C, z), then we define the intersection multiplicity of $\{f=0\}\subset Z$ and C at $z\in C$ as $I_z(f,C)= {\text {ord}}_{\tau }(f\circ \varphi (\tau ))$ ($\tau$ is the parameter in ${{\mathbb {C}}}$). Notice that, for C fixed, $I_z(-,C)$ is the valuation defined by the irreducible germ C. Obviously the above definition could be extended by linearity to define the intersection multiplicity of a f with a (local) divisor $\sum _{i=1}^k n_i C_i$ as $I_{z}(f,\sum n_i C_i) = \sum n_i I_z(f, C_i)$.

Let $\rho : (X,E)\rightarrow (Z,z)$ be a good resolution of the normal singularity (Z, z) and let $E = \bigcup _{\alpha \in G(\rho )}E_{\alpha }$ be the exceptional divisor. Let $\widetilde{C} := \overline{\rho ^{-1}(C\setminus \{z\})}$ be the strict transform of C by $\rho$. Then (see [15])

$$\begin{aligned} I_{z}(f,C) = (\overline{f})\cdot \widetilde{C} = (\widetilde{f})\cdot \widetilde{C} + \sum _{\alpha \in G(\rho )}\nu _{\alpha }(f) (E_{\alpha }\cdot \widetilde{C})\; . \end{aligned}$$

Let us take now a good resolution $\rho$ such that $\widetilde{C}$ is smooth and transversal to E at a smooth point P and also with the condition $(\widetilde{f})\cdot \widetilde{C}=0$. This resolution could be obtained by a finite number of blowing ups of points starting on (say) the minimal good resolution of (Z, z). Let $\alpha (C)\in G(\rho )$ be such that $E_{\alpha (C)}$ is the (unique) component of E with $\widetilde{C}\cap E_{\alpha (C)} = P$. Then one has $I_{z}(f,C) = \nu _{\alpha (C)}(f) = I_{P}(f\circ \rho ,\widetilde{C})$. Here $I_{P}(-,-)$ denotes the usual local intersection multiplicity of two germs at the smooth local surface (X, P). Notice that $\widetilde{C}$ is a curvetta at the point $P\in E_{\alpha (C)}$ (it means $\widetilde{C}$ is an irreducible smooth curve germ transverse to $E_{\alpha (C)}$ at P), $\widetilde{C}$ is the normalization of C and $\rho |_{\widetilde{C}}: \widetilde{C}\rightarrow C$ is a uniformization of C.

Let f, g be analytic functions on (Z, z) and let $\Lambda = \langle f, g \rangle = \{\phi _w = w_2 f - w_1 g \, | \, w=(w_1:w_2)\in {{\mathbb {C}}}{{\mathbb {P}}}^1\}$ be the pencil of analytic functions defined by f and g. As in the case of plane branches (see [7]), one has the following easy and useful result:

Proposition 1

Let $C\subset (Z,z)$ be an irreducible germ of curve. Then there exists a unique $w_0\in {{\mathbb {C}}}{{\mathbb {P}}}^1$ such that $I_{z}(\phi _w,C)$ is constant for all $w\in {{\mathbb {C}}}{{\mathbb {P}}}^1\setminus \{w_0\}$ and $I_{z}(\phi _{w_0},C)> I_{z}(\phi _w,C)$.

Proof

The statement is trivial taking into account that $I_z(\phi ,C)$ is the order of the series $\phi \circ \varphi (\tau )$. $\square$

2.2 Resolution of pencils

Let $\pi = (f,g): (Z,z)\rightarrow ({{\mathbb {C}}}^2,0)$ be a finite complex analytic morphism germ, let $\Lambda = \langle f, g \rangle = \{w_2 f - w_1 g \,|\, w=(w_1:w_2)\in {{\mathbb {C}}}{{\mathbb {P}}}^1\}$ be the pencil of analytic functions defined by f and g and let $h = (f/g): V \rightarrow {{\mathbb {C}}}{{\mathbb {P}}}^1$ be the meromorphic function defined by f/g in a suitable punctured neighbourhood of $z\in Z$.

A good resolution of (f, g) is a good resolution $\rho : (X,E)\rightarrow (Z,z)$ of (Z, z) such that the (reduced) divisor $|(fg\circ \rho )^{-1}(0)|$ has normal crossings. Starting on the minimal good resolution of (Z, z) it can be produced by a sequence of blowing-ups of points in the corresponding smooth surface (resolving the singularities of the reduced total transform of the curve $\{fg=0\}$).

Let $\rho : (X,E)\rightarrow (Z,z)$ be a good resolution of (Z, z) and $E_\alpha$ an irreducible component of E. The Hironaka quotient of (f, g) on $E_\alpha$ is the following rational number:

$$\begin{aligned} q(E_\alpha ):= \displaystyle \frac{\nu _{\alpha }(f)}{\nu _{\alpha }(g)}. \end{aligned}$$

If $q(E_\alpha )>1$ (resp. $q(E_\alpha )<1$) then the component $E_\alpha$ belongs to the zero divisor (resp. pole divisor) of $h\circ \rho$. Note that if $E_\alpha$ is a dicritical component of E then $q(E_\alpha )=1$. However, there may exist irreducible components $E_\alpha$ of E which are not dicritical and for which $q(E_\alpha )=1$, namely all the components for which the restriction of $h\circ \rho$ on $E_\alpha$ is constant (and so is not dicritical) and it’s neither zero nor infinity.

Proposition 2

There exists a (unique) minimal good resolution of $\Lambda$.

Proof

Let $\rho ' :(Y', E')\rightarrow (Z,z)$ be the minimal good resolution of (f, g). The indetermination points of $h\circ \rho '$ are the intersection points of irreducible components $E_{\alpha }$ and $E_{\beta }$ of the total transform $| (fg\circ {\rho '})^{-1}(0)|$ for which one has $q(E_\alpha )>1$ and $q(E_\beta )<1$. Here, one of the components, $E_{\alpha }$ or $E_{\beta }$, is allowed to be the strict transform $\widetilde{\xi }$ of a branch $\xi$ of $\{f=0\}$ (in such a case we put $q(\widetilde{\xi })>1$) or $\{g=0\}$ (respectively $q(\widetilde{\xi })<1$). Let P be such an indetermination point. By blowing-up at P one creates a divisor $E_\eta$ of genus 0 and one has that $\nu _{\eta }(f)=\nu _{\alpha }(f)+\nu _{\beta }(f)$ and $\nu _{\eta }(g)=\nu _{\alpha }(g)+\nu _{\beta }(g)$. (If $E_{\beta }$ is a branch $\xi$ of $\{f=0\}$ of multiplicity r, we have $\nu _{\beta }(f)=r$ and $\nu _{\beta }(g)=0$. We use similar conventions for the case in which $E_{\beta }$ is a branch of $\{g=0\}$.) If $q(E_\eta )=1$, then neither $E_\alpha \cap E_\eta$ nor $E_\beta \cap E_\eta$ is an indetermination point and moreover $E_{\eta }$ is a dicritical divisor. Else, if $q(E_\eta )>1$ (resp. $q(E_\eta )<1)$ then $E_\beta \cap E_\eta$ (resp. $E_\alpha \cap E_\eta$) is an indetermination point.

As $q(E_\alpha )>1$ and $q(E_\beta )<1$ we have $q(E_\alpha )> q(E_\eta )> q(E_\beta )$. So, by iterating the process, after a finite number of blow-ups there does not subsist indetermination points and so we have constructed a good resolution $\rho '':(Y'',E'')\rightarrow (Z,z)$ of $\Lambda$.

Now, to obtain a minimal good resolution of $\Lambda$, we have to contract some rational components of self-intersection $-1$ of the exceptional divisor (see Theorem 5.9 of [9]). By the above construction the new components (specially the last one which is dicritical and with self-intersection $-1$) can not be contracted because in such a case we have an indetermination point. As a consequence a minimal good resolution of $\Lambda$ is obtained from $\rho ''$ by iterated contractions of the rational component of self-intersection $-1$ of the exceptional divisor which are not dicritical. Uniqueness follows as in the case of the usual minimal resolution (see for example [5] th. 6.2 p. 86). $\square$

Let consider $\rho : (Y,E)\rightarrow (Z,z)$ the minimal good resolution of the pencil $\Lambda$ and ${{\widehat{h}}}= h\circ \rho$. For $w\in {{\mathbb {C}}}{{\mathbb {P}}}^1$ let ${\widehat{h}}^{-1}(w) = \widetilde{\Phi _w}$ be the strict transform of the fiber $\Phi _w$. For a dicritical component D of E, we will denote by deg$({{\widehat{h}}}_{\mid D})$ the degree of the restriction of ${{\widehat{h}}}$ to D, ${\widehat{h}}_{\mid D}: D\rightarrow {{\mathbb {C}}}{{\mathbb {P}}}^1$. Let recall that ${{\mathcal {D}}}$ denotes the union of the dicritical components of E.

Proposition 3

Let w be a generic value for the pencil $\Lambda$, then:

• (a) The resolution $\rho$ is a good resolution of $\phi _w$.

• (b) $\widetilde{\Phi _w}$ intersects E only at smooth points of ${{{\mathcal {D}}}}$.

• (c) If $D\subset {{{\mathcal {D}}}}$ is a dicritical component, then the number of intersection points of $\widetilde{\Phi _w}$ and D is equal to deg$({\widehat{h}}_{\mid D})$.

Moreover, the minimal good resolution of $\Lambda$ is the minimal good resolution of any pair of generic elements of $\Lambda$.

Proof

By definition of a generic value, $\widetilde{\Phi _w}$ meets the exceptional divisor E only at smooth points of ${{\mathcal {D}}}$. Let $D\subset {{{\mathcal {D}}}}$ be a dicritical component and P a point of $\widetilde{\Phi _w}\cap D$. Then, as P is not a critical point for ${{\widehat{h}}}$, $\widetilde{\Phi _w}$ is smooth and transversal to D at P. This implies also that

$$\begin{aligned} \text{ deg }\ ({{\widehat{h}}}_{\mid D})= \displaystyle \sum _{P\in D} I_{P}(\widetilde{\phi _w},D)\; . \end{aligned}$$

So, one has deg$({\widehat{h}}_{\mid D}) = \# (\widetilde{\Phi _w}\cap D)$.

Now, let $w'$ be another generic value. Notice that the strict transforms of $\widetilde{\Phi _w}$ and $\widetilde{\Phi _{w'}}$ intersect in the same number of points each dicritical divisor D, so both fibers have the same number of branches, just $\sum _{D\in {{{\mathcal {D}}}}} \text{ deg }({\widehat{h}}_{\mid D})$. Moreover, $\widetilde{\Phi _w}$ and $\widetilde{\Phi _{w'}}$ do not intersect ${{\mathcal {D}}}$ at the same points because ${{\widehat{h}}}$ is a morphism. As a consequence the minimal good resolution of $\Lambda$ is a good resolution of any pair of generic fibers. It leaves to show that it is the minimal one.

In the minimal good resolution of $\Lambda$ all the components of the exceptional divisor that can be contracted (i.e. those with self-intersection $-1$) are dicritical (see the proof of Proposition 2). But contracting a dicritical component we create an indetermination point. Consequently, the minimal good resolution of $\Lambda$ is the minimal good resolution of the pair $(\phi _w,\phi _{w'})$. $\square$

2.3 Hironaka quotients

In 2.2 we have defined the Hironaka quotient of (f, g) on an irreducible component $E_\alpha$ of the exceptional divisor of a good resolution of (Z, z). In the same way we can define the Hironaka quotient of $(\phi _w,\phi _{w'})$ on $E_\alpha$ for any pair $(\phi _w,\phi _{w'})$ of elements of $\Lambda = \langle f, g \rangle$. It is the rational number

$$\begin{aligned} q^w_{w'}(E_\alpha ):= \displaystyle \frac{\nu _{\alpha }(\phi _{w})}{\nu _{\alpha }(\phi _{w'})}. \end{aligned}$$

In this way $q(E_\alpha )= q^0_\infty (E_\alpha )$ (here $0=(0:1)\in {{\mathbb {C}}}{{\mathbb {P}}}^1$, $\infty =(1:0)\in {{\mathbb {C}}}{{\mathbb {P}}}^1$) but to simplify the notations we will still write $q(E_\alpha )$ for the Hironaka quotient of (f, g).

Notice that an irreducible component $E_\alpha$ of E is dicritical if and only if $q^w_{w'}(E_\alpha )=1$ for any pair $(w,{w'})$ of elements of ${{\mathbb {C}}}{{\mathbb {P}}}^1$. Indeed, if for some $(w,{w'})$ we have $q^w_{w'}(E_\alpha )>1$ (resp. $q^w_{w'}(E_\alpha )<1$) then $E_\alpha$ lies in the zero locus of $\phi _w$ (resp. $\phi _{w'}$). So, $E_\alpha$ is not dicritical. Conversely, if $E_\alpha$ is not dicritical then there exists w such that $q^w_{w'}(E_\alpha )>1$, for any $w'\ne w$.

As a consequence of Proposition 3 we have the following result:

Corollary 1

The Hironaka quotient of any pair of generic elements of $\Lambda$ associated to any irreducible component of the exceptional divisor of the minimal good resolution of $\Lambda$ is equal to one.

Proof

Let $w, w'\in {{\mathbb {C}}}{{\mathbb {P}}}^1$ be a pair of generic values of $\Lambda$. If $D\subset {{{\mathcal {D}}}}$ is a dicritical component, then $(\widetilde{\phi _w}) \cdot D = (\widetilde{\phi _{w'}}) \cdot D = \text{ deg }({\widehat{h}}_{\mid D})$ (see Proposition 3). On the other hand, if $E_{\beta }$ is a non-dicritical component of E then one has $(\widetilde{\phi _w}) \cdot E_{\beta } = (\widetilde{\phi _{w'}}) \cdot E_{\beta } = 0$. Now, the system of linear equations given by the Mumford formula (1) at the beginning of Sect. 2 is the same for $\phi _w$ and $\phi _{w'}$ and so the solutions $\{\nu _{\alpha }(\phi _w)\}$ and $\{\nu _{\alpha }(\phi _{w'})\}$ are also the same. Thus, $\nu _\alpha (\phi _w)=\nu _{\alpha }(\phi _{w'})$ and $q^w_{w'}(E_\alpha )=1$ for any $\alpha \in G(\rho )$. $\square$

Remark

Let $E_{\alpha }$ be a non-dicritical component of the exceptional divisor of the minimal good resolution of the pencil $\Lambda$ and let $C \subset (Z,z)$ be an irreducible curve such that its strict transform $\widetilde{C}$ is a curvetta at the point P of $E_{\alpha }$. Assume that $P= \widetilde{C}\cap E_{\alpha }$ does not belong to the strict transform of any fiber $\Phi$ of $\Lambda$. Then, by Proposition 1, there exists a unique $e\in {{\mathbb {C}}}{{\mathbb {P}}}^1$ such that $I_z(\phi _w, C)= \nu _{\alpha }(\phi _w)$ is constant for all $w\in {{\mathbb {C}}}{{\mathbb {P}}}^1\backslash \{e\}$ and $\nu _{\alpha }(\phi _e) > \nu _{\alpha }(\phi _w)$. Moreover, the above value $e\in {{\mathbb {C}}}{{\mathbb {P}}}^1$ must be a special value of $\Lambda$.

Let $b: ( Z_{I},E_{I})\rightarrow (Z,z)$ be the normalized blow-up of the ideal $I=(f,g)$. In [2, 3] an element $\phi \in I$ is defined to be general if it is superficial (it means that its divisorial value is minimal for each irreducible component of $E_I$) and the strict transform of $\Phi = \{\phi =0\}$ by b is smooth and transverse to the exceptional divisor at smooth points. (See definition 2.1 of [2].) Proposition 2.2 of [2] allows to characterize general elements in terms of any good resolution of $Z_I$, in particular one can use a good resolution $\rho : (X,E)\rightarrow (Z,z)$ of the pencil $\Lambda$. In these terms one has that $\phi \in \Lambda$ is general if for each $\alpha \in G(\rho )$

$$\begin{aligned} \nu _{\alpha }(\phi ) = \nu _{\alpha }(I) = \min _{\psi \in I}\{\nu _\alpha (\psi )\} = \min _{\psi \in \Lambda }\{\nu _\alpha (\psi )\} \end{aligned}$$

and, moreover, the strict transform of $\Phi$ by $\rho$ is smooth and transversal to E. By using the definition of the Milnor number of a germ of curve given in [6], from Theorems 1 and 2 of [3] one has that $\phi \in \Lambda$ is general if and only if

$$\begin{aligned} \mu (\phi ) = \mu (I) := \min _{\psi \in I}\{\mu (\psi )\} = \min _{\psi \in \Lambda } \{\mu (\psi )\}\; . \end{aligned}$$

Using Proposition 3 and the above results about Hironaka quotients we have that $\Phi _{w}$ is a generic fiber if and only if $\phi _w$ is general. Moreover, one has also that $\mu (\phi _{w})= \min _{\phi \in \Lambda }\{\mu (\phi )\}$ if and only if $\phi _{w}$ is generic, and therefore $\mu (\phi _{w_0})> \min _{\phi \in \Lambda }\{\mu (\phi )\}$ if and only if $w_{0}$ is a special value of $\Lambda$. This is the equivalence of 1 and 3 in Theorem 4.

3 Topology of special fibers

3.1 Dual graph and topology

Assume $(Z,z)\subset ({{\mathbb {C}}}^n,0)$ and let $M:= Z \cap S^{2n-1}_\varepsilon$ where $S^{2n-1}_\varepsilon$ is the boundary of the small ball $B_\varepsilon$ of radius $\varepsilon$ of ${{{\mathbb {C}}}^n}$ centered at z. The manifold M is called the link (see [15]) of the singularity (Z, z).

Let $\phi _w$ be an element of $\Lambda$ and ${ K}_{\phi _w}:= \phi ^{-1}_w(0)\cap M$. The multilink $\mathbf{K}_{\phi _w}$ of $\phi _w$ is the oriented link ${K}_{\phi _w}$ weighted by the multiplicities of the irreducible components of $\phi _w$. The topological type of $\phi _w$ is given by the isotopy class of $\mathbf{K}_{\phi _w}$ (see [14] Sect. 5).

Let $\rho _w: (X,E)\rightarrow (Z,z)$ be the minimal good resolution of (Z, z) such that the divisor $(\phi _w\circ \rho _w)$ has normal crossings. From Neumann (see [17]), the topology of $\phi _w$ determines the minimal good resolution $\rho _w$, where the irreducible components of the strict transform of $\Phi _w$ by $\rho _w$ are weighted with their multiplicity and the irreducible components of the exceptional divisor by their self-intersection and genus. Conversely, (see [13, 21]) as the intersection matrix $(E_\alpha \cdot E_\beta )_{\alpha , \beta \in G(\rho _w)}$ is negative definite (so invertible) this implies that the linear system associated to the Mumford formulas (1) admits a unique solution, namely the set $\{\nu _\alpha (\phi _w), \alpha \in G(\rho _w)\}$. As a direct consequence, the divisor $(\overline{\phi _w})$ (see Sect. 2) is completely determined on X from the set $\{(\widetilde{\phi _w})\cdot E_{\alpha } \, | \, \alpha \in G(\rho _w)\}$ and so from the minimal good resolution $\rho _w$.

Let $\rho : (X,E)\rightarrow (Z,z)$ be a good resolution of the normal surface singularity (Z, z) and let $E=\bigcup _{\alpha \in G(\rho )}E_{\alpha }$ be its exceptional divisor. It is useful to encode the information of the resolution $\rho$ by means of the so called dual graph of $\rho$. The set of vertices of this graph is the set $G(\rho )$, each vertex $\alpha$ is weighted by $(\alpha ,E_{\alpha }^2,g(E_{\alpha }))$ where $E_{\alpha }^2$ is the self-intersection of $E_{\alpha }$, and $g(E_{\alpha })$ its genus. An intersection point between $E_\alpha$ and $E_\beta$ is represented by an edge linking the vertices $\alpha$ and $\beta$.

If we take $\rho$ as a good resolution of the local curve $C=\sum _{i=1}^{\ell }n_i C_i$ (in particular if $C=\{\varphi =0\}$ for some function $\varphi$) one adds an arrow for each irreducible component $C_i$ of C weighted by the multiplicity $n_i$. In the case in which we deal with a good resolution of a pair of functions (f, g), in the graph of $\{fg=0\}$ one marks with different colors the arrows corresponding to branches of $\{f=0\}$ and those of $\{g=0\}$ (another possibility is to use different kinds of marks, say for example arrows for f and stars for g). The sharp extremities of the arrows are considered as somekind of special vertices of the graph. The notations ${{{\mathcal {G}}}}(\rho )$, ${{{\mathcal {G}}}}(\rho , \varphi )$ and ${{{\mathcal {G}}}}(\rho , f, g)$ will be used for the dual graph in each situation. Note that a good resolution $\rho$ of the pencil $\Lambda =\langle f, g\rangle$ is encoded by the dual graph ${{{\mathcal {G}}}}(\rho ,\phi _w,\phi _{w'})$ for a pair of generic functions $(\phi _w, \phi _{w'})$.

Following Neumann, one has:

Statement The fibers $\Phi _{w}$ and $\Phi _{w'}$ are topologically equivalent if and only if the graphs ${{{\mathcal {G}}}}(\rho _w, \phi _w)$ and ${{{\mathcal {G}}}}(\rho _{w'}, \phi _{w'})$ are the same.

Let $\rho : (X,E)\rightarrow (Z,z)$ be a good resolution of (f, g) and let $E_\alpha$ be an irreducible component of E. Let ${{\mathop {E}\limits ^{\circ }}}_\alpha$ denote the set of smooth points of ${E}_\alpha$ in the reduced total transform $|(fg\circ \rho )^{-1}(0)|$. An irreducible component ${E}_\alpha$ of E (or its corresponding vertex $\alpha$ in ${{{\mathcal {G}}}}(\rho ,f,g)$) is a nodal component if $\chi ({{\mathop {E}\limits ^{\circ }}}_\alpha ) <0$, where $\chi$ is the Euler characteristic. Note that $\chi ({{\mathop {E}\limits ^{\circ }}}_\alpha )$ is equal to $2-2g(E_{\alpha })-v(\alpha )$, where $v(\alpha )$ is the number of intersection points of $E_{\alpha }$ with other components of the total transform of $\{fg=0\}$. Thus, the nodal components are all the rational ones that meet at least three other components of the total transform and all the non-rational irreducible components. We say that the irreducible component $E_\alpha$ (or its corresponding vertex in ${{{\mathcal {G}}}}(\rho ,f,g)$) is an end when $\chi ({\mathop {E}\limits ^{\circ }}_{\alpha })=1$. Obviously $E_\alpha$ is an end if and only if $E_{\alpha }$ is rational and meets exactly one other component of the exceptional divisor.

The neighbouring-set of $E_\alpha$ in X is the set constituted of $E_\alpha$ union with the irreducible components of the exceptional divisor and of the strict transform of $\{fg=0\}$ that intersect $E_\alpha$. We denote it $\textsf {Ng}(E_\alpha )$, thus $\textsf {Ng}(E_\alpha )= \bigcup _{E_\beta \cap E_\alpha \ne \emptyset } E_{\beta }$.

A chain of length r, $r\ge 3$, in E is a finite set of irreducible components $\{E_{\alpha _1}, \ldots , E_{\alpha _r}\}$ such that, for $2\le i \le r-1$:

$$\begin{aligned} \chi ({{\mathop {E_{\alpha _i}}\limits ^{\circ }}})=0 \ \text { and }\ \textsf {Ng}(E_{\alpha _i})= E_{\alpha _{i-1}}\cup E_{\alpha _i}\cup E_{\alpha _{i+1}} \; . \end{aligned}$$

Notice that $\bigcup _{i=1}^{r}E_{\alpha _i}$ is connected and the strict transform of $\{ fg=0\}$ does not intersect $E_{\alpha _2}\cup \ldots \cup E_{\alpha _{r-1}}$.

A cycle of length r, $r\ge 3$, in E is a chain such that $\textsf {Ng}(E_{\alpha _r})= E_{\alpha _{r-1}}\cup E_{\alpha _r}\cup E_{\alpha _1}$. A cycle of length 2 in E is a connected part of E constituted by two irreducible components $E_{\alpha _1}, E_{\alpha _2}$ such that $\chi ({{\mathop {E_{\alpha _2}}\limits ^{\circ }}})=0$ and $\textsf {Ng}(E_{\alpha _2})= E_{\alpha _{1}}\cup E_{\alpha _2}$.

Remark

Above definitions and terminology could also be stated in terms of the dual graph ${{{\mathcal {G}}}}(\rho ,f,g)$ and, in some sense, the names used are more natural there. Here we have chosen to do so in terms of the exceptional divisor for reasons of simplicity in the next proofs, however the dual graph provides an essential and synthetic guide to visualize all the elements involved in a simple way (see e.g. the examples in Sect. 5).

Nodal components are called rupture components in several papers as an analogy to the terminology used in the case of plane curves. This is the case in [13] which is an essential reference in Sect. 4.

Proposition 4

Let $\rho : (X,E)\rightarrow (Z,z)$ be a good resolution of (f, g). Let $E_\alpha$ be an irreducible component of the exceptional divisor such that the strict transform of $\{ fg=0\}$ does not intersect $E_\alpha$. Then there exists $E_\beta \subset \textsf {Ng}(E_\alpha )$ such that $q(E_\beta )>q(E_\alpha )$ if and only if there exists $E_\gamma \subset \textsf {Ng}(E_\alpha )$ such that $q(E_\gamma )<q(E_\alpha )$.

Moreover, if $\{ E_{\alpha _1}, \ldots , E_{\alpha _r}\}$, $r\ge 3$, is a chain, then one of the following facts is true:

• $q(E_{\alpha _i})<q(E_{\alpha _{i+1}})$ for $1\le i \le r-1$.

• $q(E_{\alpha _i})>q(E_{\alpha _{i+1}})$ for $1\le i \le r-1$.

• $q(E_{\alpha _i})$ is constant for $1\le i \le r$.

In particular, if $E_{\alpha _r}$ is an end, then $q(E_{\alpha _i})$ is constant for $1\le i \le r$ and if $\{ E_{\alpha _1}, \ldots , E_{\alpha _r}\}$ is a cycle, then $q(E_{\alpha _i})$ is constant for $1\le i \le r$.

The result previously stated is a direct generalization of Proposition 1 and Corollary 1 of [8] and the proof is similar. However, here the framework and the notations are slightly different. As Proposition 4 is also a key result in the remaining proofs of this paper, we will give an outline of the proof here.

Proof

By using Eq. (1) for the functions f and g with respect to the same divisor $E_{\alpha }$ we have:

$$\begin{aligned} \begin{aligned} \sum _{E_\eta \subset \textsf {Ng}(E_\alpha ), \eta \ne \alpha }\nu _\eta (f)(E_\eta \cdot E_\alpha )&= (-E_{\alpha }^2) \; \nu _\alpha (f)\\ \sum _{E_\eta \subset \textsf {Ng}(E_\alpha ), \eta \ne \alpha }\nu _\eta (g)(E_\eta \cdot E_\alpha )&= (-E_{\alpha }^2) \; \nu _\alpha (g) \; . \end{aligned} \end{aligned}$$

(2)

Let suppose that $q(E_\eta ) \ge q (E_\alpha )$ for each $E_\eta \subset \textsf {Ng}(E_\alpha )$. This condition is equivalent to: $(E_\eta \cdot E_\alpha ) \nu _\eta (f)\nu _\alpha (g)\ge (E_\eta \cdot E_\alpha ) \nu _\alpha (f)\nu _\eta (g)\, .$ As $q (E_\beta ) > q (E_\alpha )$, we obtain:

$$\begin{aligned} \nu _\alpha (g)\displaystyle \sum _{E_\eta \subset \textsf {Ng}(E_\alpha ), \eta \ne \alpha }(E_\eta \cdot E_\alpha ) \nu _\eta (f)>\nu _\alpha (f)\sum _{E_\eta \subset \textsf {Ng}(E_\alpha ), \eta \ne \alpha }(E_\eta \cdot E_\alpha ) \nu _\eta (g)\,. \end{aligned}$$

However, by using Eq. (2), one can see that both sides of the above inequality are equal to $(-E_{\alpha }^2) \nu _{\alpha }(f)\nu _{\alpha }(g)$ and thus, we reach a contradiction. $\square$

The other statements of the Proposition are easy consequences of this result.

3.2 Proof of Theorems 1 and 2

Let $\rho : (Y,E)\rightarrow (Z,z)$ be the minimal good resolution of the pencil $\Lambda$, ${{\widehat{h}}}= h\circ \rho$. Proposition 3, together with the above Statement, gives:

Corollary 2

Let $w, w'\in {{\mathbb {C}}}{{\mathbb {P}}}^1$ be generic values of $\Lambda$. Then, the fibers $\Phi _{w}$ and $\Phi _{w'}$ are topologically equivalent.

Thus, in order to finish the proof of Theorem 1, it only remains to show that a special fiber $\Phi _{e}$ is not topologically equivalent to a generic one.

Let $\Delta$ be an element of $SZ(\Lambda )$ and $e = {{\widehat{h}}} (\Delta )$. We denote by $\Phi _e$ the fiber of $\Lambda$ associated to e and by $\widetilde{\Phi _{e}}$ its strict transform by $\rho$. The remaining part of Theorems 1 and 2 are direct consequences of the three following lemmas, one for each possibility of $\Delta$. Unless otherwise specified, w denotes a generic value of $\Lambda$.

Lemma 1

If e is the special value of $\Lambda$ associated to a connected component $\Delta$ of $\overline{E\backslash {{{\mathcal {D}}}}}$, then the strict transform of $\Phi _e$ by $\rho$ intersects $\Delta$.

Proof

Let assume that $\widetilde{\Phi _{e}}\cap \Delta = \emptyset$. Notice that, if we enlarge $\rho$ (by additional blow-ups) in order to have a good resolution of $\Phi _e$ and $\Lambda$, then the connected set $\Delta$ remains unchanged. So, we can assume that $\rho$ is also a resolution of $\Phi _e$.

For any component $E_\alpha \subset \Delta$, we have $q^e_w(E_\alpha )>1$. Let $E_\beta$ be an irreducible component of $\Delta$ such that $q^e_w (E_{\beta })\ge q^e_w (E_\alpha )$ for any $E_\alpha \subset \Delta$ and let $\Delta '$ be the maximal connected subset of E such that $E_\beta \subset \Delta '$ and $(q^e_{w})_{\mid \Delta '}$ is constant and equal to $q^e_w (E_{\beta })$. Notice that $E_{\beta }\subset \Delta '\subset \Delta$ because $q^e_w(E_{\alpha })=1$ for any $E_{\alpha }$ such that $E_{\alpha }\cap \Delta \ne \emptyset$ and $E_{\alpha }\not \subset \Delta$ (in fact such an $E_{\alpha }$ is a dicritical divisor). Now, let $E_{\gamma }\subset \Delta '$ be such that $\textsf {Ng}(E_{\gamma })\not \subset \Delta '$. Let $E_{\alpha }\subset \textsf {Ng}(E_{\gamma })$ be such that $E_{\alpha }\not \subset \Delta '$ then, one has $q^e_w(E_{\beta })> q^e_w(E_\alpha ) >1$ if $E_{\alpha }\subset \Delta$ and $q^e_w(E_{\beta }) > q^e_w(E_\alpha ) = 1$ otherwise. However, being $\Delta '\subset \Delta$, this contradicts Proposition 4 for the irreducible component $E_{\gamma }$.

As a consequence $\widetilde{\Phi _e}\cap \Delta \ne \emptyset$ and so $\Phi _e$ can not be topologically equivalent to $\Phi _w$ for a generic value w. $\square$

Lemma 2

If e is the special value of $\Lambda$ associated to a smooth point P of the dicritical component $D\subset {{{\mathcal {D}}}}$ which is a critical point of ${{\widehat{h}}}$, then the strict transform $\widetilde{\Phi _e}$ of $\Phi _e$ by $\rho$ intersects D at P. Moreover, $\widetilde{\Phi _e}$ cannot be smooth and transversal to D at P.

Proof

Blowing-up at P we create a divisor $E_\alpha$. As ${{\widehat{h}}} (P)=e$, then P lies in the zero locus of $(\phi _e/\phi _{w})\circ \rho$ for any value $w\ne e$ and so we have $q^{e}_w (E_\alpha )>1$. Moreover, as D is a dicritical component, then $q^{e}_w (D)=1$. Now, if we assume that $P\notin \widetilde{\Phi _e}$ then one can use Proposition 4 for the new divisor $E_{\alpha }$ and reach a contradiction.

Assume that $\widetilde{\Phi _e}$ is smooth and transversal to D at the point P. In this case we can choose local coordinates $\{u,v\}$ on Y at P in such a way that $\widetilde{\Phi _{e}} = \{v=0\}$ and $D =\{u=0\}$ on a neighbourhood V of P. So, the function $\phi _e\circ \rho$ is $u^{a}v$ on V and, for a generic value w, $\phi _w\circ \rho$ is $u^{b}\eta (u,v)$ for a unit $\eta$. Note that $a = \nu _{D}(\phi _e) = \nu _{D}(\phi _w)=b$, being D dicritical, and so the expression of ${\widehat{h}}$ at P is $v\eta (u,v)^{-1}$. Now, the restriction of $(\phi _e/\phi _w)\circ \rho$ to D is given locally at P as the map $v\mapsto v$. Thus, the point P is not a critical (ramified) point of ${\widehat{h}}_{\mid D}: D\rightarrow {{\mathbb {C}}}{{\mathbb {P}}}^1$.

As a consequence, $\widetilde{\Phi _e}$ is either not smooth or tangent to D at P, in particular $\Phi _e$ can not be topologically equivalent to $\Phi _w$ for a generic value w. $\square$

Lemma 3

If e is the special value of $\Lambda$ associated to an intersection point P between two irreducible components of ${{{\mathcal {D}}}}$, then the strict transform of $\Phi _e$ by $\rho$ intersects ${{\mathcal {D}}}$ at P.

Proof

Let $P =E_{\alpha _1}\cap E_{\alpha _2}$ such that $E_{\alpha _1}$ and $E_{\alpha _2}$ are dicritical components. Let us assume that $P\notin \widetilde{\Phi _{e}}$. Blowing-up at P we create a divisor $E_\alpha$ satisfying $\{E_{\alpha _1}\cup E_\alpha \cup E_{\alpha _2}\} = \textsf {Ng}(E_\alpha )$. As $q^{e}_w(E_{\alpha _1}) =q^{e}_w(E_{\alpha _2}) =1$ and $q^{e}_w(E_{\alpha })>1$, we reach a contradiction with Proposition 4.

As a consequence, ${\Phi _e}$ is not resolved by $\rho$ and so could not be topologically equivalent to a generic fiber $\Phi _w$. $\square$

4 Behaviour of the critical locus

Let $\pi =(f,g): (Z,z)\rightarrow ({{\mathbb {C}}}^2,0)$ be a finite complex analytic morphism. Following Teissier [19], the critical locus of $\pi$ is the analytic subspace defined by the zeroth Fitting ideal $F_0(\Omega _{\pi })$ of the module $\Omega _{\pi }$ of relative differentials. The critical locus can have embedded components, however, we are only interested in the components of dimension one. We denote by $C(\pi )$ the divisorial part of the critical set with its non-reduced structure (i.e. with its multiplicity) and we refer to $C(\pi )$ as the critical locus of $\pi$. Note that, out of the singular point $z\in Z$, $C(\pi )$ is defined by the vanishing of the jacobian determinant J(f, g). For a different pair of functions $f', g'\in \Lambda$, one has $J(f',g') = a J(f,g)$ for some $a\in {{\mathbb {C}}}^*$ (a is just the determinant of the linear change between both basis). As a consequence, $C(\pi )$ depends on $\Lambda$ and not on the pair of functions of $\Lambda$ fixed to define the corresponding finite morphism. If we denote by $\Gamma _k$, (resp. $n_k$), $k=1,\ldots , \ell$, the irreducible components (branches) of $C(\pi )$ (resp. their multiplicity) then $C(\pi )$ is the local (Weil) divisor $C(\pi )= \sum _{k=1}^{\ell }n_k \Gamma _k$.

Before proving Theorems 3 and 4, let first recall two results from [13, 14].

Let $(\phi _w,\phi _{w'})$ be any pair of germs of the pencil $\Lambda$, let $\rho ': (Y',E') \rightarrow (Z,z)$ be the minimal good resolution of $(\phi _w,\phi _{w'})$, and let $\Gamma (w,w'):= \bigcup _{k\in K} \Gamma _k, K\subset \{1, \ldots , \ell \}$ denote the curve consisting of the irreducible components of $C(\pi )$ which are not components of $\{\phi _w\phi _{w'}=0\}$. Let $Z_r\subset E'$ be the set of points $P\in E'$ such that, for any irreducible exceptional component $E_{\alpha }$ with $P\in E_{\alpha }$, one has $q^{w}_{w'}(E_{\alpha })=r$. The set $Z_r$ is called the r-zone of $E'$. A connected component of $Z_r$ which contains at least one nodal component is called an r-nodal zone. Then from [13] we have:

Theorem A

The set $\left\{ \displaystyle \frac{I_z(\phi _{w}, \Gamma _k)}{I_z(\phi _{w'}, \Gamma _k)}, k\in K\right\}$ is equal to the set of Hironaka quotients of $(\phi _w,\phi _{w'})$ on the nodal components of $E'$.

In [14] a repartition in bunches of the branches of $\Gamma (w,w')$ is given as follows:

Theorem B

The intersection of the strict transform of $\Gamma (w,w')$ with a connected component of $Z_r$ is not empty if and only if it is an r-nodal zone. Moreover, if $\Gamma$ is an irreducible component of $\Gamma (w,w')$ whose strict transform intersects an r-nodal zone, then $\displaystyle \frac{I_z(\phi _{w}, \Gamma )}{I_z(\phi _{w'}, \Gamma )}=r$.

Remark

Notice that a direct consequence of these theorems is the following: let P be a point that does not belong to a $Z_r$ for any r, then the strict transform of ${\Gamma (w,w')}$ by $\rho '$ does not go through P. Indeed, in this case P is the intersection point between two irreducible components of $E'$ with distinct Hironaka quotient. Let suppose that there exists an irreducible component $\Gamma$ of $\Gamma (w,w')$, such that its strict transform intersects the exceptional diviosr at P. Then, blowing-up at P til the strict transform of $\Gamma$ intersects the exceptional divisor at a smooth point, we obtain, either $\displaystyle \frac{I_z(\phi _{w}, \Gamma )}{I_z(\phi _{w'}, \Gamma )}$ does not belong to the set of Hironaka quotients of $(\phi _w,\phi _{w'})$ on the nodal components of $E'$, which contradicts Theroem A, or $\displaystyle \frac{I_z(\phi _{w}, \Gamma )}{I_z(\phi _{w'}, \Gamma )}$ is equal to a Hironaka quotient of $(\phi _w,\phi _{w'})$ but the strict transform of $\Gamma$ intersects the exceptional divisor in a zone which is not an r-nodal zone. This contradicts Theorem B.

The following Lemma treats the case of irreducible components of the critical locus which are also components of a fiber.

Lemma 4

Let $\Phi _e=\sum _{i=1}^t r_i\xi _i$, where $\xi _1,\ldots , \xi _t$ are the irreducible components of the fiber $\Phi _e$, $\xi _i\ne \xi _j$ if $i\ne j$. Then, for $i=1,\ldots , t$, one has $r_i>1$ if and only if $\xi _i$ is an irreducible component of $C(\pi )$.

Proof

Let $\xi$ be an irreducible component of a fiber $\Phi _e$. Let $w\in {{\mathbb {C}}}{{\mathbb {P}}}^1$ be a generic value and let $\rho ': (Y',E')\rightarrow (Z,z)$ be the minimal good resolution of $(\phi _e ,\phi _w)$. Let $\widetilde{\xi }$ be the strict transform of $\xi$ by $\rho '$ and let P be the intersection point of ${{\widetilde{\xi }}}$ with the exceptional divisor $E'$, $P=\widetilde{\xi }\cap E_{\alpha }= \widetilde{\xi }\cap E'$. We can choose a local system of coordinates (u, v) in a neighbourhood $U\subset Y'$ of $P=(0,0)$ such that $u=0$ is an equation of $E_\alpha$, $v=0$ is an equation of $\widetilde{\xi }$ and the equation of the total transform $\overline{\Phi _e}$ of $\Phi _{e}$ at P is $u^a v^k$, where $a = \nu _{\alpha }(\phi _{e})$ and k is the multiplicity of the branch $\xi$ in $\Phi _{e}$. On the other hand, the equation of $\overline{\Phi _w}$ at P is $u^{b}\eta (u,v)$, with $b=\nu _{\alpha }(\phi _w)$ and $\eta (u,v)$ being a unit. So, the expression of $(\phi _e/\phi _w)\circ \rho$ at $P\in U$ is $u^{a-b}v^k (\eta (u,v))^{-1}$.

Let first suppose that $\xi$ belongs to $C(\pi )$. Let Q be a point of ${{\widetilde{\xi }}} \backslash \{ P\}$, say Q has local coordinates $(u_0, 0)$. The restriction of ${\widehat{h}}$ to a small disc $D(u_0,0)$ centered at Q in $u=u_0$ is $v^k \eta _0(u_0,v)$ with $\eta _0(u_0,v)$ a unit and $k>1$ because $\xi$ lies in the ramification locus. So, as k is the multiplicity of $\xi$ in ${\Phi _{e}}$, $\xi$ is non-reduced.

Conversely, if $\xi$ is an irreducible component of a fiber $\Phi _e$ which is not reduced, the multiplicity k of $\xi$ in ${\Phi _{e}}$ satisfies $k>1$. Moreover the local equation of ${\widehat{h}}$ on any small disc D(t, 0) centered at any point of local coordinates (t, 0) in U is $v^k \eta (t,v)$ with $\eta (t,v)$ a unit. As $k>1$, each point (t, 0) is a ramification point and so $\widetilde{\xi }$ lies in the ramification locus. Hence, $\xi$ is an irreducible component of $C(\pi )$. $\square$

4.1 Proof of Theorem 3 for singular points of ${\mathcal {D}}$ and critical points of the restriction of ${\widehat{h}}$ to ${\mathcal {D}}$

Hereafter, let $\rho : (Y,E)\rightarrow (Z,z)$ denote the minimal good resolution of $\Lambda$ and ${{{\mathcal {D}}}}$ the dicritical locus of E.

Proposition 5

Let $P\in {{{\mathcal {D}}}}$ be such that $P\notin \overline{E\backslash {{{\mathcal {D}}}}}$. Then, P is a singular point of ${{\mathcal {D}}}$ or a critical point of $\widehat{h} _{| {{{\mathcal {D}}}}}$ if and only if there exists an irreducible component $\Gamma$ of $C(\pi )$ whose strict transform intersects ${{\mathcal {D}}}$ at P. Moreover, if $\widehat{h}(P)=e$, then $I_z(\phi _e, \Gamma )>I_z(\phi _w,\Gamma )$ for any $w\in {{\mathbb {C}}}{{\mathbb {P}}}^1$, $w\ne e$.

Proof

Let assume that there exists an irreducible component $\Gamma$ of $C(\pi )$ whose strict transform intersects ${{\mathcal {D}}}$ at P and let $e= \widehat{h} (P)$. If $\Gamma$ is a branch of $\Phi _e$ then, by the above Lemma, it must be a multiple component and, as a consequence, the point P is a critical point of $\widehat{h}|{{{\mathcal {D}}}}$.

So, let consider the case in which $\Gamma$ is not a branch of $\Phi _{e}$ and assume that P is not a singular point of ${{{\mathcal {D}}}}$, i.e. P is a smooth point of ${{{\mathcal {D}}}}$ in the exceptional divisor E. Let D denote the irreducible component of ${{{\mathcal {D}}}}$ such that $P\in D$.

If the strict transform $\widetilde{\Phi }_e$ of $\Phi _e$ at P has normal crossings with ${{{\mathcal {D}}}}$, then there exists an irreducible branch $\xi$ of $\Phi _e$ such that its strict transform $\widetilde{\xi }$ coincides with $(\widetilde{\Phi }_e)_{P}$, i.e. $\widetilde{\xi }$ is smooth, transversal to D and $\xi$ is not a multiple branch of $\Phi _e$ by Lemma 4. By Theorem B there exists a r-nodal zone R in the minimal good resolution of $(\phi _e,\phi _w)$ (here w is assumed to be a generic value) such that the strict transform of $\Gamma$ intersects R and moreover, $I_z(\phi _e,\Gamma )/I_z(\phi _w,\Gamma ) = r$ with $r>1$, because $P\in \widetilde{\Gamma }\cap \widetilde{\Phi _e}$. Taking into account that $\widetilde{\Phi _e}$ is smooth and transversal to the dicritical divisor D, then one has that $P = \widetilde{\Gamma }\cap E \subset D\subset R$ and so, by Theorem A,

$$\begin{aligned} \frac{I_z(\phi _e,\Gamma )}{I_z(\phi _w,\Gamma )} = q^{e}_{w}(D) = \frac{\nu _{D}(\phi _e)}{\nu _{D}(\phi _w)}\;. \end{aligned}$$

However, this is impossible because the last quotient is equal to 1, being D dicritical. Thus, as a consequence, $(\widetilde{\Phi _e})_P$ must be singular or tangent to D. In both cases P is a critical point of $\widehat{h}|_{{{\mathcal {D}}}}$ (i.e. $\phi _e$ is a special function of $\Lambda$). $\square$

Conversely, let P be a singular point of ${{\mathcal {D}}}$ or a smooth point of ${{\mathcal {D}}}$ which is a critical point of $\widehat{h}_{| {{{\mathcal {D}}}}}$ and let $e={{\widehat{h}}}(P)$, then from Theorem 2, $\Phi _e$ is a special fiber of $\Lambda$. If the irreducible component of ${{\widetilde{\Phi }}} _{e}$ that intersects ${{\mathcal {D}}}$ at P is non-reduced, then from Lemma 4 we have finished. Thus, we assume that ${{\widetilde{\Phi }}}_e$ is reduced at P.

Before all, note that $\widetilde{\Phi }_e$ has not normal crossings with E at P: if P is a singular point of ${{\mathcal {D}}}$, then there are at least three components of the total transform intersecting at P. Otherwise, if P is smooth on ${{{\mathcal {D}}}}$ then $\widetilde{\Phi }_e$ is either singular or tangent to ${{\mathcal {D}}}$.

Let $w,w'\in {{\mathbb {C}}}{{\mathbb {P}}}^1$ be generic values and let $\rho ': (Y',E')\rightarrow (Z,z)$ be the minimal good resolution of $\{\phi _{w}\phi _{w'}\phi _{e}=0\}$. Note that $\rho ' = \rho \circ \sigma$, where $\sigma$ is a sequence of blowing-ups of points, each of them produces a new irreducible rational exceptional component. In particular, $\Delta = \sigma ^{-1}(P)\subset E'$ is a connected exceptional part and must contain a nodal component $E_{\alpha }\subset E'$. Notice that no component of $\Delta$ is contracted in the minimal good resolution $\rho '': (Y'',E'')\rightarrow (Z,z)$ of the pair $(\phi _e, \phi _w)$; i.e. $\Delta \subset E''$. As a consequence, $E_{\alpha }\subset E''$ is also a nodal component of $E''$. Let R be the corresponding nodal zone in $E''$ which contains $E_{\alpha }$. Note that, for each $E_{\beta }\subset R\subset \Delta$ one has $r= q^e_{w}(E_{\beta }) = \nu _{\beta }(\phi _e)/\nu _{\beta }(\phi _w)>1$.

Now, Theorem B implies that there exists a branch $\Gamma$ of $C(\pi )$ such that its strict transform by $\rho ''$ intersects $\Delta$ and also that

$$\begin{aligned} \frac{I_{z}(\phi _e,\Gamma )}{I_{z}(\phi _w,\Gamma )} = \frac{\nu _{\alpha }(\phi _e)}{\nu _{\alpha }(\phi _w)} = r >1. \end{aligned}$$

Taking into account that $R\subset \Delta$ and $\sigma (\Delta )=P$, one has that the strict transform of $\Gamma$ by $\rho$ intersects E at the point P and moreover $I_{z}(\phi _e,\Gamma ) > I_{z}(\phi _w,\Gamma )$. Note that the above inequality is true for any irreducible component $\Gamma$ of $C(\pi )$ whose strict transform by $\rho$ intersects ${{\mathcal {D}}}$ at P. Thus, the special fiber $\phi _e$ is the unique fiber with the condition $I_{z}(\phi _e,\Gamma ) > \min _{w}I_{z}(\phi _w,\Gamma )$.

Remark

Notice that, if $\{P\}\subset {{{\mathcal {D}}}}$ is an isolated special zone, $\widehat{h}(P)=e$ and $\Gamma$ is the corresponding irreducible component of $C(\pi )$ then, for any fibers $\Phi _a$ and $\Phi _{a'}$ different from $\Phi _{e}$ we have $I_{z}(\phi _{a},\Gamma )=I_{z}(\phi _{a'}, \Gamma )$ (see Proposition 1).

4.2 Proof of Theorem 3 for the connected components of $\overline{E\backslash {{{\mathcal {D}}}}}$

Let recall that $\rho : (Y,E)\rightarrow (Z,z)$ is the minimal good resolution of $\Lambda$ and ${{{\mathcal {D}}}}$ the dicritical locus of E. Let $\Delta$ be a connected component of $\overline{E\backslash {{{\mathcal {D}}}}}$ such that $(h\circ \rho )(\Delta ) = e$. Let $w, w'$ be generic values of $\Lambda$ and let denote by $\rho ' : (Y',E')\rightarrow (Z,z)$ the minimal good resolution of $\{\phi _w \phi _{w'} \phi _e=0\}$. Let $\tau : (Y',E')\rightarrow (Y,E)$ denote the composition of blowing-ups of points which produces $Y'$ from (Y, E):

$$\begin{aligned} (Y',E') \overset{\tau }{\rightarrow } (Y,E) \overset{\rho }{\rightarrow } (Z,z) \; . \end{aligned}$$

Lastly, let $\Delta '$ be the pull-back of $\Delta$ by $\tau$. Note that $\Delta '$ is a connected component of $\overline{E'\backslash {{{\mathcal {D}}}'}}$ because the dicritical locus ${{{\mathcal {D}}}'}$ on $E'$ is just the strict transform of ${{{\mathcal {D}}}}$ by $\tau$. We distinguish two cases, according to the existence of a nodal component $E_{\alpha }\subset \Delta '$ (with respect to $\phi _w$ and $\phi _{e}$).

Case 1 There exist a nodal component $E_{\alpha }$, with $E_{\alpha }\subset \Delta '$.

For each component $E_{\beta }\subset \Delta '$ one has $q_{w'}^{w}(E_{\beta })=1$ and $q^e_{w}(E_{\beta })>1$. Let R be the nodal zone of $E'$ such that $E_\alpha \subset R$. Then $R\subset \Delta '$ because $q^e_{w}$ is constant and $> 1$ on R and $q^{e}_{w}(D)=1$ for any dicritical divisor, in particular for any dicritical D such that $D\cap \Delta '\ne \emptyset$.

Now, from Theorem B, there exists a branch $\Gamma$ of the critical locus $C(\pi )$ whose strict transform by $\rho '$, denoted by ${\widetilde{\Gamma }}$, intersects R. As a consequence, the strict transform of $\Gamma$ by $\rho$, $\tau ({\widetilde{\Gamma }})$, intersects $\Delta$. Again Theorem B implies that $q^{e}_{w}(E_\alpha ) = I_{z}(\phi _{e}, \Gamma )/I_{z}(\phi _{w}, \Gamma )$ and hence, the special value e is the unique one such that $I_{z}(\phi _{e}, \Gamma ) > I_{z}(\phi _{w'}, \Gamma )$ for any generic value $w'$.

Case 2 There are no nodal components in $\Delta '$.

In this case one has that $\Delta ' = E_{\alpha _1}\cup \ldots \cup E_{\alpha _r}$ and there exists a dicritical component $D \subset { {{\mathcal {D}}}'}$ such that $\{D = E_{\alpha _0}, E_{\alpha _1}, \ldots , E_{\alpha _r}\}$ is a chain and $\chi ({\mathop {E}\limits ^{\circ }}_{\alpha _r})\ge 0$. Now, note that the strict transform $\widetilde{\Phi _e}$ of $\Phi _{e}$ by $\rho '$ intersects $\Delta '$ (see Theorem 2). Therefore, the only way to avoid the existence of a nodal component with respect to $\phi _w \phi _e$ is if $E_{\alpha _r}$ is an end (i.e. it is rational and is connected only with the previous one $E_{\alpha _{r-1}}$) and $\widetilde{\Phi _{e}}$ intersects $E_{\alpha _r}$. Moreover, $\widetilde{\Phi _{e}}$ with its reduced structure must be smooth and transversal to $E_{\alpha _r}$. It means that the minimal good resolution of $\Lambda$ is also a resolution of the reduced irreducible component $\xi _{e}$ of $\Phi _{e}$ whose strict transform meets $\Delta$ at $E_{\alpha _r}$. Indeed, otherwise in order to resolve $\xi _{e}$, one has to blow-up at $\xi _{e}\cap E_{\alpha _r}$ and by this process a nodal component is produced.

Fig. 1

Graph in Case 2

Lemma 5

Let $v_{0},\,\ldots ,v_{r}$, $e_{1},\,\ldots ,e_{r}$ be sequences of integers such that $v_{i-1} = e_i v_i -v_{i+1}$ for $i=1,\ldots , r-1$. Let $q_{0},\,\ldots ,q_{r-1}\in {{\mathbb {Z}}}$ be defined recursively as $q_0=1$, $q_1=e_1$ and, for $i\ge 2$, $q_{i}=e_iq_{i-1}-q_{i-2}$. Then, for $i\ge 1$ one has $\gcd (q_i, q_{i-1})=1$ and $v_0=q_{i} v_{i}-q_{i-1}v_{i+1}$.

Proof

Obviously $\gcd (q_0,q_1)=1$ and from the definition of $q_{i}$, if $\gcd (q_{i-1},q_{i-2})=1$ then $\gcd (q_{i-1},q_i)=1$. The equality $v_0=q_{i}v_i-q_{i-1}v_{i+1}$ is obvious for $i=1$ and, by induction, using the equality $v_{i-1} = e_i v_i -v_{i+1}$ in the inductive hypothesis $v_0=q_{i-1}v_{i-1}-q_{i-2}v_{i}$, one has:

$$\begin{aligned} v_0= q_{i-1}v_{i-1}-q_{i-2}v_{i} = q_{i-1}(e_i v_i -v_{i+1}) -q_{i-2}v_{i} = q_{i}v_{i}-q_{i-1}v_{i+1}\; . \end{aligned}$$

$\square$

Now, the proof of the case 2 is a consequence of the following:

Proposition 6

The irreducible curve $\xi _{e}$ is a branch of $\Phi _{e}$ with multiplicity bigger than 1. As a consequence $\xi _e$ is also a branch of $C(\pi )$ and thus, $C(\pi )$ intersects $\Delta$.

Proof

Recall that w is a generic element of $\Lambda$. For the sake of simplicity let denote $v_i=\nu _{\alpha _i}(\phi _w)$ and $e_i = - E_{\alpha _i}^2$ for $i=0,\ldots ,r$. Then, using the formula

$$\begin{aligned} \left( (\widetilde{\phi _{w}}) + \sum _{\alpha \in G(\rho ')} \nu _{\alpha }(\phi _w) E_{\alpha } \right) \cdot E_{\alpha _i} = 0 \end{aligned}$$

(3)

for $i=1,\ldots , r$ one has that

$$\begin{aligned} \begin{aligned} v_0&= e_1 v_1 - v_2\\ v_1&= e_2 v_2 - v_3 \\&\cdots \\ v_{r-2}&= e_{r-1} v_{r-1} - v_r\\ v_{r-1}&= e_r v_r \end{aligned} \end{aligned}$$

(4)

By Lemma 5 one has that $v_0 = q_rv_r$. Moreover, taking into account that $e_i = - E_{\alpha _i}^2 \ge 2$, one can easily prove that $q_r> q_{r-1}> \cdots> q_1 > q_0 = 1$.

Now, let consider the special fiber $\Phi _{e}$ and write $v'_i=\nu _{\alpha _i}(\phi _{e})$ for $i=0,\ldots , r$. Equations (3) applied for $\phi _e$ (instead of $\phi _w$) gives a sequence of equalities $v'_{i-1} = e_i v'_i -v'_{i+1}$, for $i=1,\ldots , r-1$ (like in (4) above with $v'_i$ instead of $v_i$) together with the last one:

$$\begin{aligned} v'_{r-1} = e_r v'_{r} - (\widetilde{\phi _{e}})\cdot E_{\alpha _r} =e_r v'_{r} - k \; . \end{aligned}$$

Lemma 5 implies that $v'_0 = q_r v'_r - q_{r-1} k$. As $E_{\alpha _0}=D$ is a dicritical divisor, one has that $v'_0 = \nu _{\alpha _0}(\phi _{e}) = \nu _{\alpha _0}(\phi _{w}) = v_0$, i.e.

$$\begin{aligned} q_r v_r = q_r v'_r -q_{r-1} k \; . \end{aligned}$$

By Lemma 5 again, $\gcd (q_r,q_{r-1})=1$ and so $q_r$ divides k. In particular $k = (\widetilde{\phi _{e}})\cdot E_{\sigma }>1$ and the irreducible germ $\xi _e$ appears repeated k times in $\Phi _{e}$. $\square$

4.3 Special fibers and critical locus

Let $C(\pi )= \sum _{i=1}^{\ell }n_i \Gamma _i$ be the decomposition of the critical locus in irreducible components. For each $i\in \{1,\ldots , \ell \}$ the intersection multiplicity $I_z(\phi , \Gamma _i)$ is constant, except for the unique special value $\varepsilon (\Gamma _i) (=\varepsilon (i))$ such that $I_z(\phi _{\varepsilon (i)}, \Gamma _i) > I_z(\phi , \Gamma _i)$, for $\phi \ne \phi _{\varepsilon (i)}$. So, as in [8], one has a surjective map $\varepsilon : {{{\mathcal {B}}}}(C(\pi ))\rightarrow Sp(\Lambda )$ from the set of branches of the critical locus to the set of special values of $\Lambda$.

If $w\in {{\mathbb {C}}}{{\mathbb {P}}}^1$ is a generic value one has that

$$\begin{aligned} I_{z}(\phi _w, C(\pi )) = \sum _{i=1}^{\ell }n_i I_z(\phi _w,\Gamma _i) = \min \{ I_z(\phi , C(\pi )) \, , \, \phi \in \Lambda \} \end{aligned}$$

and, on the other hand, for a special value $e\in {{\mathbb {C}}}{{\mathbb {P}}}^1$ there is

$$\begin{aligned} I_{z}(\phi _e, C(\pi )) = \sum _{i=1}^{\ell }n_i I_z(\phi _e,\Gamma _i) > \sum _{i=1}^{\ell }n_i I_z(\phi _w,\Gamma _i) = \min \{ I_z(\phi , C(\pi )) \, , \, \phi \in \Lambda \}\; . \end{aligned}$$

As a consequence, one has the following result which gives the equivalence of items 1 and 2 in Theorem 4.

Corollary 3

$\Phi _e$ is a special fiber of $\Lambda$ if and only if

$$\begin{aligned} I_z (\phi _e, C(\pi )) > \min \left\{ I_z(\phi , C(\pi ) ), \phi \in \Lambda \right\} \; . \end{aligned}$$

Remark

As in [8], the map $\varepsilon : {{{\mathcal {B}}}}(C(\pi ))\rightarrow Sp(\Lambda )$, defined above, can be factorized through the set of special zones $SZ(\Lambda )$ as $\varepsilon = {\widehat{h}}\circ \psi$:

$$\begin{aligned} {{{\mathcal {B}}}}(C(\pi ))\overset{\psi }{\rightarrow } SZ(\Lambda ) \overset{{\widehat{h}}}{\rightarrow } Sp(\Lambda )\; . \end{aligned}$$

The map $\psi$ associates to the branch $\Gamma$ the special zone $\Delta$ such that the strict transform of $\Gamma$ in the minimal good resolution intersects $\Delta$. Obviously one can decompose the branches of $C(\pi )$ in bunches by means of $\psi$.

Let $\rho ' : (Y',E')\rightarrow (Z,z)$ be a good resolution of all the fibers of $\Lambda$ (i.e. a good resolution of the product of all the special fibers and a pair of generic ones). Let e (resp. w) be a special value (resp. a generic one). An 1-nodal zone for the pair $(\phi _e, \phi _w)$ can be decomposed in some different zones when we use different special values. This fact can be used to determine a finer decomposition in bunches of the branches of the critical locus $C(\pi )$. To do that, one can use the determination of all the nodal zones in $E'$ with respect to all the pairs $(\phi _e,\phi _w)$, when e varies in the set of special values and w is a fixed generic value.

5 Examples

As seen in Sect. 3.1, to the minimal good resolution $\rho$ of the pencil $\Lambda$, one can associate its dual graph ${{{\mathcal {G}}}}(\rho )$. The following examples illustrate Theorems 1, 2 and 3 in terms of the dual graph. To construct ${{{\mathcal {G}}}}(\rho )$, we follow the method of Laufer described in [10, 12] and also [13]. It consists, first, in establishing the graph of the minimal embedded resolution of the discriminant curve, which is the image by $\pi$ of the critical locus $C(\pi )$ of $\pi$. This graph is constructed as follows. To each irreducible component $E'_i$ of the exceptional divisor of the minimal resolution of the discriminant curve, we associate a vertex weighted by $(i, (E'_i)^2, v(E_i'))$, where $(E'_i)^2$ is the self-intersection of $E_i'$ and $v(E_i')$ is the valuation of the discriminant function along $E_i'$. An intersection point between two irreducibe components of the exceptional divisor is represented by an edge linking the associated vertices. Second, from the dual graph of the discriminant curve, we deduce the graph of the minimal good resolution of (Z, z) and so the one of $\rho$, using in particular Propositions 2.6.1, 2.7.1 (examples 2 and 3) 3.6.1 and 3.7.1 (example 1) of [12]. As in Fig. 1 of Sect. 4.2, we use a different mark for the vertices representing dicritical divisors.

5.1 Example 1

Let (Z, z) be defined by $z^3 =h(x,y)$ with $h(x,y)= (y+x^2)(y-x^2)(y+2x^2)(x+y^2)(x-y^2)(x+2y^2)$ and let $\pi$ be the projection on the (x, y)-plane. In this way $(u,v)=(x,y)$, $f=u\circ \pi = x$ and $g=v\circ \pi =y$.

The discriminant curve of $\pi$ is the curve $h(u,v)=0$. The dual graph of its minimal embedded resolution is represented in Fig. 2.

Fig. 2

Graph of the minimal resolution of the discriminant of $\pi$

From Proposition 3.6.1 and 3.7.1 of [12] we deduce the graph of the minimal good resolution of (Z, z) (see Fig. 3).

Fig. 3

The graph of the minimal good resolution of (Z, z)

As the minimal embedded resolution of the discriminant curve $h(u,v)=0$ of $\pi$ is also the minimal good resolution of the product $uv(\lambda u+\mu v)h(u,v)=0$, for $(\lambda :\mu )\in {{\mathbb {C}}}{{\mathbb {P}}}^1\backslash \{ (1:0), (0:1)\}$, from Propositions 3.6.1 and 3.7.1 of [12] we can deduce the dual graph of the minimal good resolution of $\Lambda$ (Fig. 4), the one of (f, g) and as a consequence the one of the minimal good resolution of $(\phi _w\phi _{w'}fg)^{-1}(0)$ where w and $w'$ are generic values of $\Lambda$ (Fig. 5). Notice that the minimal good resolution of $\Lambda$ is also the minimal good resolution of (f, g).

Fig. 4

The graph of the minimal good resolution of $\Lambda$

The dicritical components of E are $E_0^1, E_0^2, E_0^3$. We have $SZ(\Lambda )=\{\Delta _1,\Delta _2\}$ with $\Delta _1= \{E_1\}$ and $\Delta _2=\{ E_2\}$. The map $(f/g)\circ \rho$ has no critical point on ${{\mathcal {D}}}$ and ${{\mathcal {D}}}$ has no singular point neither. The special fiber associated to $\Delta _1$ is $\{ f=0\}$ and the one associated to $\Delta _2$ is $\{ g=0\}$. We conclude that $\Lambda$ admits two special elements f and g; the special value associated to $\Delta _1$ is (0 : 1) and the one associated to $\Delta _2$ is (1 : 0). The Hironaka quotients are $q(E_1)=2$ and $q(E_2)=1/2$.

Moreover, using the minimal resolution of the discriminant curve (see Fig. 2), we deduce that, for each $\Delta _i$, there exists three irreducible components of the reduced critical locus of $\pi$ whose strict transform intersects $\Delta _i$.

Fig. 5

Minimal good resolution of (f, g)

5.2 Example 2

Let (Z, z) be the $D_6$ singularity defined by the equation $z^2 =y(x^2+y^4)$. The graph of its minimal resolution is shown in Fig. 6.

Fig. 6

The graph of the minimal good resolution of $D_6$

On this surface we make two examples for two different projections (pencils). Firstly, let $\pi =(f,g): (Z,z)\rightarrow ({{\mathbb {C}}}^2,0)$ be defined by $f(x,y,z)= u\circ \pi = x$ and $g(x,y,z)= v\circ \pi =y$. The discriminant curve of $\pi$ is the curve $v(u^2+v^4)=0$. Notice that this projection is not a generic one because the image of the curve $\{ g= 0\}$ is an irreducible component of the discriminant curve and the image of $\{ f=0\}$ is tangent to the discriminant curve.

Furthermore, using the minimal good resolution of the discriminant curve (Fig. 7) and Proposition 2.6.1 and 2.7.1 of [12], we obtain that the minimal good resolution of $\Lambda$ is the one of (Z, z) and there exists a unique dicritical component $E_1$: the divisor with weight $(1,-2,0)$. Thus, $\Delta _0=\{ E_0\}$ and $\Delta _1= E_2\cup E_3\cup E_4\cup E_5$ (see Fig. 8 for the notations) are two special zones. Moreover $C(\pi )$ intersects the exceptional divisor at $E_0,E_4,E_5$ and so $SZ(\Lambda )=\{\Delta _0 ,\Delta _1\}$. The Hironaka quotients corresponding to each vertex are: $q(E_0)=1/2$, $q(E_1)=1$, $q(E_2)=3/2$ and $q(E_3)=q(E_4)=q(E_5)=2$.

Fig. 7

Graph of the minimal resolution of the discriminant of $\pi$

Fig. 8

The graph of the minimal good resolution of $\Lambda$

The connected component $\Delta _0$ does not contain any nodal component and $\Delta _1$ has a nodal component with Hironaka quotient equal to 2. The special fiber associated to $\Delta _1$ is $\{ f=0\}$ whose strict transform meets $\Delta _1$ at $E_3$, and there are two irreducible components of $C(\pi )$ intersecting $\Delta _1$ at $E_4$ and $E_5$. The special fiber of $\Lambda$ associated to $\Delta _0$ is $\{g=0\}$ which is also a non reduced irreducible component of the critical locus and intersects $\Delta _0$ at $E_0$. The minimal good resolution of the pencil $\Lambda$ is also the minimal good resolution of (f, g), so the corresponding graph of the minimal good resolution of $fg=0$ is represented in Fig. 9.

Fig. 9

The graph of the minimal good resolution of (f, g)

Let us consider the morphism $\pi =(f,g): (Z,z)\rightarrow ({{\mathbb {C}}}^2,0)$ defined on $D_6$ by $f(x,y,z)= x+2iy^2=u$ and $g(x,y,z)= x^2+y^3=v$. In this case the minimal good resolution of (Z, z) coincides with the one of $\{fg=0\}$. It is not the case of the minimal good resolution of $\Lambda$ whose graph of resolution is represented in Fig. 10.

There exists a unique dicritical component represented by $E_*$ and exactly one special zone consisting of $E_0\cup E_1 \cup E_2\cup E_3\cup E_4\cup E_5$. The Hironaka quotients of (f, g) corresponding to each vertex are: $q(E_*)=1$, $q(E_0)=q(E_1)=q(E_2)=1/2$, $q(E_3)=q(E_4)=q(E_5)=2/3$.

Fig. 10

The graph of the minimal good resolution of $\Lambda$

In this case f is a generic element of the pencil $\Lambda$ and g is the unique special element of $\Lambda$.

The graph of the minimal good resolution of $\{ fg=0\}$ is represented in Fig. 11.

Fig. 11

The graph of the minimal good resolution of (f, g)

5.3 Example 3

With this example, issued from [13], we illustrate the case where a special zone is a singular point of the dicritical locus.

Let (Z, z) be defined by $z^2 =(x^2+y^5)(y^2+x^3)$ and let $\pi =(f,g): (Z,z)\rightarrow ({{\mathbb {C}}}^2,0)$ be the projection on the (x, y)-plane. The dual graph of the minimal embedded resolution of the discriminant curve $(u^2+v^5)(v^2+u^3)=0$ of $\pi$ and the coordinate axes is shown in Fig. 12. The arrows representing the strict transforms of the coordinate axes are depicted by double-arrows.

Fig. 12

Graph of the minimal resolution of the discriminant of $\pi$ and the coordinates axes

The graph of the minimal good resolution of $\Lambda$ is in Fig. 13. The components $E_{0^1}$ and $E_{0^2}$ are dicritical. Thus, there exists two special zones $\Delta _0$ and $\Delta _1$ with $\Delta _0= E_{1^1}\cup E_{1^2}$ and $\Delta _1= E_{0^1}\cap E_{0^2}= \{P\}$ where P is the singular point of ${{\mathcal {D}}}$.

Fig. 13

The graph of the minimal good resolution of $\Lambda$

The special fibers associated to $\Delta _0$ and $\Delta _1$ are respectively $\{f=0\}$ and $\{ g=0\}$. The graph of the minimal good resolution of (f, g) is shown Fig. 14.

The Hironaka quotients of the rational components (of self-intersection $-1$) $E_2$ and $E_3$ are respectively 2/3 and 5/2 and there exists two irreducible components of $C(\pi )$ whose strict transform intersects $E_2$ and two others whose strict transform intersects $E_3$.

Fig. 14

The graph of the minimal good resolution of (f, g)