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Sbornik: Mathematics, 2024, Volume 215, Issue 2, Pages 206–233
DOI: https://doi.org/10.4213/sm9894e (Mi sm9894) 		 
On quasi-generic covers of the projective plane

Vik. S. Kulikov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
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DOI: https://doi.org/10.4213/sm9894e
Abstract: The Chisini Theorem for almost generic covers of the projective plane, whose proof is contained in the article “A Chisini Theorem for almost generic covers of the projective plane” (Sb. Math. 213:3 (2022), 341–356}), is generalized to the case of quasi-generic covers of the projective plane branched in curves with $\mathrm{ADE}$-singularities.
Bibliography: 18 titles.
Keywords: finite covers of the projective plane, Chisini's conjecture.
Received: 03.02.2023 and 17.11.2023
Bibliographic databases:
Document Type: Article
MSC: 14B05, 14E22
Language: English
Original paper language: Russian

Introduction

Let $S$ be a nonsingular irreducible projective surface defined over the field of complex numbers $\mathbb C$ and $f\colon S\to\mathbb P^2$ a finite morphism to the projective plane $\mathbb P^2$ branched in an irreducible curve $B_f\subset\mathbb P^2$. The morphism $f$ induces a monodromy homomorphism $f_*\colon \pi_1(\mathbb P^2\setminus B_f,q)\to \mathbb S_{\operatorname{deg} f}$ to the symmetric group $\mathbb S_{\operatorname{deg} f}$ acting on the fibre $f^{-1}(q)=\{q_1,\dots,q_{\operatorname{deg} f}\}$. Its image $G_f:=f_*(\pi_1(\mathbb P^2\setminus B_f,q))\subset \mathbb S_{\operatorname{deg} f}$ is called the monodromy group of $f$.

Let $\gamma$ be a so-called geometric generator of the fundamental group $\pi_1(\mathbb P^2\setminus B_f,q)$, that is, an element of $\pi_1(\mathbb P^2\setminus B_f,q)$ represented by a simple loop around the curve $B_f$ near a nonsingular point $p\in B_f$. We denote by $\overline{r}_f=(r_1,\dots ,r_k)$ the cyclic type of the permutation $f_*(\gamma)$, that is, the set of lengths of nontrivial cycles included in the factorization of $f^*(\gamma)$ into a product of disjoint cycles. The set $\overline r_f$ of integers $r_j\geqslant 2$, $j=1,\dots,k$, is called the ramification data of $f$.

Let $p$ be a point of the curve $B_f\subset \mathbb P^2$. It is well known that the group $\pi_1^{\mathrm{loc}}(B_f,p):=\pi_1(V_p\setminus B_f)$ does not depend on $V_p$, where $V_p\subset \mathbb P^2$ is a sufficiently small complex analytic neighbourhood of the point $p$ which is biholomorphic to a ball of radius $r\ll 1$ centred at $p$. The image $G_{f,p}:=\operatorname{im} f_{p*}:=\operatorname{im} f_*\circ \iota_*$ is called the local monodromy group of $f$ at the point $p$, where $\iota_*\colon \pi_1^{\mathrm{loc}}(B_f,p)=\pi_1(V_p\setminus B_f,\widetilde q\,)\to \pi_1(\mathbb P^2\setminus B_f,q)$ is a homomorphism defined uniquely up to conjugation in $\pi_1(\mathbb P^2\setminus B_f,q)$ by the embedding $\iota\colon V_p\hookrightarrow \mathbb P^2$. The collection

$$ \begin{equation*} \mathcal G_f=\{f_{p*}\colon \pi_1^{\mathrm{loc}}(B_f,p)\to G_{f,p} \mid p\in \operatorname{Sing} B_f\} \end{equation*} \notag $$
of homomorphisms $f_{p*}\colon \pi_1^{\mathrm{loc}}(B_f,p)\to G_{f,p}$ considered up to internal automorphisms of the groups $G_{f,p}$, is called the local monodromy data1 of $f$, and the triple $\mathrm{pas}(f)=(B_f,\overline r_f,\mathcal G_f)$ is called the passport of $f$.

We say that two finite morphisms $f_i\colon S_i\to \mathbb P^2$, $i=1,2$, are equivalent if there exists an isomorphism $\varphi\colon S_1\to S_2$ such that $f_1=f_2\circ\varphi$.

Let $\mathcal F_{\overline r}$ be the set of finite morphisms $f\colon S\to\mathbb P^2$ of smooth irreducible surfaces $S$ such that $\overline r_f=\overline r$.

In [14] the term Chisini Theorem was introduced. In this article we change slightly the definition of the Chisini Theorem. Namely, a statement is called a Chisini Theorem for morphisms in a subset $\mathcal M$ of $\mathcal F_{\overline r}$ if it states that there is a constant $\mathfrak{d}=\mathfrak{d}(\mathcal M)\in\mathbb N$ such that if $f_1$ and $f_2\in \mathcal M$ satisfy the conditions $\mathrm{pas}(f_1)=\mathrm{pas}(f_2)$ and $\max(\operatorname{deg} f_1,\operatorname{deg} f_2)\geqslant \mathfrak{d}$, then the morphisms $f_1$ and $f_2$ are equivalent.

For example, a finite morphism $f\colon S\to\mathbb P^2$ of a nonsingular irreducible surface $S$ is called a generic cover of the projective plane if it satisfies the following conditions:

Note that if $S$ is embedded in a projective space $\mathbb P^n$, then it is well known (see, for example, [3]) that the restriction $f:=\mathrm{pr}_{\mid S}\colon S\to \mathbb P^2$ to $S$ of a linear projection $\mathrm{pr}\colon \mathbb P^n\to\mathbb P^2$ which is generic with respect to the embedding of $S$ in $\mathbb P^n$, is a generic cover.

Chisini’s conjecture (see [2]) claims that the Chisini Theorem with constant $\mathfrak{d}=5$ is true for the set $\mathcal F_{(2), G}\subset\mathcal F_{(2)}$ of generic covers of the projective plane.2 Note (see, for example, [6]) that there are examples of nonequivalent generic covers of the projective plane that have the same passport, but whose degrees are ${\leqslant 4}$. Chisini’s conjecture was proved in [8] for generic linear projections, and the following theorem was proved in [15] using the results of [5].

Theorem 1. The Chisini Theorem with constant $\mathfrak d=12$ is true for the generic covers of the projective plane.

A finite morphism $f\colon S\to\mathbb P^2$ of a nonsingular irreducible surface $S$ is an almost generic cover of the projective plane if it satisfies conditions (i)–(iii) and the following condition:

The following theorem was proved in [14].

Theorem 2. The Chisini Theorem with constant $\mathfrak d=12$ is true for almost generic covers of the projective plane.

In particular, if $f_1\colon S_1\to\mathbb P^2$ and $f_2\colon S_2\to\mathbb P^2$, $\max(\operatorname{deg} f_1,\operatorname{deg}, f_2)\geqslant 12$, are two almost generic covers of the projective plane branched in the same curve $B\subset\mathbb P^2$, which does not have singular points of singularity types $A_{6k-1}$, $k\in\mathbb N$, then the covers $f_1$ and $f_2$ are equivalent.

Note that if we omit condition (iii) in the definition of generic covers, then a theorem similar to Theorem 1 does not hold. For example, it is well known that if $S$ is an Abelian surface whose endomorphism ring is $\mathbb{Z}$, then for each prime number $n$ there are $(n^{4}-1)/(n-1)$ finite etâle cyclic morphisms $f_i\colon S_i\to S_0$ of degree $n$ such that $S_{i_1}$ and $S_{i_2}$ are not isomorphic for $i_1\neq i_2$. Therefore, if $f_0\colon S\to \mathbb P^2$ is a generic projection, then the branch curve of at least $(n^{4}-1)/(n-1)$ morphisms $\widetilde f_i=f_0\circ f_i\colon S_i\to\mathbb P^2$ of degree $n\cdot\operatorname{deg} f_0$ satisfies conditions (i), (ii) and (iv).

Let $V_p\subset \mathbb P^2$ be a sufficiently small simply connected neighbourhood of a point $p\in B_f$ such that $\pi_1(V_p\setminus B_f,p)=\pi_1^{\mathrm{loc}}(B_f,p)$ and $U_{p,j}\subset f^{-1}(V_p)\subset S$ a connected neighbourhood of a point $p_j\in f^{-1}(p)$. We call $f_{\mid U_{p,j}}\colon U_{p,j}\to V_p$ the germ of $f$ at the point $p_j$, and a germ $f_{\mid U_{p,j}}$ is called nontrivial if $\operatorname{deg} f_{\mid U_{p,j}}\geqslant 2$. We denote by $(B_j,p)\subset (B_f,p)\subset V_p$ the branch curve germ of the nontrivial germ $f_{\mid U_{p,j}}\colon {U_{p,j}\to V_p}$.

We denote by $\mathcal F_{(2),Q}$ the subset of $\mathcal F_{(2)}$ consisting of the quasi-generic covers.

In § 2.3.3 (see Definition 3) we define a subset $\mathcal F_{(2),E}$ of $\mathcal F_{(2),Q}$ elements of which are called extra-quasi generic covers and the aim of this article is to prove the following result.

Theorem 3. The Chisini Theorem with constant $\mathfrak d=12$ is true for the extra-quasi-generic covers of the projective plane.

Theorem 4. A cover $f\in \mathcal F_{(2)}$ belongs to $\mathcal F_{(2),E}$ if for each point $p\in \operatorname{Sing} B_f$ the singularity type of the branch curve germ $(B_j,p)\subset (B_f,p)$ of each nontrivial germ $f_{\mid U_{p_j}}\colon U_{p_j}\to V_p$ at $p_j$ of the cover $f$, is of some $\mathrm{ADE}$-type, and in this case the singularity type of $(B_j,p)$ is either $A_0$ (that is, $(B_j,p)$ is a curve germ smooth at $p$), or $A_{3n-1}$ for some $n\geqslant 1$, or $E_6$.

We denote by $\mathcal F_{(2),\mathrm{ADE}}$ the subset of $\mathcal F_{(2)}$ consisting of covers $f\colon S\to\mathbb P^2$ branched in curves $B_f$ with singular points of $\mathrm{ADE}$-type only.

Corollary 1. The Chisini Theorem with constant $\mathfrak d=12$ is true for covers belonging to $F_{(2),\mathrm{ADE}}$.

The branch curve $B_f$ of $f\in \mathcal F_{(2),\mathrm{ADE}}$ can have singular points of the following types only: $A_{n}$, $D_{3n+2}$ $n\in\mathbb N$, $E_6$ and $E_7$.

If $\max(\operatorname{deg} f_1,\operatorname{deg}, f_2)\geqslant 12$ for $f_1$ and $f_2\in\mathcal F_{(2),\mathrm{ADE}}$ branched in the same branch curve $B\subset\mathbb P^2$, then the covers $f_1$ and $f_2$ are equivalent if $B$ has no singular points of type $A_{6n-1}$ or $D_{6n+2}$, $n\geqslant 1$.

In [9] the notion of so-called dualizing covers of the plane associated with plane curves was introduced. The set $\mathcal F_{(2),D_g}$ of dualizing covers associated with irreducible projective curves $C$ of genus $g$ immersed in the projective plane is a subset of $\mathcal F_{(2)}$. In § 5 we prove the following result.

Theorem 5. Let $\iota\colon C\hookrightarrow \widehat{\mathbb P}^2$ be an immersion of an irreducible projective curve $C$ of genus $g$ and $f_{\iota(C)}\colon X_{\iota(C)}\to \mathbb P^2$ the dualizing cover associated with $\iota(C)$, and let the cover $f\colon X\to \mathbb P^2$ belong to the set $\mathcal F_{(2)}$. If $\mathrm{pas}(f)=\mathrm{pas}(f_{\iota(C)})$, then the covers $f$ and $f_{\iota(C)}$ are equivalent if either $\operatorname{deg} \iota(C)\geqslant 8$ and $g\geqslant 1$, or $\operatorname{deg} \iota(C)\geqslant 12$ and ${g=0}$.

The main steps of the proof of Theorem 3 coincide with the ones of the proof of Theorem 2 in [14]. In §§ 1 and 2 the properties of quasi-generic covers and the properties of the fibre product of two nonequivalent quasi-generic covers branched in the same curve are investigated. In § 3, applying results obtained in §§ 1 and 2, the proof of Theorem 3 is completed using Hodge’s index theorem and the Bogomolov–Miaoka–Yau inequality. In § 4 the proofs of Theorem 4 and Corollary 1 are given.

§ 1. Properties of covers belonging to $\mathcal F_{(2)}$

We denote by $B\subset \mathbb P^2$ the branch curve and by $R\subset S$ the ramification curve of a cover $\{f\colon S\to\mathbb P^2\} \in \mathcal F_{(2)}$, $\operatorname{deg} f=N$, and let $f^{-1}(B)=R\cup C$, where the curve $C\subset S$ is the additional curve to the curve $R$ in the proper inverse image $f^{-1}(B)$ of the branch curve $B$. By condition (ii) the image of the divisor $B\in \operatorname{Pic} \mathbb P^2$ under the homomorphism $f^*\colon \operatorname{Pic} \mathbb P^2\to \operatorname{Pic} S$ is $f^*(B)=2R+C\in \operatorname{Pic} S$.

1.1. The monodromy groups of covers belonging to $\mathcal F_{(2)}$

Lemma 1. The monodromy group $G_f\subseteq \mathbb S_N$ of $f\in \mathcal F_{(2)}$ coincides with $\mathbb S_{N}$.

Proof. The group $\pi_1(\mathbb P^2\setminus B,q)$ is generated by geometric generators. For a geometric generator $\gamma\in\pi_1(\mathbb P^2\setminus B,q)$ it follows from conditions (ii) and (iii) that $f_*(\gamma)\in \mathbb S_N$ is a transposition. Therefore, the group $G_f\subset\mathbb S_N$ is generated by transpositions and it acts transitively on the fibre $f^{-1}(q)$, since $S$ is an irreducible surface. Consequently, $G_f=\mathbb S_N$. The lemma is proved.

1.2. The local properties of covers belonging to $\mathcal F_{(2)}$

1.2.1.

Let $\mathbb S_N$ be the symmetric group acting on a set $P$ consisting of $N$ elements. We denote by $\mathbb S_m\subset \mathbb S_N$, $m\leqslant N$, a subgroup of $\mathbb S_N$ that is generated by several transpositions belonging to $\mathbb S_N$, acts transitively on a subset $P_1$ of $P$ consisting of $m$ elements and leaves fixed the elements of $P\setminus P_1$. Denote by $V_p\subset \mathbb P^2$ a sufficiently small neighbourhood of a point $p\in B$ such that $\pi_1({V_p\setminus B},q)=\pi_1^{\mathrm{loc}}(B,p)$, and let $\mu_p(B)$ be the multiplicity at the point $p$ of the curve $B$.

Lemma 2. For a point $p\in B$ the local monodromy group $G_{f,p}\subset \mathbb S_N$ of a cover $f\in\mathcal F_{(2)}$, $\operatorname{deg} f=N$, branched in $B\subset \mathbb P^2$, has the following form:

$$ \begin{equation} G_{f,p}=\mathbb S_{m_{p,1}}\times\dots\times \mathbb S_{m_{p,k_p}}\times \mathbb S_{m_{p,k_p+1}}\times \dots\times\mathbb S_{m_{p,M_p}}\subset \mathbb S_N, \end{equation} \tag{1} $$
where $m_{p,1}\geqslant 2$ for $1\leqslant j\leqslant k_p$, $m_{p,j}=1$ for $j>k_p$, and
$$ \begin{equation} M_p=N- \sum_{j=1}^{k_p}(m_{p,j}-1) \end{equation} \tag{2} $$
is the number of orbits of the action of $G_{f,p}$ on the set $f^{-1}(q)=\{q_1,\dots, q_N\}$.

Proof. By the Zariski–van Kampen theorem (see, for example, [7]), the group $\pi_1^{\mathrm{loc}}(B,p)$ is generated by $\mu_p(B)$ geometric generators and by Lemma 1, for geometric generators $\gamma\in \pi_1^{\mathrm{loc}}(B,p)$, $f_{p*}(\gamma)\in \mathbb S_N$ are transpositions. Therefore, $G_{f,p}$ is generated by at most $\mu_p(B)$ transpositions. Hence the local monodromy group $G_{f,p}$ has the form (1) and
$$ \begin{equation*} f^{-1}(V_p)= \bigsqcup_{j=1}^{M_p} U_{p,j} \end{equation*} \notag $$
is a disjoint union of $M_p$ connected germs of a smooth surface, where $M_p$ is the number of orbits of the action of $G_{f,p}$ on the set $f^{-1}(q)=\{q_1,\dots, q_N\}$. The lemma is proved.

1.2.2.

We denote by $(B_j,p)$ the branch curve germ of a nontrivial germ $f_{p,j}:=f_{\mid U_{p,j}}\colon U_{p,j}\to V_p$. Note that $(B_{j_1},p)\cap (B_{j_2},p)=p$ for $1\leqslant j_1< j_2\leqslant k_p$, since $\operatorname{deg} f_{\mid R}=1$ and $(B,p)=(B_1,p)\cup\dots\cup (B_{k_p},p)$.

For $j=1,\dots, k_p$ the embedding $\iota_j\colon (B_j,p)\hookrightarrow (B,p)$ induces an epimorphism

$$ \begin{equation*} \iota_{j_*}\colon \pi_1^{\mathrm{loc}}(B,p)=\pi_1(V_p\setminus B,q)\to \pi_1(V_p\setminus B_j,q)=\pi_1^{\mathrm{loc}}(B_j,p), \end{equation*} \notag $$
which can be included in the commutative diagram
$(3)$
in which the epimorphism $f_{p,j*}\colon \pi_1^{\mathrm{loc}}(B_j,p)\to\mathbb S_{m_{p,j}}$ is the monodromy homomorphism of the $m_{p,j}$-sheeted cover $f_{p,j}\colon U_{p,j}\to V_p$.

So the germs $f_{p,j}:=f_{\mid U_{p,j}}\colon U_{p,j}\to V_p$ of the cover $f$ are nontrivial $m_{p,j}$-sheeted covers for $1\leqslant j\leqslant k_p$ and for $j=k_p+1,\dots,M_p$, the germs $f_{p,j}:=f_{\mid U_{p,j}}\colon U_{p,j}\to V_p$ are biholomorphic maps.

The following lemma is well known.

Lemma 3. Let $f_{p,j}\colon (U_{p,j},p_j)\to V_p$ be a two-sheeted germ over a point $p\in B$ of a cover $\{f\colon S\to \mathbb P^2\} \in\mathcal F_{(2)}$ branched in a curve $B$. Then there are local coordinates $z_j,w_j$ in $U_{p,j}$ and $u_j,v_j$ in $V_p$ such that $f_{p,1}$ is given by $u_j=z_j$, $v_j=w_j^2$. In particular, the branch curve germ $(B_j,p)$ of $f_{p,j}$ defined by the equation $v_j=0$ is nonsingular at $p$, and therefore the nontrivial germ $f_{p,j}$ of the cover $f$ satisfies condition (IV).

1.2.3.

Consider a point $p\in \operatorname{Sing} B$. For $j\leqslant k_p$ each germ $(B_j,p)$ at the point $p$ splits into several irreducible germs: $(B_j,p)=(B_{j,1},p)\cup\dots\cup (B_{j,k_{p,j}},p)$. We have

$$ \begin{equation*} \mu_{p}(B_j)= \sum_{l=1}^{k_{p,j}}\mu_{p}(B_{j,l}). \end{equation*} \notag $$

By definition (see [10]) the integers

$$ \begin{equation*} c_{v,p}(B):=\sum_{j=1}^{k_p}\sum_{l=1}^{k_{p,j}} (\mu_{p}(B_{j,l})-1)=\mu_p(B)-\sum_{j=1}^{k_p}k_{p,j} \end{equation*} \notag $$
and
$$ \begin{equation*} {n}_{v,p}(B):=\delta_p(B)-c_{v,p}(B), \end{equation*} \notag $$
where $\delta_{p}(B)$ is the $\delta$-invariant of the singularity $(B,p)$, are called, respectively, the numbers of virtual cusps and virtual nodes of the curve germ $(B,p)$, and
$$ \begin{equation} c_v(B):=\sum_{p\in \operatorname{Sing} B}c_{v,p}(B) \quad\text{and}\quad n_v(B):=\sum_{p\in \operatorname{Sing} B}n_{v,p}(B) \end{equation} \tag{4} $$
are called, respectively, the numbers of virtual cusps and virtual nodes of the curve ${B\subset\mathbb P^2}$.

Let $\widehat{B}$ be the dual curve of $B$ of genus $g$. It follows from generalized Plücker’s formulae (see [10]) that

$$ \begin{equation*} \operatorname{deg} \widehat B = 2\operatorname{deg} B-c_v(B) +2g-2, \end{equation*} \notag $$
and since $\operatorname{deg} \widehat B > 0$, we have
$$ \begin{equation} c_v(B)< 2(\operatorname{deg} B+g-1). \end{equation} \tag{5} $$

Lemma 4. The number $M_p$ of the points in a fibre $f^{-1}(p)$ of a cover $\{f\colon {S\to\mathbb P^2}\}\in\mathbb F_{(2),Q}$, $\operatorname{deg} f=N$, is equal to

$$ \begin{equation} M_p=N -c_{v,p}(B)- \sum_{j=1}^{k_p}k_{p,j}. \end{equation} \tag{6} $$

Proof. By condition (IV), for each nontrivial germ $f_{p,j}\colon U_{p,j}\to V_p$ we have
$$ \begin{equation*} m_{p,j}-1=\mu_{p}(B_j)=k_{p,j}+c_{v,p}(B_j). \end{equation*} \notag $$
Therefore, Lemma 4 follows from equality (2).

1.3. Invariants of the branch and ramification curves

Lemma 5. The degree $\operatorname{deg} B:=2d$ of the branch curve $B$ of a cover $\{f\colon {S\to\mathbb P^2}\}\in \mathcal F_{(2)}$, is an even integer.

Proof. By Lemma 1 the monodromy homomorphism $f_*\colon \pi_1(\mathbb P^2\setminus B)\to \mathbb S_N$ is an epimorphism. Therefore, the induced homomorphism
$$ \begin{equation*} f_{*\mathrm{ab}}\colon \pi_1(\mathbb P^2\setminus B)\,/\,[\pi_1(\mathbb P^2\setminus B),\pi_1(\mathbb P^2\setminus B)]=H_1(\mathbb P^2\setminus B,\mathbb Z)\to \mathbb S_N/\,[\mathbb S_N,\mathbb S_N]\simeq \mathbb Z_2 \end{equation*} \notag $$
is an epimorphism too. Hence $\operatorname{deg} B$ is an even integer, since it is well known that $H_1(\mathbb P^2\setminus B,\mathbb Z) \simeq \mathbb Z_{\operatorname{deg} B}$ for an irreducible curve $B\subset\mathbb P^2$. The lemma is proved.

We denote by $\delta(R)$ the $\delta$-invariant of the ramification curve $R$ of a cover $\{f\colon {S\to\mathbb P^2}\}\in \mathcal F_{(2)}$ and by $g$ the genus of $R$ and the branch curve $B$ of $f$.

Lemma 6. The self-intersection number in $S$ of the ramification curve $R$ of a cover $\{f\colon S\to\mathbb P^2\}\in \mathcal F_{(2)}$ is equal to

$$ \begin{equation*} (R^2)_S=3d+g+\delta(R)-1. \end{equation*} \notag $$

Proof. The canonical class of $S$ is
$$ \begin{equation*} K_S=R+f^*(K_{\mathbb P^2})= R-3f^*(L), \end{equation*} \notag $$
where $L$ is a line in $\mathbb P^2$. We have $(f^*(L),R)_S=\operatorname{deg} B=2d$, since $f_{\mid R}\colon R\to B$ is a birational morphism. Therefore,
$$ \begin{equation*} 2(g+\delta(R)-1)=(R+K_S,R)_S=2(R^2)_S-3(f^*(L),R)_S=2(R^2)_S-6d; \end{equation*} \notag $$
hence $(R^2)_S=3d+g+\delta(R)-1$. The lemma is proved.

1.4. Invariants of covering surfaces

Let $N=\operatorname{deg} f$ be the degree of a cover $\{f\colon S\to\mathbb P^2\}\in\mathcal F_{(2)}$ branched in a curve $B\subset\mathbb P^2$.

Proposition 1. The self-intersection number of the canonical class $K_S$ of $S$ is equal to

$$ \begin{equation*} (K_S^2)_S=9N-9d+g+\delta(R)-1. \end{equation*} \notag $$

Proof. It follows from Lemma 6 and its proof that
$$ \begin{equation*} \begin{aligned} \, (K^2_S)_S &=(R+f^*(K_{\mathbb P^2}),R+f^*(K_{\mathbb P^2}))_S= (R-3f^*(L),R-3f^*(L))_S \\ &=(R^2)_S-6(R,f^*(L))_S+9(f^*(L),f^*(L))_S= g+\delta(R)-1-9d+9N. \end{aligned} \end{equation*} \notag $$
The proposition is proved.

Proposition 2. For a cover $\{f\colon S\to\mathbb P^2\}\in\mathcal F_{(2),Q}$ the topological Euler characteristic $e(S)$ of $S$ is equal to

$$ \begin{equation*} e(S)=3N+ 2(g-1)-c_{v}(B). \end{equation*} \notag $$

Proof. In the notation introduced above we have
$$ \begin{equation} e(S)=Ne(\mathbb P^2\setminus B)+(N-1)e(B\setminus \operatorname{Sing} B) +e(f^{-1}(\operatorname{Sing} B)) \end{equation} \tag{7} $$
and Proposition 2 follows from (7), the equalities
$$ \begin{equation*} \begin{gathered} \, e(\mathbb P^2)=3, \qquad e(\mathbb P^2\setminus B)=e(\mathbb P^2)-e(B), \\ e(B) = 2-2g- \sum_{p\in\operatorname{Sing} B}\biggl(\sum_{j=1}^{k_p}k_{p,j}+1\biggr), \\ e(B\setminus \operatorname{Sing} B) = 2-2g- \sum_{p\in\operatorname{Sing} B}\sum_{j=1}^{k_p}k_{p,j}, \\ e(f^{-1}(\operatorname{Sing} B)) = \sum_{p\in\operatorname{Sing} B}M_p \stackrel{(6)}{=} \sum_{p\in\operatorname{Sing} B} \biggl(N- c_{v,p}(B)-\sum_{j=1}^{k_p}k_{p,j}\biggr) \end{gathered} \end{equation*} \notag $$
and equality (4). The proposition is proved.

§ 2. On fibre products of nonequivalent covers belonging to $\mathcal F_{(2)}$

Let $\{f_1\colon S_1\to \mathbb P^2\}$ and $\{f_2\colon S_2 \to\mathbb P^2\}$ be two covers belonging to $\mathcal F_{(2)}$ and branched in the same curve $B\subset \mathbb P^2$, let $\operatorname{deg} f_i=N_i$ for $i=1,2$, and let $\mathrm{pas}(f_1)=\mathrm{pas}(f_2)$. We have $f^{*}_1(B)=2R_1+C_1$ and $f^{*}_2(B)=2R_2+C_2$, where $R_i\subset S_i$ is the ramification curve of $f_i$, $i=1,2$, and $f_i^*\colon \operatorname{Div}(\mathbb P^2)\to\operatorname{Div}(S_i)$ is the map from the set of divisors in $\mathbb P^2$ to the set of divisors in $S_i$ which is induced by the cover $f_i$.

2.1. The irreducibility of the fibre product of two non-equivalent covers belonging to $\mathcal F_{(2)}$

Consider the fibre product

$$ \begin{equation*} S_1\times _{\mathbb P^2}S_2=\{(x,y)\in S_1\times S_2\mid f_1(x)=f_2(y)\} \end{equation*} \notag $$
of $S_1$ and $S_2$ over $\mathbb P^2$. Let $\nu\colon \widetilde X=\widetilde{S_1\times _{\mathbb P^2}S_2}\to S_1\times _{\mathbb P^2}S_2$ be the normalization of $S_1\times _{\mathbb P^2}S_2$. Let $g_{1}\colon \widetilde X\to S_1$, $g_{2}\colon \widetilde X\to S_2$ and $g_{1,2}\colon \widetilde X\to \mathbb P^2$ denote the corresponding natural morphisms. Then we have $\operatorname{deg} g_1=N_2$, $\operatorname{deg} g_2=N_1$ and $\operatorname{deg} g_{1,2}=N_1N_2$.

We use the following notation:

$$ \begin{equation*} \begin{gathered} \, \widetilde R=g_1^{-1}(R_1)\cap g_2^{-1}(R_2) \subset \widetilde X, \\ \widetilde C=g_1^{-1}(C_1)\cap g_2^{-1}(C_2), \ \ \widetilde C_1=g_1^{-1}(R_1)\cap g_2^{-1}(C_2)\ \ \!\text{and}\! \ \ \widetilde C_2= g_2^{-1}(R_2)\cap g_1^{-1}(C_1). \end{gathered} \end{equation*} \notag $$

Proposition 3. If $f_1\colon S_1\to \mathbb P^2$ and $f_2\colon S_2 \to\mathbb P^2$ are nonequivalent, then $\widetilde X$ is irreducible.

The proof is literally the same as the proof of Proposition 2 in [5].

2.2. Singular points of the fibre product of two nonequivalent covers belonging to $\mathcal F_{(2)}$

Let $\mathrm{pr}_i\colon S_1\times_{\mathbb P^2}S_2\to S_i$, $i=1,2$, be the projections onto the factors. Set $\mathcal R_{1,2}=\mathrm{pr}_1^{-1}(R_1)\cap \mathrm{pr}_2^{-1}(R_2)\subset S_1\times_{\mathbb P^2}S_2$.

Lemma 7. We have $\operatorname{Sing} (S_1\times_{\mathbb P^2}S_2)=\mathcal R_{1,2}$.

Proof. Consider points $x\in S_1$ and $y\in S_2$ such that $f_1(x)=f_2(y)$, and let $U_1\subset S_1$ and $U_2\subset S_2$ be sufficiently small neighbourhoods of $x$ and $y$ such that $f_1(U_1)=f_2(U_2)=V$.

If $x\notin R_1$, then $f_1\colon U_1\to V$ is a biholomorphic map. Therefore, $U_1\times_{V}U_2\subset S_1\times_{\mathbb P^2}S_2$ is biholomorphic to the graph of $f_2\colon U_2\to V$, and so $S_1\times_{\mathbb P^2}S_2$ is nonsingular at the point $(x,y)$. Similarly, $S_1\times_{\mathbb P^2}S_2$ is nonsingular at $(x,y)$ if ${y\notin R_2}$. Therefore, $(S_1\times_{\mathbb P^2}S_2)\setminus \mathcal R_{1,2}$ is a nonsingular surface.

If $x\in R_1\setminus \operatorname{Sing} R_1$ and $y\in R_2\setminus \operatorname{Sing} R_2$, then $(x,y)\in \mathcal R_{1,2}$ and by Lemma 3 there are local coordinates $(z_i,w_i)$ in $U_{i}$, $i=1,2$, and local coordinates $(u,v)$ in $V$ such that the $f_i\colon U_{i}\to V$ are two-sheeted covers given by the functions $u=z_i$ and $v=w_i^2$. Therefore, $U_1\times_{V}U_2$ is defined in $U_1\times U_2$ by the equations $z_1=z_2$ and $w_1^2=w_2^2$. Consequently, $(x,y)\in \operatorname{Sing} (S_1\times_{\mathbb P^2}S_2)$ if $x\in R_1\setminus \operatorname{Sing} R_1$ and $y\in R_2\setminus \operatorname{Sing} R_2$, that is, $\operatorname{Sing} (S_1\times_{\mathbb P^2}S_2)=\mathcal R_{1,2}$, since $\operatorname{Sing} (S_1\times_{\mathbb P^2}S_2)$ and $\mathcal R_{1,2}$ are closed subsets of $S_1\times_{\mathbb P^2}S_2$. The lemma is proved.

2.3. Resolution of singular points of $\widetilde X$

Let $\rho \colon X\to \widetilde X$ be a resolution of the singular points of the surface $\widetilde X$, which we describe below (the resolution we use is not minimal; moreover, it blows up some nonsingular points of $\widetilde X$).

2.3.1.

We denote by

$$ \begin{equation*} \overline R=\rho^{-1}(\widetilde R), \qquad \overline C=\rho^{-1}(\widetilde C)\quad\text{and} \quad \overline C_i=\rho^{-1}(\widetilde C_i), \quad i=1,2, \end{equation*} \notag $$
the proper inverse images of the curves $\widetilde R$, $\widetilde C$ and $\widetilde C_i$.

Consider the morphism $h_1=g_1\circ \rho\colon X \to S_1$.

Lemma 8. We have $\operatorname{deg} h_{1\mid \overline R}=2$.

The proof follows directly from the proof of Lemma 7, since $\widetilde R$ is the proper inverse image $\nu^{-1}(\mathcal R_{1,2})$ of $\mathcal R_{1,2}$ under the normalization of the surface $S_1\times_{\mathbb P^2}S_2$.

In §§ 2.3.2 and 2.3.3 we investigate the properties of the divisor $h_1^*(R_1)=\rho^*({\widetilde R+\widetilde C_1})$ at points in

$$ \begin{equation*} \operatorname{Supp}(\rho^*(\widetilde R))\cap \operatorname{Supp}(\rho^*(\widetilde C_1)). \end{equation*} \notag $$

2.3.2.

Since $\mathrm{pas}(S_1)=\mathrm{pas}(f_2)$, by the Riemann–Stein theorem (see [16] and [4]), for each point $p\in \operatorname{Sing} B$ we can identify the germ

$$ \begin{equation*} f_{1,p}:=f_{1\mid Z_{1,p}}\colon Z_{1,p}:=f_{1}^{-1}(V_p)\to V_p \end{equation*} \notag $$
of the cover $f_1$ with the germ
$$ \begin{equation*} f_{2,p}:=f_{2\mid Z_{2,p}}\colon Z_{2,p}:=f_{2}^{-1}(V_p)\to V_p \end{equation*} \notag $$
of the cover $f_2$ over $p\in V_p$ and we use results obtained in § 1.1 by modifying slightly the notation from there. Namely, we denote by $U_{p,i,j}\subset S_i$ the neighbourhood $U_{p,j}$ in $f^{-1}(V_p)=\bigsqcup U_j\subset S$ in the case when $f=f_i$, $i=1,2$. In particular, neighbourhoods $Z_{1,p}\times_V Z_{2,p}\subset S_1\times_{V_p} S_2$ and $Z_{1,p}\times_{V_p} Z_{1,p}\subset S_1\times_{V_p} S_1$ are naturally biholomorphic to each over.

We denote by $\widetilde W_{p,j_1,j_2}=\widetilde{U_{p,1,j_1}\times_{V_p} U_{p,2,j_2}}\subset \widetilde X$ the normalizations of the fibre products of neighbourhoods $U_{p,1,j_1}$ and $U_{p,2,j_2}$ over $V_p$.

It follows from the proof of Lemma 7 that

$$ \begin{equation*} h_1(\operatorname{Supp}(\rho^*(\widetilde R))\cap \operatorname{Supp}(\rho^*(\widetilde C_1)))\subset f^{-1}_1(\operatorname{Sing} B). \end{equation*} \notag $$

Note that

$$ \begin{equation} \widetilde R\cap\widetilde W_{p,j,j}=\varnothing \quad \text{for } j>k_{p}, \end{equation} \tag{8} $$
since $\widetilde R\cap\widetilde W_{p,j,j} =((R_1\cap U_{p,1,j})\times_V (R_2\cap U_{p,2,j}))$ and $R_1\cap U_{i,j}=\varnothing$ for $j>k_{p}$. Note also that
$$ \begin{equation} \widetilde R_1\cap\widetilde W_{p,j_1,j_2}=\varnothing \quad \text{for } j_1\neq j_2, \end{equation} \tag{9} $$
since $g_1(\widetilde R\cap \widetilde W_{p,j_1,j_2})=R_1\cap U_{p,1,j_1}$ and $f_1(R_1\cap U_{p,1,j_1})=B_{j_1}\subset V_p$, but
$$ \begin{equation*} g_2(\widetilde R\cap \widetilde W_{p,j_1,j_2})= R_2\cap U_{p,2,j_2} \quad\text{and}\quad f_2(R_2\cap U_{p,2,j_2})=B_{j_2}\subset V_p, \end{equation*} \notag $$
and $B_{j_1}\cap B_{j_2}=\{p\}$ for $j_1\neq j_2$. Therefore, by (8) and (9)
$$ \begin{equation*} \operatorname{Supp}(\rho^*(\widetilde R))\cap \operatorname{Supp}(\rho^*(\widetilde C_1))\cap g_{1,2}^{-1}(V_p)\subset \rho^{-1}\biggl(\bigcup_{j=1}^{k_p}\widetilde W_{p,j,j}\biggr). \end{equation*} \notag $$

2.3.3.

To describe $\widetilde W_{p,j,j}$, consider the monodromy homomorphisms

$$ \begin{equation*} f_{i\mid U_{p,i,j}*} \colon \pi_1(V_p\setminus B_j) \to \mathbb S_{m_{p,j}} \end{equation*} \notag $$
of $m_{p,j}$-sheeted covers $f_{i\mid U_{p,i,j}}\colon U_{p,i,j}\to V_p$ branched in $V_p\cap B_j$. Since we identify the covers $f_{i\mid U_{p,i,j}}$, namely,
$$ \begin{equation*} f_{\mid U_{p,j}}:=f_{1\mid U_{p,1,j}}=f_{2\mid U_{p,2,j}}\colon U_{p,j}=U_{p,1,j}=U_{p,2,j}\to V_p, \end{equation*} \notag $$
we can assume that $f_{\mid U_{p,j}*}:=f_{1\mid U_{p,1,j}*}=f_{2\mid U_{p,2,j}*}$, since for equivalent covers the monodromy homomorphisms $f_{i\mid U_{p,i,j}*}$ coincide up to conjugations in $\mathbb S_{m_{p,j}}$, and the monodromy homomorphism of $g_{1,2\mid \widetilde W_{p,j,j}}\colon \widetilde W_{p,j,j}\to V_p$ is
$$ \begin{equation*} g_{1,2\mid\widetilde W_{p,j,j}*} = (f_{\mid U_{p,j}*},f_{\mid U_{p,j}*}) \colon \pi_1(V_p\setminus B_j,q) \to \mathbb S_{m_{p,j}}\times \mathbb S_{m_{p,j}}\subset \mathbb S_{m_{p,j}^2}, \end{equation*} \notag $$
which is defined by the action of $\gamma\in\pi_1(V_p\setminus B_j,q)$ on the set
$$ \begin{equation*} f_{\mid U_{p,j}}^{-1}(q)\times f_{\mid U_{p,j}}^{-1}(q)=\{q_1,\dots,q_{m_{p,j}}\}\times \{q_1,\dots,q_{m_{p,j}}\}\subset U_{p,j}\times_{V_p} U_{p,j} \end{equation*} \notag $$
as follows:
$$ \begin{equation*} g_{1,2\mid\widetilde W_{p,j,j}*}(\gamma)((q_i,q_j))=(f_{\mid U_{p,j}*}(\gamma)(q_i),f_{\mid U_{p,j}*}(\gamma)(q_j)). \end{equation*} \notag $$

Therefore, it is easy to see that the set $f_{\mid U_{p,j}}^{-1}(q)\times f_{\mid U_{p,j}}^{-1}(q)$ is the union of two orbits $O'$ and $O''$ of the action of the fundamental group $g_{1,2\mid\widetilde W_{p,j,j}*}(\pi_1(V_p\setminus B_j,q))\simeq\mathbb S_{m_{p,j}}$ on $f^{-1}_{\mid U_{p,j}}(q)\times f_{\mid U_{p,j}}^{-1}(q)$, namely,

$$ \begin{equation*} O'=\{(q_1,q_1),\dots,(q_{m_{p,j}}, q_{m_{p,j}})\}\ \ \text{and} \ \ O''=\{(q_{j_1},q_{j_2})\mid 1\leqslant j_1, j_2\leqslant m_{p,j},\,j_1\!\neq\! j_2\}. \end{equation*} \notag $$
Consequently, $\widetilde W_{p,j,j}=\widetilde W'_{p,j,j}\sqcup \widetilde W_{p,j,j}''$ is a disjoint union of two neighbourhoods such that $g_{i,p\mid\widetilde W_{p,j,j}'}\colon \widetilde W_{p,j,j}'\to U_{p,i,j}$ is a biholomorphic isomorphism and $g_{1,2\mid\widetilde W_{p,j,j}'}$: $\widetilde W_{p,j,j}'\to V_p$ coincides with $f_{1\mid U_{p,1,j}}$ and $f_{2\mid U_{p,2,j}}$ (because the restrictions of $g_{1,p}$ and $g_{2,p}$ to $\widetilde W_{j,j}'$ are isomorphisms). Therefore, $g^{-1}_{1\mid \widetilde W_{p,j,j}'}(R_1)=g^{-1}_{2\mid \widetilde W_{p,j,j}'}(R_2)=\widetilde R\cap\widetilde W_{p,j,j}'$, so that
$$ \begin{equation} \widetilde R\cap \widetilde C_1\cap\widetilde W_{p,j,j}'=\varnothing. \end{equation} \tag{10} $$
By Lemma 8,
$$ \begin{equation} \operatorname{deg} g_{1\mid \widetilde W_{p,j,j}'\cap\widetilde R}=\operatorname{deg} g_{1\mid \widetilde W_{p,j,j}''\cap\widetilde R}=1. \end{equation} \tag{11} $$
The degree of the map $g_{1\mid \widetilde W_{p,j,j}''}\colon \widetilde W_{p,j,j}''\to U_{p,1,j}$ is equal to $m_{p,j}-1$. Therefore, $g_{1\mid \widetilde W_{p,j,j}''}\colon \widetilde W_{p,j,j}''\to U_{p1,j}$ is a biholomorphic isomorphism for $m_{p,j}=2$ too, and similarly to equality (10) we obtain
$$ \begin{equation} \widetilde R\cap \widetilde C_1\cap\widetilde W_{p,j,j}''=\varnothing \end{equation} \tag{12} $$
for $m_{p,j}=2$.

Put $W_{p,j_1,j_2}''=\rho^{-1}(\widetilde W_{p,j_1,j_2}'')$. In the case when $m_{p,j}\geqslant 3$, the divisor $h^*(R_1)$ in $W_{p,j,j}''$ is equal to

$$ \begin{equation*} h^*(R_1)=\overline R+\overline C_1 +E''_{p,j}, \end{equation*} \notag $$
where $E''_{p,j}$ is a divisor with support in $\rho^{-1}(g_{1,2\mid \widetilde W_{p,j,j}''}^{-1}(p))$.

Let $p_{1,j}=f_1^{-1}(p)\cap U_{p,1,j}$, and $\delta_{p_{1,j}}(R_1)$ be the $\delta$-invariant of the curve $R_1$ at the point $p_{1,j}$. Note that by Lemma 3, $\delta_{p_{1,j}}(R_1)=0$ if $m_{p,1,j}=2$.

Definition 2. We say that a nontrivial germ $f_{\mid U_{p,j}}\colon U_{p,j}\to V_p$ of a cover $\{f\colon {S\to \mathbb P^2}\}\in \mathcal F_{(2)}$ has an extra property over a point $p\in\operatorname{Sing} B$ if either $\operatorname{deg} f_{1\mid U_{p,1,j}}=2$, or

$$ \begin{equation} \begin{gathered} \, E''_{p,j}= E_{p,j,\overline R}+E_{p,j,\overline C_1}, \\ (\overline R+E_{p,j\overline R},\overline C_1+E_{p,j,\overline C_1})_{W_{p,j,j}''} \leqslant 2 \delta_{p_{1,j}}(R_1)+c_{v,p}(B_j) \end{gathered} \end{equation} \tag{13} $$
for $\operatorname{deg} f_{1\mid U_{p,1,j}}>2$, where $(\operatorname{Div}_1,\operatorname{Div}_2)_{W_{p,j,j}''}$ is the intersection number of the divisors $\operatorname{Div}_1$ and $\operatorname{Div}_2$ in $W_{p,j,j}''$ (we assume here that in the definition of the germ $W_{p,j,j}''$ the cover satisfies $f_2=f_1=f$).

Definition 3. A quasi-generic cover $f\colon S\to\mathbb P^2$ is extra-quasi generic if all of its nontrivial germs over each point $p\in \operatorname{Sing} B$ have an extra property.

Remark 1. Note that if $\mathrm{pas}(f_1)=\mathrm{pas}(f_2)$ and the cover $f_1$ is extra-quasi generic, then $f_2$ is an extra-quasi generic cover too.

For an extra-quasi generic cover $f\colon S\to \mathbb P^2$ put

$$ \begin{equation*} E_{\overline R}= \sum_{p\in\operatorname{Sing} B}\sum_{j=1}^{k_p}E_{p,j,\overline R} \quad\text{and}\quad E_{\overline C_1}= \sum_{p\in\operatorname{Sing} B}\sum_{j=1}^{k_p}E_{p,j,\overline C_1}, \end{equation*} \notag $$
where by definition $E_{p,j,\overline R}=E_{p,j,\overline C_1}=0$ if $m_{p,j}=2$. It follows from (8)–(13) that
$$ \begin{equation} \begin{aligned} \, \notag (\overline R+E_{\overline R},\overline C_1+E_{\overline C_1})_X &= \sum_{p\in\operatorname{Sing} B}\sum_{j=1}^{k_p}(\overline R+E_{p,j,\overline R},\overline C_1+E_{p,j,\overline C_1})_{W_{p,j,j}''} \\ &\leqslant \sum_{p\in\operatorname{Sing} B} \sum_{j=1}^{k_p}(2 \delta_{p_{1,j}}(R_1)+c_{v,p}(B_j))=2 \delta(R)+c_{v}(B). \end{aligned} \end{equation} \tag{14} $$

§ 3. Proof of Theorem 3

Note that if $f\colon S\to \mathbb P^2$ is an extra-quasi-generic cover, then

$$ \begin{equation*} h_1^*(R_1)=\overline R+\overline C_1+E_{\overline R}+E_{\overline C_1}+E', \end{equation*} \notag $$
where $E'$ is a divisor with support in
$$ \begin{equation*} \rho^{-1}(g_{1,2}^{-1}(\operatorname{Sing} B))\cap\biggl(\bigcup_p \biggl(\bigcup_{j_1\neq j_2}W_{p,j_1,j_2}\biggr)\biggr). \end{equation*} \notag $$
We have
$$ \begin{equation} (E_{\overline R},E')_X=(E_{\overline C_1},E')_X=0, \end{equation} \tag{15} $$
since the divisors $E_{\overline R}$ and $E_{\overline C_1}$ have support in $\rho^{-1}(g_{1,2}^{-1}(\operatorname{Sing} B))\cap\bigl(\bigcup_p \bigcup_{j=1}^{k_p}W''_{p,j,j}\bigr)$. Since $\widetilde R\cap \bigl( \bigcup_p \bigcup_{j_1\neq j_2}W_{p,j_1,j_2}\bigr)=\varnothing$, we have
$$ \begin{equation} (\overline R,E')_X=0. \end{equation} \tag{16} $$

Set

$$ \begin{equation*} D=\overline R+E_{\overline R} \quad\text{and}\quad D_1=\overline C_1+E_{\overline C_1}+E'. \end{equation*} \notag $$
Then we have $h_1^*(R_1)=D+D_1$.

Lemma 9. If $f\colon S\to \mathbb P^2$ is an extra-quasi generic cover, then

$$ \begin{equation*} \begin{gathered} \, (D,D)_X = 2(3d+g-1+\delta_R)-(D,D_1)_X, \\ (D_1,D_1)_X = (N_2-2)(3d+g-1+\delta_R)-(D,D_1)_X \end{gathered} \end{equation*} \notag $$
and
$$ \begin{equation*} (D,D_1)_X \leqslant 2\delta(R)+c_v(B). \end{equation*} \notag $$

Proof. It follows from (14)–(16) that
$$ \begin{equation*} (D,D_1)_X= (\overline R+E_{\overline R},\overline C_1+E_{\overline C_1}+E')_X \leqslant 2\delta(R)+c_v(B). \end{equation*} \notag $$

Applying Lemmas 6 and 8 we obtain

$$ \begin{equation*} (h_1^*(R_1),D)_X=(h_1^*(R_1),\overline R+E_D)_X=\operatorname{deg} h_{1\mid \overline R}(R_1,R_1)_{S_1}=2(3d+g-1+\delta(R)). \end{equation*} \notag $$

On the other hand

$$ \begin{equation*} (h_1^*(R_1),D)_X= (D+D_1,D)_X= (D,D)_X+(D,D_1)_X, \end{equation*} \notag $$
and therefore $(D,D)_X= 2(3d+g-1+\delta_R)-(D,D_1)_X$.

In a similar way it follows from Lemma 6 that

$$ \begin{equation*} (h_1^*(R_1),h_1^*(R_1))_X=\operatorname{deg} h_1(R_1,R_1)_{S_1}=N_2(3d+g-1+\delta(R)). \end{equation*} \notag $$
Therefore,
$$ \begin{equation*} \begin{aligned} \, (D_1,D_1)_X &=(h_1^*(R_1),h_1^*(R_1))_X-2(h_1^*(R_1),D)_X+(D,D)_X \\ &=(N_2-2)(3d+g-1+\delta_R)-(D,D_1)_X, \end{aligned} \end{equation*} \notag $$
since $D_1=h_1^*(R_1)-D$. The lemma is proved.

The following proposition (an analogue of Theorem 1 in [5]) plays the crucial role in the proof of Theorem 3.

Proposition 4. Let $f_i\colon S_i\to \mathbb P^2$, $i=1,2$, be two extra-quasi generic covers of the projective plane branched along a curve $B\subset \mathbb P^2$, and let $\operatorname{deg} f_i=N_i$. If $\mathrm{pas}(f_1)=\mathrm{pas}(f_2)$, but $f_1$ and $f_2$ are not equivalent, then

$$ \begin{equation} N_i\leqslant \frac{4(3d+g-1+\delta(R))}{2(3d+g-1+\delta(R))-(2\delta(R)+c_v(B))}. \end{equation} \tag{17} $$

Proof. By (5) we have
$$ \begin{equation*} 2\delta(R)+c_v(B)<2(3d+g-1+\delta(R)). \end{equation*} \notag $$
Therefore, by Lemma 9
$$ \begin{equation*} (D,D)_X = 2(3d+g-1+\delta(R))- (D,D_1)_X\geqslant 2(3d+g-1+\delta(R))-(2\delta(R)+c_v(B))\,{>}\, 0. \end{equation*} \notag $$
Applying Hodge’s index theorem and Lemma 9 to the divisors $D$ and $D_1$ we obtain
$$ \begin{equation*} \begin{aligned} \, \begin{vmatrix} (D,D)_X & (D,D_1)_X \\ (D_1,D)_X & (D_1,D_1)_X \end{vmatrix} &= 2(N_2-2)(3d+g-1+\delta(R))^2 \\ &\qquad -N_2(3d+g-1+\delta(R))(D,D_1)_X \leqslant 0, \end{aligned} \end{equation*} \notag $$
and so
$$ \begin{equation*} \begin{aligned} \, &N_2[2(3d+g-1+\delta(R))-(2\delta(R)+c_v(B))] \\ &\qquad \leqslant N_2[2(3d+g-1+\delta(R))- (D,D_1)_X] \leqslant 4(3d+g-1+\delta(R)) . \end{aligned} \end{equation*} \notag $$
Renumbering $f_1$ and $f_2$, we obtain
$$ \begin{equation*} N_1[2(3d+g-1+\delta(R))-(2\delta(R)+c_v(B))] \leqslant 4(3d+g-1+\delta(R)) . \end{equation*} \notag $$
Thus, if $f_1\colon S_1\to \mathbb P^2$ and $f_2\colon S_2\to\mathbb P^2$ are nonequivalent extra-quasi generic covers branched over the same curve $B$, and if $\mathrm{pas}(f_1)=\mathrm{pas}(f_2)$, then their degrees satisfy inequalities (17). The proposition is proved.

To complete the proof of Theorem 3 it remains to apply the arguments used in [15].

If $f\colon S\to\mathbb P^2$ is a quasi-generic cover, then by Proposition 1 the self-intersection number of the canonical class $K_S$ of $S$ is equal to

$$ \begin{equation} K_S^2=9N-9d+g-1+\delta(R), \end{equation} \tag{18} $$
and by Proposition 2 the topological Euler characteristic $e(S)$ of $S$ is equal to
$$ \begin{equation} e(S)=3N+ 2(g-1)-c_{v}(B)=3N+ 2(g-1+\delta(R))-(2\delta(R)+c_{v}(B)). \end{equation} \tag{19} $$

Lemma 10. If $S$ satisfies the Bogomolov–Miaoka–Yau inequality $K^2_S\leqslant 3e(S)$, then

$$ \begin{equation} 2\delta(R)+c_v(B)\leqslant 3d+\frac{5}{3}(g-1+2\delta(R)). \end{equation} \tag{20} $$

Proof. We have the inequality
$$ \begin{equation*} 9N-9d+g-1+\delta(R)\leqslant 3(3N+ 2(g-1+\delta(R))- (2\delta(R)+c_v(B)), \end{equation*} \notag $$
which is equivalent to (20). The lemma is proved.

Therefore, in the case when $K^2_S\leqslant 3e(S)$ we have

$$ \begin{equation} \frac{4(3d+g-1+\delta_R)}{2(3d+g-1+\delta(R))-(2\delta(R)+c_v(B))}\leqslant 4+\frac{8(g-1+\delta(R))}{9d+(g-1+\delta(R))}<12. \end{equation} \tag{21} $$

Lemma 11. If $S$ does not satisfy the Bogomolov–Miaoka–Yau inequality $K^2_S\leqslant 3e(S)$, then

$$ \begin{equation} 2\delta(R)+c_v(B)\leqslant \frac{3}{2}(3d+g-1+\delta(R)). \end{equation} \tag{22} $$

Proof. If $S$ does not satisfy the Bogomolov–Miaoka–Yau inequality, then $S$ is an irregular linear surface (see, for example, [1]), and therefore $K^2_S\leqslant 2e(S)$. Applying (18) and (19) we obtain inequality (22). The lemma is proved.

Therefore, in the case when $S$ does not satisfy Bogomolov-Miaoka-Yau inequality, we have

$$ \begin{equation} \frac{4(3d+g-1+\delta(R))}{2(3d+g-1+\delta(R))-(2\delta(R)+c_v(B))} \leqslant 8. \end{equation} \tag{23} $$

Now Theorem 3 follows from inequalities (17), (21) and (23).

§ 4. On nontrivial germs of quasi-generic covers branched in curve germs with singular points of $\mathrm{ADE}$-type

In what follows we assume that $p\in \operatorname{Sing} B$ belongs to the set $\mathcal S_{\mathrm{ADE}}$ of singular points of $\mathrm{ADE}$-type and use the notation introduced above.

To prove Theorem 4 and Corollary 1 it suffices to prove that any quasi-generic cover $f\colon S\to\mathbb P^2$ with nontrivial germs branched in curve germs with singular points of $\mathrm{ADE}$-type are extra-quasi generic.

4.1. The local monodromy groups of a quasi-generic cover branched in a curve having $\mathrm{ADE}$ singularities

Recall that if $p\in \operatorname{Sing} B$ belongs to the set $\mathcal S_{\mathrm{ADE}}$, then $B$ can locally be given by one of the following equations:

(If a curve $B$ has a singularity of type $A_0$ at $p$, then $B$ is nonsingular at $p$.)

Lemma 12. Let $p\in \operatorname{Sing} B$ belong to the set $\mathcal S_{\mathrm{ADE}}$. Then the local monodromy group $G_{f,p}\subset \mathbb S_N$ of a quasi-generic cover $f\colon S\to\mathbb P^2$ branched in $B\subset \mathbb P^2$ is one of the following subgroups:

case (2): $\mathbb S_2$;

case (2, 2): $\mathbb S_2\times\mathbb S_2$;

case (2, 2, 2): $\mathbb S_2\times\mathbb S_2\times\mathbb S_2$;

case (3): $\mathbb S_3$;

case (3, 2): $\mathbb S_3\times\mathbb S_2$;

case (4): $\mathbb S_4$.

The proof follows from Lemma 2 (and its proof) since the multiplicity $\mu_p(B)$ at the point $p$ of the curve $B$ satisfies $1\leqslant \mu_p(B)\leqslant 3$.

It follows from Lemma 12 that there are six possible cases:

case (2): $f^{-1}(V)=\bigsqcup_{j=1}^{N-1}U_j$, $k_p=1$ and $m_{p,1}=2$;

case (2, 2): $f^{-1}(V)=\bigsqcup_{j=1}^{N-2}U_j$, $k_p=2$ and $m_{p,1}=m_{p,2}=2$;

case (2, 2, 2): $f^{-1}(V)=\bigsqcup_{j=1}^{N-3}U_j$, $k_p=3$ and $m_{p,1}=m_{p,2}=m_{p,3}=2$;

case (3): $f^{-1}(V)=\bigsqcup_{j=1}^{N-2}U_j$, $k_p=1$ and $m_{p,1}=3$;

case (3, 2): $f^{-1}(V)=\bigsqcup_{j=1}^{N-3}U_j$, $k_p=2$, $m_{p,1}=3$ and $m_{p,2}=2$;

case (4): $f^{-1}(V)=\bigsqcup_{j=1}^{N-3}U_j$, $k_p=1$ and $m_{p,1}=4$.

4.1.1.

In case (2), by Lemma 3, $f_{\mid U_{p,1}}\colon U_{p,1}\to V_p$ is a two-sheeted germ over a point $p\in B$ of the quasi-generic cover $f\colon S\to \mathbb P^2$ branched in a nonsingular curve germ $(B_1,p)$. By Definition 2 the germ $f_{\mid U_{p,1}}\colon U_{p,1}\to V_p$ has the extra property over the point $p$.

4.1.2.

In case (2, 2), by Lemma 3 the intersection $f^{-1}(V_p)\cap R$ lies in $ U_{p,1}\cup U_{p,2}$ and there are local coordinates $z_j,w_j$ in $U_{p,j}$ and $u_j,v_j$ in $V_p$, $j=1,2$, such that $f_{\mid U_{p,j}}\colon U_{p,j}\to V_p$ is given by the functions $u_j=z_j$ and $v_j=w_j^2$. The intersection $U_{p,j}\cap R$ is defined by the equation $w_j=0$ for $j=1,2$ and $B\cap V$ is defined by $v_1v_2=0$, where the irreducible branches $B_j=f(R\cap U_{p,j})$ defined by $v_j=0$, $j=1,2$, are nonsingular and $B_1\neq B_2$, since $\operatorname{deg} f_{\mid R}=1$. Consequently, $p\in B\cap V$ is a singular point of $B$ of singularity type $A_{2n-1}$, where $n$ is the intersection number $(B_1,B_2)_p$ of the branches $B_1$ and $B_2$ at the point $p$.

Note that by Definition 2, the nontrivial germs of $f_{\mid U_{p,j}}\colon U_{p,j}\to V_p$ have the extra property over the point $p$.

4.1.3.

Case (2, 2, 2) is similar to case (2, 2). In this case, by Lemma 3 there are local coordinates $z_j,w_j$ in $U_{p,j}$ and $u_j,v_j$ in $V$, $j=1,2,3$, such that $f_{\mid U_{p,j}}\colon U_{p,j}\to V_p$ is given by the functions $u_j=z_j$ and $v_j=w_j^2$. The intersection $f^{-1}(V_p)\cap R\subset U_{p,1}\cup U_{p,2}\cup U_{p,3}$, and the intersection $U_{p,j}\cap R$ is given by the equation $w_j=0$ for $j=1,2,3$. The intersection $B\cap V_p$ is given by $v_1v_2v_3=0$, where the irreducible branches $B_j$ of $V_p\cap B$, given by $v_j=0$, $j=1,2,3$, are nonsingular and $B_{j_1}\neq B_{j_2}$ for $1\leqslant j_1< j_2\leqslant 3$, since $\operatorname{deg} f_{\mid R}=1$. Since $p\in V_p\cap B$ is a singular point of $B$ of some $\mathrm{ADE}$-type and $V_p\cap B$ consists of three irreducible components, the singularity type of $B$ at $p$ is $D_{2n+2}$ for some $n\geqslant 1$.

Note that by Definition 2 the nontrivial germs of $f_{\mid U_{p,j}}\colon U_{p,j}\to V_p$ have an extra property over the point $p$.

4.1.4.

Case (3) was considered in [14], and in this case we have the following result.

Proposition 5. If $\operatorname{deg} f_{\mid U_{p,1}}=3$ for the nontrivial germ $f_{\mid U_{p,1}}\colon U_{p,1}\to V_p$ of a quasi-generic cover $f\colon S\to\mathbb P^2$ branched in a curve germ $(B_1,p)\subset (B,p)\subset V_p$, then the singularity type of $(B_1,p)$ is $A_{3n-1}$ for some $n\in \mathbb N$.

If $U_{p,1}$ and $V_p$ are irreducible germs of nonsingular surfaces and $f_{\mid U_{p,1}}\colon {U_{p,1}\,{\to}\, V_p}$ is a finite cover germ branched in a curve germ $(B_1,p)\subset V_p$ with singular point of type $A_{3n-1}$, $n\geqslant 1$, then $\operatorname{deg} f_{\mid U_{p,1}}=\mu_p(B_1)+1=3$ and the germ $f_{\mid U_{p,1}}$ has an extra property over $p$.

Proof. It follows from Theorem 2 in [12] (see also [14]) that there are local coordinates $z,w$ in $U_{p,1}$ and $u,v$ in $V_p$ such that $f_{\mid U_{p,1}}\colon U_{p,1}\to V_p$ is given by
$$ \begin{equation*} \begin{gathered} \, u=z, \\ v=w^3-3z^nw \end{gathered} \end{equation*} \notag $$
for some $n\in \mathbb N$. The ramification curve germ $R\cap U_{p,1}$ is given by $w^2-z^n=0$, that is, $R$ has a singular point $p_1=f^{-1}(p)\cap U_{p,1}$ of singularity type $A_{n-1}$. Therefore, the $\delta$-invariant of the curve $R$ at $p_1$ is
$$ \begin{equation} \delta_{p_1}(R)=k \quad\text{if } n=2k-1 \text{ or } n=2k. \end{equation} \tag{24} $$

By Claim 4 in [14] the branch curve $V_p\cap B$ is defined by the equation

$$ \begin{equation*} v^2-4u^{3n}=0, \end{equation*} \notag $$
that is, $p$ is a singular point of $B$ of singularity type $A_{3n-1}$ and the number of virtual cusps of $B$ at $p$ is
$$ \begin{equation} c_{v,p}(B)=0 \quad\text{for } n=2k-1 \quad\text{and}\quad c_{v,p}(B)=1 \quad\text{for } n=2k. \end{equation} \tag{25} $$
It follows from (24) and (25) that
$$ \begin{equation} n=2\delta_{p_{1}}(R)+c_{v,p}(B). \end{equation} \tag{26} $$

It was shown in [14] (see § 2.2.3 in [14]) that $p_{1,1}''=g_{1,2}^{-1}(p)\cap \widetilde W_{p,1,1}''$ is a singular point of type $A_{n-1}$ of $\widetilde W_{p,1,1}''$,

$$ \begin{equation*} g_{i\mid\widetilde W_{p,1,1}''}^{-1}(R_1\cap U_{p,1,1})=(\widetilde R\cap \widetilde W_{p,1,1}'')\cup (\widetilde C_1\cap \widetilde W_{p,1,1}''), \end{equation*} \notag $$
and the curves $g_{1,2}^{-1}(V_p)\cap\widetilde R$ and $g_{1,2}^{-1}(V_p)\cap\widetilde C_1$ intersect only at $p_{1,1}''$. The resolution of singularities $\rho\colon W_{p,1,1}\to\widetilde W_{p,1,1}$ described in [14] has the following properties: in the neighbourhood $W_{p,1,1}''=\rho^{-1}(\widetilde W_{p,1,1}'')$ the divisor has the form
$$ \begin{equation} h_1^*(U_1\cap R_1)= \overline R+\overline C_1 +2E_{p}, \end{equation} \tag{27} $$
where $E_{p}$ is a divisor with support in $\rho^{-1}(g_{1,2}^{-1}(p)\cap W''_{p,1,1})$,
$$ \begin{equation} (E_p,E_p)_X=-n, \qquad (\overline R,E_p)_{X}=(\overline C_1, E_p)_{X}=n \end{equation} \tag{28} $$
and
$$ \begin{equation} (\overline R,\overline C_1)_{W_{p,1,1}}=0. \end{equation} \tag{29} $$

Put $E_{p,\overline R}=E_{p,\overline C_1}=E_{p}$. Then it follows from (26)–(29) that a nontrivial germ $f_{\mid U_{p,1}}\colon U_{p,1}\to V_p$ of a quasi-generic cover branched in a germ of a curve $(B_1,p)\subset V_p$ with singular point of type $A_{3n-1}$, $n\geqslant 1$, has an extra property over the point $p$.

To complete the proof of Proposition 5 it remains to notice that $\mu_p(B_1)=2$ for a singular point of type $A_{3n-1}$. Therefore, by Lemma 2 (and its proof) we have $\operatorname{deg} f_{\mid U_{p,1}}\leqslant 3$ and by Lemma 3 the case when the germ of a two-sheeted cover $f_{\mid U_{p,1}}\colon U_{p,1}\to V_p$ is branched in a curve germ $(B_1,p)\subset V_p$ having a singular point of type $A_{3n-1}$, $n\geqslant 1$, is impossible since $U_{p,1}$ is the germ of a nonsingular surface. The proposition is proved.

4.1.5.

Case (3, 2) is a combination of cases (2) and (3).

Proof. By Proposition 5, $f_{\mid U_{p,1}}\colon U_{p,1}\to V_p$ is a three-sheeted cover branched in a curve germ $(B_1,p)$ of singularity type $A_{3n-1}$, and by Lemma 3, $f_{\mid U_{p,2}}\colon U_{p,2}\to V$ is a two-sheeted cover branched in a smooth curve germ $(B_2,p)$. Therefore, we have $(B,p)=(B_1,p)\cup (B_2,p)$.

Since $B$ has some $\mathrm{ADE}$ singularity type at $p$, we have two possibilities depending on the intersection number $(B_1,B_2)_p$, which can take value $2$ or $3$. So the singularity type of $B$ at $p$ is either $D_{3n+2}$ if $(B_1,B_2)_p=2$ or $E_7$ if $(B_1,B_2)_p=3$.

Statement (2) of Proposition 6 follows directly from Definition 2 and Proposition 5. The proposition is proved.

4.1.6.

In case (4) we have $f^{-1}(V_p)\cap R=U_{p,1}\cap R$ and assume that the singularity type of $V_p\cap B$ at $p$ is some $\mathrm{ADE}$-type. It follows from Theorem 1 in [11] and its proof that up to equivalence, there is a unique four-sheeted germ $f_{\mid U_{p,1}}\colon U_{p,1}\to V_p$ of a quasi-generic cover branched in a curve germ $(B_1,p)\subset V_p\cap B$ with singular point of some $\mathrm{ADE}$-type. It is only the case when the singularity type of $(B_1,p)$ is $E_6$ (the case $F_{4_2,0,1}$ in the notation used in [11]) and when there are local coordinates $z,w$ in $U_{p,1}$ and $u,v$ in $V_p$ such that the germ $f_{\mid U_{p,1}}\colon U_{p,1}\to V_p$ is given by

$$ \begin{equation*} u=z, \qquad v=w^4-4zw. \end{equation*} \notag $$

Proposition 7. If $\operatorname{deg} f_{\mid U_{p,1}}=4$ for a nontrivial germ $f_{\mid U_{p,1}}\colon U_{p,1}\to V_p$ of a quasi-generic cover $f\colon S\to\mathbb P^2$ branched in a curve germ $(B_1,p)\subset (B,p)\subset (V,p)$ with singular point of some $\mathrm{ADE}$-type, then the singularity type of $(B_1,p)$ at $p$ is $E_6$.

A nontrivial germ $f_{\mid U_{p,1}}\colon U_{p,1}\to V_p$ of a quasi-generic cover branched in a curve germ $(B_1,p)\subset V_p$ with singular point of type $E_6$ has the extra property over the point $p$ and $\operatorname{deg} f_{\mid U_{p,1}}=4$.

Proposition 7 is a particular case of Proposition 11, which is proved in § 5.

Theorem 4 and Corollary 1 follow directly from Theorem 3 and Propositions 5–7.

§ 5. Proof of Theorem 5

5.1. The properties of dualizing covers of the plane which are associated with curves immersed in the plane

Let $\iota\colon C\to {\mathbb P}^2$ be a morphism of the smooth irreducible reduced projective curve $C$ to the projective plane such that $\iota_{\mid C} \colon C\to \iota(C)$ is a birational morphism, $\operatorname{deg} \iota(C)=d\geqslant 2$.

Consider a point $p\in C$ and its image $P=\iota(p)\in {\mathbb P}^2$. We choose a local parameter $w$ in a complex-analytic neighbourhood $U\subset C$ of $p$, where $U\simeq \{{w\in \mathbb C}\mid |w|<\varepsilon\}$, and we choose homogeneous coordinates $(x_1,x_2,x_3)$ in ${\mathbb P}^2$ so that $P=(0,0,1)$ and the morphism $\iota$ is given by

$$ \begin{equation} x_1=w^{d_p}+\sum_{i=d_p+1}^{\infty} a_iw^i, \qquad x_2=-d_pw^{r_p}\quad\text{and} \quad x_3=-1, \end{equation} \tag{30} $$
where $a_{d_p+1}\neq 0$ and $d_p>r_p\geqslant 1$. The number $r_p$ is called the multiplicity of singularity of the germ $\iota(U)$ of $\iota(C)$ at the point $P=\iota(p)$, the straight line $l_p=\{x_1=0\}$ is called the tangent line to the curve $\iota(C)$ at $P$, and the number $d_p$ is called the tangent multiplicity of the curve germ $\iota(U)$ at $P$. If $\iota$ is an immersion of $C$, then $r_p=1$ for all $p\in C$.

Let $B\subset \widehat{\mathbb P}^2$ be the dual curve to the curve $\iota(C)$ (the curve $B$ consists of lines $l_p\in \widehat{\mathbb P}^2$, $p\in C$, tangent to the curve $\iota(C)$). The correspondence graph between the curves $\iota(C)$ and $B$ is a curve $\check C$ in ${\mathbb P}^2\times \widehat{\mathbb P}^2$ (the so-called Nash blowup of $\iota(C)$), which lies in the incidence variety $I=\{(P,l)\in {\mathbb P}^2\times \widehat{\mathbb P}^2\mid P\in l\}$:

$$ \begin{equation*} \check C=\{(\iota(p),l_p)\in I\mid p\in C \text{ and } l_p \text{ is a tangent line to } \iota(C)\text{ at } \iota(p)\in C\}. \end{equation*} \notag $$

Below we denote by $L_P\subset \widehat{\mathbb P}^2$ the line dual to a point $P\in \iota(C)\subset \mathbb P^2$.

Let $\mathrm{pr}_1\colon \mathbb P^2\times \widehat{\mathbb P}^2\to \mathbb P^2$ and $\mathrm{pr}_2\colon \mathbb P^2\times \widehat{\mathbb P}^2\to \widehat{\mathbb P}^2$ be the projections onto the factors, let $X'=\mathrm{pr}_1^{-1}(\iota(C))\cap I$, and let $h\colon X'\to \widehat{\mathbb P}^2$ be the restriction of the projection $\mathrm{pr}_2$ to $X'$. In is obvious that $h^{-1}(l)$ consists of the points $(P,l)\in\mathbb P^2\times \widehat{\mathbb P}^2$ such that $P\in \iota(C)\cap l$ and so $\operatorname{deg} h=\operatorname{deg} \iota(C)=d$.

Let $\nu\colon X\to X'$ be the normalization of the surface $X'$. The morphism $f_{\iota(C)}=h\circ \nu\colon X\to \widehat{\mathbb P}^2$ is called the dualizing cover of the plane associated with the curve $\iota(C)\subset \mathbb P^2$. We have $\operatorname{deg} f_{\iota(C)}=d$. It is easy to see that $X$ is isomorphic to the fibre product $C\times_{\iota(C)}X'$ of the morphism $\iota\colon C\to \iota(C)$ and the projection $\mathrm{pr}_1\colon X'\to \iota(C)$. The projection $\mathrm{pr}_1\colon X'\to \iota(C)$ defines the structure of a ruled surface on $X'$ and this structure induces a ruled structure on $X$ over $C$. Note that $\widetilde {C}=\nu^{-1}(\check C)\subset X$ is a section of this ruled structure, and the image $f_{\iota(C)}(F_p)$ of a fibre $F_p$ is a line $L_{\iota(p)}\subset \widehat{\mathbb P}^2$ dual to the point $\iota(p)\in \iota(C)\subset \mathbb P^2$.

It follows from the proof of Theorem 1 in [9] that the ramification curve of the cover $f_{\iota(C)}$ is $R=\widetilde C\cup \widetilde F$, where $\widetilde F= \bigcup_{r_p\geqslant 2} F_p$ and the union is taken over all points $p\in C$ for which the multiplicity at the point $P=\iota(p)$ satisfies $r_p\geqslant 2$. The restriction of $f_{\iota(C)}$ to each component of $R$ is a birational morphism onto the image of this component and $f_{\iota(C)}$ branches with multiplicity two at the general point of $\widetilde C$ and with multiplicity $r_p$ at the general point of $F_p\subset R$. The branch curve of $f_{\iota(C)}$ is $B=\widehat C\cup \widehat L$, where $\widehat L= \bigcup_{r_p\geqslant 2} L_{\nu(p)}$ and the union is taken over all $p\in C$ for which $r_p\geqslant 2$ at the point $P=\iota(p)$.

Below we assume that $\iota\colon C\to\mathbb P^2$ is an immersion given by the parametrization (30) in a neighbourhood $U\subset C$ of a point $p\in C$,

$$ \begin{equation} x_1=w^{n+1}+\sum_{i=n+2}^{\infty} a_iw^i, \qquad x_2=-(n+1)w, \qquad x_3=-1, \end{equation} \tag{31} $$
where $d_p=n+1$ is the tangent multiplicity of the curve germ $\iota(U)$ at the point $P$. If $(y_1,y_2,y_3)$ are the homogeneous coordinates in $\widehat{\mathbb P}^2$ dual to the homogeneous coordinates $(x_1,x_2,x_3)$ in $\mathbb P^2$, then the surface $X$ in the neighbourhood $U\times \widehat{\mathbb P}^2\subset C\times \widehat{\mathbb P}^2$ is defined by the equation
$$ \begin{equation*} y_1\biggl(w^{n+1}+\sum_{i=n+2}^{\infty} a_iw^i\biggr)-(n+1)y_2w-y_3=0. \end{equation*} \notag $$

In particular, $X_{U,1}:=X\cap (U\times \widehat{\mathbb P}^2)$ lies in $U\times {\mathbb C}^2$, where $\mathbb C^2=\{y_1\neq 0\}$ is an affine plane in $\widehat{\mathbb P}^2$, and $X_{U,1}$ is defined by the equation

$$ \begin{equation} w^{n+1}+\sum_{i=n+2}^{\infty} a_iw^i -(n+1)zw-v=0, \end{equation} \tag{32} $$
where $z=y_2/y_1$ and $v=y_3/y_1$. Consequently, $X$ is a smooth surface and $(z,w)$ are local coordinates in $X_{U,1}$.

The restriction $f_{\iota(C)\mid X_{U,1}}\colon X_{U,1}=U\times \mathbb C^2\to \mathbb C^2$ of $f_{\iota(C)}$ to $X_{U,1}$ is the restriction of the projection $(z,w,v)\mapsto (z,v)$, and therefore it is given by the functions

$$ \begin{equation} u = z\quad\text{and}\quad v = w^{n+1}+ \sum_{i=n+2}^{\infty} a_iw^i -(n+1)zw. \end{equation} \tag{33} $$
It is easy to see that $\operatorname{deg} f_{\iota(C)\mid X_{U,1}}=n+1$ and the Jacobian of $f_{\iota(C)\mid X_{U,1}}$ is
$$ \begin{equation*} J(f_{\iota(C)\mid X_U})=(n+1)\biggl(w^{n}+\frac{1}{n+1}\sum_{i=n+2}^{\infty}i a_it^{i-1} -z\biggr), \end{equation*} \notag $$
so that $f_{\iota(C)\mid X_{U,1}}$ is ramified with multiplicity $2$ in the germ $R_p$ of the ramification curve $R_{f_{\iota(C)}}$ defined by the equation
$$ \begin{equation} w^{n}+\frac{1}{n+1}\sum_{i=n+2}^{\infty} ia_iw^{i-1} -z=0. \end{equation} \tag{34} $$
It follows from (33) and (34) that the germ $(B,l_p)=(f_{\iota(C)\mid X_{U,1}}(R_p),l_p)$ of the branch curve $B_{f_{\iota(C)}}$ is given parametrically by
$$ \begin{equation} u= w^{n}+\frac{1}{n+1}\sum_{i=n+2}^{\infty} ia_iw^{i-1}\quad\text{and}\quad v = -nw^{n+1}- \sum_{i=n+2}^{\infty}(i-1) a_iw^i. \end{equation} \tag{35} $$

5.2. Deformations of the germs of dualizing covers

5.2.1.

Consider the family

$$ \begin{equation*} F\colon \mathcal U_{l_p}=U_{l_p}\times \Delta \to \mathcal V_{l_p}=V_{l_p}\times \Delta \end{equation*} \notag $$
of germs of the finite covers given by
$$ \begin{equation} u = z \quad\text{and}\quad v = w^{n+1}+ (1-\tau)\sum_{i=n+2}^{\infty} a_iw^i -(n+1)zw, \end{equation} \tag{36} $$
where $\Delta=\{\tau\in \mathbb C\mid |\tau|<1+\varepsilon\}$. For $\tau_0\in\Delta$ set $V_{l_p,\tau_0}=V_{l_p}\times \{\tau=\tau_0\}$, $U_{l_p,\tau_0}=U_{l_p}\times \{\tau=\tau_0\}$ and $f_{\tau_0}=F_{\mid U_{l_p,\tau_0}}\colon U_{l_p,\tau_0}\to V_{l_p,\tau_0}$.

Proposition 8. A family $F\colon \mathcal U_{l_p}\to \mathcal V_{l_p}$ of finite covers given by (36) is a strong deformation of the cover $f_{0}\colon U_{l_p,0} \to V_{l_p,0}$ given by (33).

Proof. It it is easy to see that the branch curve germs $(B_{\tau_0},l_p)$ of finite covers $f_{\tau_0}\colon U_{l_p,\tau} \to V_{l_p,\tau}$, $\tau_0\in \Delta$, are given parametrically ($\tau=\tau_0$) by
$$ \begin{equation} u= w^{n}+\frac{1-\tau}{n+1}\sum_{i=n+2}^{\infty} ia_iw^{i-1}\quad\text{and}\quad v = -nw^{n+1}-(1-\tau)\sum_{i=n+2}^{\infty}(i-1) a_iw^i. \end{equation} \tag{37} $$

Note that $\mathcal U_{l_p}$ is a smooth complex threefold and the differential form $F^*(d\tau)$ is distinct from zero at each point in $\mathcal U_{l_p}$. Therefore, by Definitions 2 and 4 in [13] (cf. [18] and [17]), it suffices to show that the singular points of the family of curve germs $B_{\tau_0}$ can be resolved simultaneously up to divisors with normal crossings by a sequence of $\sigma$-processes.

Let $\sigma_1\colon V_{l_p,\tau_0}'\to V_{l_p,\tau_0}$ be the $\sigma$-process with centre $l_p\times \{\tau=\tau_0\}$. In one of two neighbourhoods with coordinates $u_1$ and $v_1$ covering the surface germ $V_{l_p,\tau_0}'$ the map $\sigma_1$ is given by $u=u_1$ and $v=u_1v_1$ and the proper inverse image of the curve germs $\sigma_1^{-1}(B_{\tau_0})\subset V_{l_p,\tau_0}'$ are given parametrically by

$$ \begin{equation} \begin{gathered} \, u_1= w^{n}+\frac{1-\tau_0}{n+1}\sum_{i=n+2}^{\infty} ia_iw^{i-1}, \\ v_1 = -w\frac{n-(1-\tau_0) \sum_{i=n+2}^{\infty}(i-1) a_iw^{i-n-1}}{1+((1-\tau_0)/(n+1))\sum_{i=n+2}^{\infty} ia_iw^{i-n-1}}. \end{gathered} \end{equation} \tag{38} $$
It follows from (38) that the germs $\sigma_1^{-1}(B_{\tau_0})$, $\tau_0\in\Delta$, are nonsingular, they touch the exceptional curve of $\sigma_1$ (given by $u_1=0$) and touch each other with multiplicity $n$.

Therefore, we need to perform $n$ $\sigma$-processes more to obtain as a result a divisor with normal crossings whose dual weighted graph is shown in Figure 1, where the proper inverse image of the curve germ $B_{\tau_0}$ is denoted by the same letter and the proper inverse image of the exceptional curve of the $i$th $\sigma$-process is denoted by $E_i$ for $i=1,\dots,n+1$. Proposition 8 is proved.

5.2.2.

By definition, a pair $(\mathcal V_p,\mathcal C)$, where $\mathcal C$ is a surface in $\mathcal V_p=V_p\times \Delta$ such that $\operatorname{Sing} \mathcal C=\{p\}\times \Delta$, is a strong equisingular deformation of the curve germs

$$ \begin{equation*} C_{\tau_0}=\mathrm{pr}_2^{-1}(\tau_0)\cap \mathcal C\subset V_{p,\tau_0}=\mathrm{pr}_2^{-1}(\tau_0), \qquad \tau_0\in \Delta, \end{equation*} \notag $$
if there is a finite sequence of monoidal transformations $\overline{\sigma}_i\colon \mathcal V_{p,i}\to \mathcal V_{p,i-1}$ (here $\mathcal V_{p,0}=\mathcal V_p$) with centres in sections of the projection to $\Delta$ such that $\overline{\sigma}^{-1}(\mathcal C)$ is a divisor with normal crossings in $\mathcal V_{p,n}$, where $\overline{\sigma}=\overline{\sigma}_1\circ\dots\circ\overline{\sigma}_n\colon \mathcal V_{p,n}\to\mathcal V_p$.

The following is obvious.

Lemma 13. If $\mathcal C=\mathcal C_{1}\cup \dots \cup \mathcal C_{n}$ is a union of irreducible surfaces and $(\mathcal V_p,\mathcal C)$ is a strong equisingular deformation of curve germs $C_{\tau_0}\subset V_{p,\tau_0}$, then

It is easy to show that if $\mathrm{pr}_2\colon (\mathcal V_p,\mathcal C)\to \Delta$ is a strong equisingular deformation of curve germs, then (reducing the neighbourhood $V_p$ if necessary) the map $\mathrm{pr}_2 \colon (\mathcal V_p,\mathcal C)\to \Delta$ is a $C^{0}$-locally trivial fibration of the pair $(\mathcal V_{p,\tau_0},\mathcal C_{\tau_0})$. In particular, there is a natural isomorphism $\pi_1(\mathcal V_p\setminus \mathcal C)\simeq \pi_1(V_{p,\tau_0}\setminus C_{\tau_0})$.

By the Riemann–Stein theorem (see [16]), if $f_p\colon U_p\to V_p$ is the germ of an $N$-sheeted finite cover of smooth surfaces and $\mathrm{pr}_2\colon (\mathcal V_p,\mathcal B)\to \Delta$ is a strong equisingular deformation of the branch curve germ $(B,p):=B_{\tau_0}\subset V_{p,\tau_0}=V_p$ of $f_p$, then the monodromy homomorphism

$$ \begin{equation*} F_*\colon \pi_1(\mathcal V_p\setminus \mathcal B)\simeq \pi_1(V_{p,\tau_0}\setminus B_{\tau_0})\xrightarrow{f_{p*}} G_{f_p,p}\subset \mathbb S_N \end{equation*} \notag $$
defines an $N$-sheeted finite cover $F\colon \mathcal U_p\to \mathcal V_p=V_p\times \Delta$ branched in $\mathcal B\subset\mathcal V_p$.

Proposition 9. The cover $F\colon \mathcal U_p\to \mathcal V_p=V_p\times \Delta$ is a strong deformation of the germ $f_p\colon U_p\to V_p$.

The map $\mathrm{pr}_2\circ F\colon (\mathcal U_p,F^{-1}(\mathcal B))\to \Delta$ is a strong equisingular deformation of the inverse image $f_p^{-1}(B)$.

The proof is literally the same as the proof of Theorem 3 in [11].

In the notation used in § 2.3.3, let $f_{\mid U_{p,1}}\colon U_{p,1}\to V_p$ branched in $(B_1,p)\subset V_p$ be a germ of a finite cover $f\colon S\to \mathbb P^2$ belonging to $\mathcal F_{(2)}$, and let $\operatorname{deg} f_{\mid U_{p,1}}=n+1$.

Proposition 10. Let $F\colon \mathcal U_{p,1}\to \mathcal V_{p}$ be a strong deformation of a germ $f_{\mid U_{p,1}} \colon U_{p,1}\to V_p$ of a cover $\{f\colon S\to\mathbb P^2\} \in\mathcal F_{(2)}$, $\operatorname{deg} f_{\mid U_{p,1}}=n+1$. Assume that $f_{\mid U_{p,1}}$ satisfies the following conditions:

Then for each $\tau_0\in\Delta$, the germs $f_{\tau_0}\colon U_{p,\tau_0}\to V_{p,\tau_0}$ satisfy condition (IV) and they have an extra property over the point $p$.

Proof. The claim that the germs $f_{\tau_0}\colon U_{p,\tau_0}\to V_{p,\tau_0}$ satisfy condition (IV) follows directly from assertion (ii) in Lemma 13.

We use the notation introduced in § 2.3.3.

To prove that the germs $f_{\tau_0}\colon U_{p,\tau_0}\to V_{p,\tau_0}$ have an extra property over the point $p$, let us consider the cover $g_{1\mid \widetilde W_{p,1,1}''}\colon \widetilde W_{p,1,1}''\to U_{p,1,1}$. This is an $n$-sheeted cover branched along $C_1\cap U_{p,1,1}$, and it is ramified along $\widetilde C_2\cap \widetilde W_{p,1,1}''$. To describe the monodromy homomorphism

$$ \begin{equation*} g_{1\mid \widetilde W_{p,1,1}''*}\colon \pi_1(U_{p,1,1}\setminus C_1, q_1)\to \mathbb S_n \end{equation*} \notag $$
we identify the group $\pi_1(U_{p,1,1}\setminus f_{1\mid U_{p,1,1}}^{-1}(B_1), q_1)$ with the stabilizer
$$ \begin{equation*} \begin{aligned} \, \pi_1(V_p\setminus B_1,q)^{q_1} &=\{\gamma\in \pi_1(V_p\setminus B_1)\mid f_{1\mid U_{p,1,1}*}(\gamma)(q_1)=q_1 \} \\ &\simeq \pi_1(U_{p,1,1}\setminus f_{\mid U_{p,1,1}}^{-1}(B_1), q_1) \end{aligned} \end{equation*} \notag $$
of $q_1\in f_1^{-1}(q)$. Let $\iota_*\colon \pi_1(U_{p,1,1}\setminus f_{\mid U_{p,1,1}}^{-1}(B_1),q_1)\to \pi_1(U_{p,1,1}\setminus C_1,q_1)$ be the epimorphism induced by the embedding $\iota \colon U_{p,1,1}\setminus f_{1\mid U_{p,1,1}}^{-1}(B_1)\to U_{p,1,1}\setminus C_1$. Then the monodromy homomorphism $g_{1\mid \widetilde W_{p,1,1}''*}\colon \pi_1(U_{p,1,1}\setminus C_1, q_1)\to \mathbb S_n$ is defined by the action of the group $\pi_1(U_{p,1,1}\setminus C_1,q_1)$ on the set $g_{1\mid \widetilde W_{p,1,1}''}^{-1}(q_1)=\{(q_1,q_2),\dots,(q_1,q_{n+1})\}$ given as follows for $\iota_*(\gamma)\in \pi_1(U_{p,1,1}\setminus C_1, q_1)$:
$$ \begin{equation} g_{1*}(\iota_*(\gamma))((q_1,q_j))=(q_1,f_*(\gamma)(q_j)), \end{equation} \tag{39} $$
where $\gamma\in \pi_1(U_{p,1,1}\setminus f^{-1}(B), q_1)^{q_1}$.

Note that $g_{1\mid \widetilde W_{p,1,1}''*}\colon \pi_1(U_{p,1,1}\setminus C_1, q_1)\to \mathbb S_n$ is an epimorphism, since $f_{1\mid U_{p,1,1}*}$: $\pi_1(V_p\setminus B_1,q)\to \mathbb S_{n+1}$ is an epimorphism.

The neighbourhood $U_{p,1,1}$ is the germ of a nonsingular surface. Therefore, it is simply connected and by the Zariski–van Kampen theorem, $\pi_1(U_{p,1,1}\setminus C_1, q_1)$ is generated by geometric generators. It is easy to see that under the identification of the group $\pi_1(U_{p,1,1}\setminus f_{1\mid U_{p,1,1}}^{-1}(B_1), q_1)$ with the stabilizer $\pi_1(V_p\setminus B_1,q)^{q_1}$ the geometric generators belonging to $\pi_1(V_p\setminus B_1,q)^{q_1}$ are the geometric generators of $\pi_1(U_{p,1,1}\setminus f_1^{-1}(B_1), q_1)$ and for $\gamma\notin \ker \iota_*$ the $\iota_*(\gamma)\in \pi_1(U_{p,1,1}\setminus C_1, q_1)$ are geometric generators of $\pi_1(U_{p,1,1}\setminus C_1, q_1)$. Therefore, by (39), for geometric generators ${\overline{\gamma}\in \pi_1(U_{p,1,1}\setminus C_1, q_1)}$ their images $g_{1*}(\overline{\gamma})\in \mathbb S_n$ are transpositions.

Similarly, the monodromy homomorphism

$$ \begin{equation*} g_{1,2\mid \widetilde W_{p,1,1}''*}\colon \pi_1(\widetilde W_{p,1,1}''\setminus g_{1\mid \widetilde W_{p,1,1}''}^{-1}(B_1))\to \mathbb S_{n(n+1)} \end{equation*} \notag $$
of the cover $g_{1,2\mid \widetilde W_{p,1,1}''}\!\colon \widetilde W_{p,1,1}''\!\to\! V_{p}$ is defined by the action of the group $\pi_1({V_{p}\!\setminus\! B_1},q)$ on the set $g_{1,2\mid \widetilde W_{p,1,1}''}^{-1}(q)=\{(q_{j_1},q_{j_2})\in \{q_1,\dots,q_{n+1}\}^2\mid q_{j_1}\neq q_{j_2}\}$ given as follows:
$$ \begin{equation} g_{1,2\mid \widetilde W_{p,1,1}''*}(\gamma)((q_{j_1},q_{j_2}))=(f_{1\mid \widetilde U_{p,1}*}(\gamma)(q_{j_1}),f_*(\gamma)(q_{j_2})). \end{equation} \tag{40} $$

Consider a deformation $F\colon \mathcal U_{p,1,1}=\mathcal U_{p,1}\to \mathcal V_p$ branched in $\mathcal B\subset \mathcal V_p$, where $\mathrm{pr}_2\colon (\mathcal V_p,\mathcal B)\to\Delta$ is a strong equisingular deformation of the curve germ $(B_{\tau_0},p_{\tau_0})\subset V_{p,\tau_0}$. It follows from Proposition 9 and Lemma 13 that the monodromy homomorphisms $g_{1\mid \widetilde W_{p,1,1}''*}\colon \pi_1(U_{p,1,1}\setminus C_1)\to \mathbb S_{n}$ and $g_{1,2\mid \widetilde W_{p,1,1}''*}\colon \pi_1(V_{p}\setminus B_1)\to \mathbb S_{n(n+1)}$, defined uniquely by (39) and (40), define strong deformations $G_1\colon \widetilde{\mathcal W}''_{p,1,1}\to \mathcal U_{p,1,1}$ of the germ $g_1\colon \widetilde W''_{p,1,1}\to U_{p,1,1}$ and $G_{1,2}\colon \widetilde{\mathcal W}''_{p,1,1}\to \mathcal V_{p}$ of the germ $g_{1,2}\colon \widetilde W''_{p,1,1}\to V_{p}$. It is easy to see that the deformations $G_1$ and $G_{1,2}$ can be included in the following commutative diagram

and to complete the proof of Proposition 10 it suffices to apply statements (i) and (iii) from Lemma 13 and Proposition 9 to the surface $G_{1,2}^{-1}(\mathcal B)$ taking into account that the map $F$ is ramified in a strong equisingular deformation ${\mathcal R_1\subset \mathcal U_{p,1,1}}$ of the ramification curve $R_1\cap U_{p,1,1}$ of the cover $f_1\colon U_{p,1,1}\to V_p$ and ${G_1^{-1}(\mathcal R_1)\subset G_{1,2}^{-1}(\mathcal B)}$. Proposition 10 is proved.

5.3. The end of the proof of Theorem 5

Consider an $(n+1)$-sheeted germ $f_{\mid U_{p,1}}\colon U_{p,1}\to V_p$ of the finite cover $\{f\colon S\to\mathbb P^2 \}\in \mathcal F_{(2)}$ given in the local coordinates $(z,w)$ in $U_{p,1}$ and $(u,v)$ in $V_p$ by

$$ \begin{equation} u=z \quad\text{and}\quad v=w^{n+1}-(n+1)wz. \end{equation} \tag{41} $$
Note that the cover germ $f_{\mid U_{p,1}}$ coincides with the germ $f_1\colon U_{l_p,1}\to V_{l_p,1}$ in the deformation $F\colon \mathcal U_{l_p}\to \mathcal V_{l_p}$ of cover germs given by (36). Therefore, Theorem 5 follows from Propositions 10 and 4, inequalities (21) and (23), Theorem 3 and the following result.

Proof. We use the notation introduced in § 2.3.3. It is easy to see that the ramification curve $R_i\subset U_{p,i,1}$ of the cover $f_i$, $i=1,2$, is given by
$$ \begin{equation} w^n-z=0, \end{equation} \tag{42} $$
and it is easy to check that the germ $(B,p)\subset V_p$ of the branch curve $B$ is given by
$$ \begin{equation} v^n+(-1)^{n+1}n^nu^{n+1}=0. \end{equation} \tag{43} $$
Therefore,
$$ \begin{equation} \mu_p(B)=n, \qquad c_{v,p}(B)=n-1 \quad\text{and}\quad \delta_{p_{i,1}}(R_i)=0, \end{equation} \tag{44} $$
where $p_{i,1}=f_{i} ^{-1}(p)\cap U_{p,i,1}$, so that the germ $f_{\mid U_{p,1}}$ satisfies condition (IV).

We have

$$ \begin{equation} \begin{aligned} \, \notag & f_i^*(v^n+(-1)^{n+1}n^nu^{n+1}) = (w^{n+1}-(n+1)wz)^n+(-1)^{n+1}n^nz^{n+1} \\ &\qquad = \sum_{i=0}^n (-1)^{n-i}C_n^i(n+1)^{n-i}w^{n(i+1)}z^{n-i}+(-1)^{n+1}n^nz^{n+1}. \end{aligned} \end{equation} \tag{45} $$
It follows from (42) and (43) that $f_i^*(v^n+(-1)^{n+1}n^nu^{n+1})$ is divisible by $({w^n-z})^2$, and it is easy to see that the polynomial on the right-hand side of (45) is a quasi-homogeneous polynomial of the variables $w$ and $z$. Therefore,
$$ \begin{equation*} f_i^*(v^n+(-1)^{n+1}n^nu^{n+1})= (w^n-z)^2\prod_{i=1}^{n-1}(w^n-\alpha_iz), \end{equation*} \notag $$
and so $C_i\subset U_{p,i,1}=C_{i,1}\cup \dots\cup C_{i,n-1}$ consists of $n-1$ irreducible components $C_{i,j}$ given by $w^n-\alpha_jz=0$ (recall that $f_i^*(B)=2R_i+C_i$ in the notation used in § 2.3.3). Note that the $C_{i,j}$, $j=1,\dots,n-1$, and $R_i$ are smooth curve germs and
$$ \begin{equation} (R_i,C_{i,j})_{p_{i,1}}=(C_{i,j_1},C_{i,j_2})_{p_{i,1}}=n \end{equation} \tag{46} $$
for $j=1,\dots, n-1$ and $1\leqslant j_1<j_2\leqslant n-1$.

In the notation used in § 2.3.3 the cover $g_{1\mid \widetilde W_{p,1,1}''}\colon \widetilde W_{p,1,1}''\to U_{p,1,1}$ is a $n$-sheeted cover branched along $C_1\cap U_{p,1,1}$ and it is ramified along $\widetilde C_2\cap \widetilde W_{p,1,1}''$. The properties of the monodromy homomorphism

$$ \begin{equation*} g_{1\mid \widetilde W_{p,1,1}''*}\colon \pi_1(U_{p,1,1}\setminus C_1, q_1)\to \mathbb S_n \end{equation*} \notag $$
were considered in the proof of Proposition 10. In particular, it was shown that $g_{1\mid \widetilde W_{p,1,1}''*}\colon \pi_1(U_{p,1,1}\setminus C_1, q_1)\to \mathbb S_n$ is an epimorphism, and for geometric generators $\overline{\gamma}\in \pi_1(U_{p,1,1}\setminus C_1, q_1)$ their images $g_{1*}(\overline{\gamma})\in \mathbb S_n$ are transpositions.

To describe the curve germs $\widetilde R\cap \widetilde W_{p,1,1}''$ and $\widetilde C_1\cap \widetilde W_{p,1,1}''$, let us resolve the singular point $p_1$ of the curve $C_1\cap U_{p,1,1}=\cup_{j=1}^{n-1}C_{1,j}$ by a sequence $\psi={\sigma_1\circ \dots \circ\sigma_n}$: ${Z_n\to U_{p,1,1}}$ of $n$ $\sigma$-processes such that $\psi^{-1}(C_1\cap U_{p,1,1})$ is a divisor with normal crossings. The dual weighted graph of the curve $\psi^{-1}(C_1\cap U_{p,1,1})$ is shown in Figure 2, in which the proper inverse images of curve germs $C_{1,j}$ are denoted by the same characters and the proper inverse image of the exceptional curve of the $i$th $\sigma$-process is denoted by $E_i$, $i=1,\dots,n-1$.

Applying Theorem 4 in [13], it is easy to show that the fundamental group

$$ \begin{equation*} \pi_1\biggl(U_{p,1,1}\setminus \biggl(\bigcup_{j=1}^{n-1}C_{1,j}\biggr)\biggr) \stackrel{\psi}\simeq\pi_1\biggl(Z_n\setminus \psi^{-1}\biggl(\bigcup_{j=1}^{n-1} C_{1,j}\biggr)\biggr) \end{equation*} \notag $$
has the following presentation:
$$ \begin{equation} \pi_1(U_{p,1,1}\setminus C_1) \nonumber \end{equation} \tag{47} $$
$$ \begin{equation} \qquad=\bigl\langle c_{1,1},\dots, c_{1,n-1},e_1,\dots, e_{n} \mid e_1^2=e_2, \ e_i^2=e_{i-1}e_{i+1}, \ 2\leqslant i\leqslant n-1, \end{equation} \notag $$
$$ \begin{equation} \qquad\qquad e_n=e_{n-1}c_{1,1}\cdots c_{1,n-1}, \end{equation} \tag{48} $$
$$ \begin{equation} \qquad\qquad [e_1,e_2] =\dots = [e_{n-1},e_{n}]= 1, \end{equation} \tag{49} $$
$$ \begin{equation} \qquad\qquad [e_n,c_{1,1}]=\dots =[e_n,c_{1,n-1}]=1\bigr\rangle, \end{equation} \tag{50} $$
where the $e_i$ ($c_{1,j}$) are some geometric generators represented by simple loops around the curves $E_i$ (the curves $C_{1,j}$, respectively). It follows from (47) and (48) that $\pi_1(U_{p,1,1}\setminus C_1)$ is generated by the elements $c_{1,j}$, $j=1,\dots, n-1$, and $e_n$. By (50) the element $e_n$ belongs to the centre of the group $\pi_1\bigl(Z_n\setminus \psi^{-1}\bigl(\bigcup_{j=1}^{n-1}C_{1,j}\bigr)\bigr)$. Therefore,
$$ \begin{equation} g_{1\mid \widetilde W_{p,1,1}''*}(e_{n})=\mathrm{id}, \end{equation} \tag{51} $$
since the centre of the group $g_{1\mid \widetilde W_{p,1,1}''*}(\pi_1(Z_n\setminus \psi^{-1}(\bigcup_{j=1}^{n-1}C_{1,j}))=\mathbb S_n$ is trivial, and the transpositions $g_{1\mid \widetilde W_{p,1,1}''*}(c_{1,j})$, $j=1,\dots, n-1$, generate the group $\mathbb S_n$ and
$$ \begin{equation} g_{1\mid \widetilde W_{p,1,1}''*}(e_{n-1})=\bigl(g_{1\mid \widetilde W_{p,1,1}''*}(c_{1,1})\dots g_{1\mid \widetilde W_{p,1,1}''*}(c_{1,n-1})\bigr)^{-1} \end{equation} \tag{52} $$
is a cycle of length $n$.

Denote by $R'=\psi^{-1}(R_1\cap U_{p,1,1})\subset Z_n$ the proper inverse image of $R_1\cap U_{p,1,1}$. Applying (46), it is easy to see that

$$ \begin{equation} (R',E_i)_{Z_n}=(R',C_{1,j})_{Z_n}=0 \end{equation} \tag{53} $$
for $1\leqslant i\leqslant n-1$ and $1\leqslant j\leqslant n-1$, and
$$ \begin{equation} (C_{1,j},E_n)_{Z_n}=(R',E_n)_{Z_n}=1 \end{equation} \tag{54} $$
for $1\leqslant j\leqslant n-1$.

Let $T_n\subset Z_n$ be a small tubular neighbourhood of $\bigcup_{i=1}^{n-1}E_i$. It is well known (see, for example, Ch. III, § 5, in [1]) that

$(1_n)$ the cycle $\bigcup_{i=1}^{n-1}E_i$ can be contracted to a normal singular point $p'\in Z_n'$ of type $A_{n,n-1}$ by a bimeromorphic map $\eta\colon Z_n\to Z_n'$, where $T_n'=\eta(T_n)$ is isomorphic to a neighbourhood of the origin in the surface in $\mathbb C^3$ defined by the equation $z^n=xy$, and, without loss of generality, $\eta(T_n\cap E_n)$ is defined in $T_n'$ by the equations $x=z=0$;

$(2_n)$ $\pi_1\bigl(T_n\setminus \bigcup_{i=1}^{n-1}E_i\bigr)=\pi_1(T_n'\setminus p')\simeq \mathbb Z_n$ and the universal unramified cover

$$ \begin{equation*} \alpha\colon \Delta_1^2\setminus \{(0,0)\}=\{(z_1,z_2)\in \mathbb C^2\mid |z_1|>1, |z_2|<1, (z_1,z_2)\neq (0,0)\}\to T_n'\setminus p' \end{equation*} \notag $$
is given by $z=z_1z_2$, $x=z_1^n$ and $y=z_2^n$; in particular, the proper inverse image $\alpha^{-1}(\eta(T_n\cap E_n))$ defined in $\Delta_1^2$ by $z_1=0$ is the germ of a nonsingular curve.

It is easy to see that the bimeromorphic map $\psi\colon Z_n\to U_{p,1,1}$ is factored into a composition, $\psi=\overline{\sigma}_n\circ \eta$, of two bimeromorphic maps $\eta\colon Z_n\to Z_n'$ and ${\overline{\sigma}_n\colon Z_n'\to U_{p,1,1}}$. Let $\widetilde Z_n$ and $\widetilde Z_n'$ be the normalizations of the fibre products $Z_n\times_{U_{p,1,1}} \widetilde W_{p,1,1}''$ and $Z_n\times_{U_{p,1,1}} \widetilde W_{p,1,1}''$. We have the commutative diagram

The monodromy homomorphisms

$$ \begin{equation*} \widetilde g_{1*}\colon \pi_1\biggl(Z_n\setminus \psi^{-1}\biggr(\bigcup_{j=1}^{n-1}C_{1,j}\biggr)\biggr)\to \mathbb S_n \quad\text{and}\quad \widetilde g_{1*}\colon \pi_1\biggl(Z_n'\setminus \overline{\sigma}_n^{-1}\biggl(\bigcup_{j=1}^{n-1}C_{1,j}\biggr)\biggr)\to \mathbb S_n \end{equation*} \notag $$
coincide with $g_{1\mid \widetilde W_{1,1}''*}$, since $\psi\colon Z_n\setminus \psi^{-1}(\bigcup_{l=j}^{n-1}C_{1,j})\to U_{p,1,1}\setminus (\bigcup_{j=1}^{n-1}C_{1,j})$ and $\overline{\sigma}_n\colon Z_n'\setminus \overline{\sigma}_n^{-1}(\bigcup_{j=1}^{n-1}C_{1,j})\to U_{p,1,1}\setminus (\bigcup_{j=1}^{n-1}C_{1,j})$ are biholomorphic maps. By (51) and (52) the cover $\widetilde{g}_n'\colon \widetilde Z_n'\to Z_n'$ is not branched in the curve $\eta(E_n)\subset Z_n'$, and it is branched at the point $p'$ with multiplicity $n$ and also in the proper inverse images (denoted by the same characters) $C_{1,j}:=\eta(C_{1,j})\subset Z_n'$, $j=1,\dots, n-1$. Note that by $(1_n)$ and (52), $\widetilde g_{n\mid \widetilde g_n^{-1}(T_n')}'\colon \widetilde g_n^{-1}(T_n')\to T_n'$ can be identified with the cover $\alpha\colon \Delta_1^2\to T_n'$ in $(2_n)$. Since the $C_{1,j}\subset Z_n'$ are smooth curve germs and $C_{1,j_1}\cap C_{1,j_2}=\varnothing$ in $Z_n'$ for $j_1\neq j_2$, applying property $(2_n)$ we obtain that $\widetilde Z_n'$ is nonsingular and the bimeromorphic holomorphic map $\widetilde{\sigma}_n'$ contracts the nonsingular curve $E=\widetilde g_n'^{-1}(\eta(E_n))$ to the point $p_{1,1}=g_{1,2}^{-1}(p)\cap \widetilde W''_{p,1,1}$. Note that $E=\widetilde{\sigma}_n^{-1}(p_1)$ is connected because $\widetilde{\sigma}_n\colon \widetilde Z_n'\to \widetilde W''_{p,1,1}$ is a holomorphic bimeromorphic map from a connected smooth surface germ and $g_{1,2}^{-1}(p)\cap \widetilde W_{p,1,1}''=p_{1,1}$.

Since the $g_{1\mid \widetilde W_{p,1,1}''*}(c_{1,j})$ are transpositions for $j=1,\dots, n-1$, the inverse images $\widetilde g_1'^*(C_{1,j})$ of the divisors $C_{1,j}$ in $Z_n'$ are equal to

$$ \begin{equation} \widetilde g_1'^*(C_{1,j})=2\widetilde C_{2,1,j}+ \sum_{l=1}^{n-1}\widetilde C_{1,2,j,l}, \end{equation} \tag{55} $$
where the $\widetilde C_{2,1,j}$ are irreducible components of $\widetilde{\sigma}_n'^{-1}(\widetilde C_2)$ and $\widetilde C_{1,2,j,l}$ are irreducible components of $\widetilde{\sigma}_n'^{-1}(\widetilde C)$. In addition, it follows from (54) that
$$ \begin{equation} (\widetilde C_{2,1,j},E)_{\widetilde Z_n'}=1 \end{equation} \tag{56} $$
for $j=1,\dots,n-1$.

To show that $\widetilde W_{p,1,1}''$ is nonsingular, consider a differential $2$-form $\omega$ given by $\omega= dz\wedge dw$ in the local coordinates $z$, $w$ in $U_{p,1,1}$. Then it is easy to see that the divisor of the form $\psi^*(\omega)$ in $Z_n$ is $(\psi^*(\omega))=\sum_{j=1}^njE_j$ and, consequently,

$$ \begin{equation*} (\widetilde g_n'^*(\sigma_n'^*(\omega))=nE+ \sum_{j=1}^{n-1}\widetilde C_{2,1,j}. \end{equation*} \notag $$

Therefore, taking equality (56) into account, by adjunction formula we have

$$ \begin{equation*} 2g(E)-2= ((\widetilde g_n'^*(\sigma_n'^*(\omega))+E,E)_{\widetilde Z_n'}=(n+1)(E^2)_{\widetilde Z_n'}+ n-1\geqslant -2. \end{equation*} \notag $$
But the curve $E$ is contracted to the point by morphism $\widetilde{\sigma}_n'$. Therefore, $(E^2)_{\widetilde Z_n'}< 0$ and hence, $(E^2)_{\widetilde Z_n'}=-1$ and $g(E)=0$. Consequently, $\widetilde{\sigma}_n\colon \widetilde Z_n'\to \widetilde W''_{p,1,1}$ is the $\sigma$-process with centre at the point $p_{1,1}$ and $E$ is its exceptional curve. Thus, $\widetilde W_{p,1,1}''$ is a nonsingular surface.

By (53) and (54) the proper inverse image $\widetilde g_1'^{-1}(R')=\bigsqcup_{j=1}^nR'_j$ is a disjoint union of $n$ irreducible smooth germs $R'_{j}$ of curves and $\widetilde{\sigma}_n(\widetilde g_1'^{-1}(R'))=g_1^{-1}(R_1)\cap \widetilde W_{p,1,1}''$; moreover,

$$ \begin{equation} (\widetilde{\sigma}_n(R'_{j_1}),\widetilde{\sigma}_n(R'_{j_2}))_{\widetilde W_{p,1,1}''}=1 \end{equation} \tag{57} $$
for $j_1\neq j_2$.

Applying (11) we see that just one curve germ, say, $\widetilde{\sigma}_n'(R'_{1})$, is the germ of $\widetilde R\cap \widetilde W_{p,1,1}''$, and the $\widetilde{\sigma}_n'(R'_{j})$ for $j\geqslant 2$ are germs of $\widetilde C_1\cap \widetilde W_{p,1,1}''$. Therefore, by (57) we have

$$ \begin{equation*} (\overline R,\overline C_1)_{\widetilde W_{p,1,1}''}=n-1, \end{equation*} \notag $$
and so, by (44) the germ $f_{\mid U_{p,1}}\colon U_{p,1}\to V_p$ has an extra property over the point $p$. Proposition 11 is proved.

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13. Vik. S. Kulikov, “On rigid germs of finite morphisms of smooth surfaces”, Sb. Math., 211:10 (2020), 1354–1381  mathnet  crossref  mathscinet  zmath  adsnasa
14. Vik. S. Kulikov, “A Chisini Theorem for almost generic covers of the projective plane”, Sb. Math., 213:3 (2022), 341–356  mathnet  crossref  mathscinet  zmath  adsnasa
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Citation: Vik. S. Kulikov, “On quasi-generic covers of the projective plane”, Sb. Math., 215:2 (2024), 206–233
Citation in format AMSBIB
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\by Vik.~S.~Kulikov
\paper On quasi-generic covers of the projective plane
\jour Sb. Math.
\yr 2024
\vol 215
\issue 2
\pages 206--233
\mathnet{http://mi.mathnet.ru//eng/sm9894}
\crossref{https://doi.org/10.4213/sm9894e}
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\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2024SbMat.215..206K}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85197424508}
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