On the almost generic covers of the projective plane (Q2214903)

In this interesting paper, the author studies almost generic covers of the complex projective plane. Let \(X\) be a smooth irreducible projective surface, a finite morphism \(f: X \rightarrow \mathbb{P}^{2}\) branched along a curve \(B \subset \mathbb{P}^{2}\) is called a generic covering of the projective plane if it has the following properties: i) for each point \(p \in \mathbb{P}^{2}\) the fibre \(f^{-1}(p)\) is supported on at least \(\mathrm{deg} \, f - 2\) distinct points, ii) \(f\) is ramified with multiplicity \(2\) at a generic point of its ramification locus \(R\), iii) the singular points of \(B\) are only the ordinary nodes and ordinary cups. The notion of generic covers of \(\mathbb{P}^{2}\) can be generalized in the following way. We say that a finite morphism \(f: X \rightarrow \mathbb{P}^{2}\) is an almost generic covering of the projective plane if it satisfied properties i)--ii) of generic covers. It is worth recalling that Chisini's conjecture claims that a curve \(B \subset \mathbb{P}^{2}\) satisfying iii) can be the branched curve of at most one generic covering \(f : X \rightarrow \mathbb{P}^{2}\) with \(\mathrm{deg} \, f \geq 5\). Due to this reason, it seems to be very natural to check whether Chisini's conjecture can be extended to a wider class of coverings, for instance almost generic coverings. As a first step towards this direction, the author studies types of singular points of the branch curve that can occur in the case of almost generic coverings and he computes the basic invariants of the covering surface \(X\) in terms of invariants of the branch curve and degree of the covering \(f\). In order to formulate the main result of the paper, let us recall basic notions devoted to the monodromy of \(f\). A dominant morphism \(f: X \rightarrow \mathbb{P}^{2}\) defines a homomorphism \[f_{*} : \pi_{1}(\mathbb{P}^{2} \setminus B, p) \rightarrow S_{\mathrm{deg}\, f},\] called the monodromy of \(f\), whose image \(G_{f} := f_{*}(\pi_{1}(\mathbb{P}^{2} \setminus B,p))\) is the monodromy group of \(f\) and it is a subgroup of the symmetric group \(S_{\mathrm{deg}\, f}\) acting on the fibre \(f^{-1}(p) = \{q_{1}, ..., q_{\mathrm{deg}\,f}\}\). Let \(o\in B\) be a point of a curve in \(\mathbb{P}^{2}\). It is well-known that the group \(\pi_{1}^{\mathrm{loc}}(B,o) := \pi_{1}(V \setminus B)\) does not depend on \(V\), where \(V \subset \mathbb{P}^{2}\) is a sufficiently small complex analytic neighbourhood biholomorphic to a ball of small radius centered at \(o\). The image \(G_{f,o} :=\mathrm{im} \, f_{*} \circ i_{*}\) is called the local monodromy group of \(f\) at the point \(o\), where \(i_{*} : \pi_{1}^{\mathrm{loc}}(B,o) \rightarrow \pi^{1}(\mathbb{P}^{2} \setminus B, p)\) is a homomorphism defined by the embedding \(V \rightarrow \mathbb{P}^{2}\). The main result of the paper can be formulated as follows. Main Result. The monodromy group \(G_{f}\) of an almost generic covering \(f : X \rightarrow \mathbb{P}^{2}\) coincides with \(S_{\mathrm{deg}\,f}\) . The branch curve \(B\) of the cover \(f\) can have only the singular points of type \(A_n\) and the points of \(B\) are divided into three types according to the types of singularities of \(B\) at these points and properties of the local monodromy groups: i) \(p \in B \setminus\mathrm{Sing} \, B\) and \(G_{f,p} \simeq \mathbb{Z}_{2}\) is generated by a transposition; ii) \(p \in\mathrm{Sing} \, B\) is of type \(A_{n,2}\), that is, \(B\) has the singularity of type \(A_{2n-1}\) at \(p\), \(n \in \mathbb{N}\), and \(G_{f,p} = \mathbb{Z}_{2} \times \mathbb{Z}_{2}\) is generated by two commuting transpositions; iii) \(p \in\mathrm{Sing} \, B\) is of type \(A_{n,3}\), that is \(B\) has the singularity of type \(A_{3n-1}\) at \(p\), \(n \in \mathbb{N}\), and \(G_{f,p} \simeq S_{3}\) is generated by two non-commuting transpositions.