Dualizing coverings of the plane (Q2832309)

This paper is devoted to study the properties of dualizing coverings of the plane which are associated to plane curves. In particular, the author gives a complete description of the set of plane curves for which all singularities of the Galoisation of the associated dualizing covering are quotient singularities.NEWLINENEWLINELet \(C\subset \mathbb{P}^2\) be an irreducible reduced curve of degree \(\deg(C)=:d\geq 2\). Let \(\widehat{\mathbb{P}}^2\) denote the dual plane and \(I:=\{(P,l) \in \mathbb{P}^2 \times \widehat{\mathbb{P}}^2 \mid P\in l \} \) the incident variety. Let \(\mathrm{pr}_1: \mathbb{P}^2 \times \widehat{\mathbb{P}}^2 \to \mathbb{P}^2 \) be the projection onto the first factor and define \(X':=\mathrm{pr}_1^{-1}(C)\cap I\). Let \(\nu': X \to X'\) be its normalization and \(h': X' \to \widehat{\mathbb{P}}^2\) the restriction of the projection onto the second factors. The morphism \(h:= h' \circ \nu': X \to \widehat{\mathbb{P}}^2 \) is a covering of degree \(d\), called the \textit{dualizing covering} associated to the curve \(C\).NEWLINENEWLINEIn the first part of the paper the author studies the properties of dualizing coverings; in particular, he shows that \(X\) is a smooth ruled surface and provides a description of the ramification locus \(\overline{R}\), of the branch locus \(\overline{B}\) and of the restriction \(h_{|\overline{R} }\).NEWLINENEWLINEIn the second part the author recalls some facts about finite coverings of surfaces. Then he introduces the concept of \textit{passport of singularities} of the curve \(C\) with respect to a line \(l\in \widehat{\mathbb{P}}^2\), which keeps trace of the local behaviour of \(C\) at the singulars points of \(l\) with respect to \(C\).NEWLINENEWLINEIn Theorem 2 he determines the local monodromy groups of dualizing coverings at a point \(l\in\widehat{\mathbb{P}}^2\): this group is the direct product of symmetric and cyclic groups.NEWLINENEWLINEFinally he states the main theorem, which gives a complete description of those plane curves for which the Galoisation of the associated dualizing covering has only quotient singularities.NEWLINENEWLINEThe Galoisation \(f: Z \to \widehat{\mathbb{P}}^2 \) of the dualizing cover \(h: X \to \widehat{\mathbb{P}}^2 \) associated to an irreducible reduced curve \(C\subset \mathbb{P}^2\) has only quotient singularities if and only if \(C\) satisfies the following.NEWLINENEWLINE(i) If an irreducible germ \((C_i,P)\) of the curve \(C\) is singular at \(P\), then the singularity is either of type \(A_2\), or of type \(E_6\). Moreover, if there are several singular germs through \(P\), then they have the same type.NEWLINENEWLINE(ii) The passport of singularities of the curve \(C\) with respect to a line \(l\in \widehat{\mathbb{P}}^2\) belongs to a prescribed list, which reports also the singularities of \(Z\) over the point \(l\).