Galois points for a plane curve and its dual curve. II. (Q899576)

The paper is a continuation of [the authors, Rend. Semin. Mat. Univ. Padova 132, 61--74 (2014; Zbl 1333.14028)]. The authors consider curves \(C\subset {\mathbb P}^2\) over an algebraically closed field \(K\) of characteristic 0, of degree at least 3. The basic definitions are as follows. For a point \(P\in {\mathbb P}^2\), let \(\pi_P\,:\, C \to {\mathbb P}^1\) be the projection from \(P\). A Galois point is a point \(P\) for which the function field extension induced by \(\pi_P\) is Galois; a Galois point is said to be extendable if any bilinear transformation induced by the Galois group can be extended to a linear transformation of the projective plane. In the previous paper, the authors showed that the Galois group at an extendable Galois point \(P\) has a natural action on the dual curve \(C^*\subset {\mathbb P}^{2*}\), which preserves the fibres of the projection \(\pi_{\bar P}\) from a suitable point \(\bar P \in {\mathbb P}^{2*}\). In the present paper they investigate the possible Galois groups of \(\pi_{\bar P}\) in several cases. In particular, they discuss when \(\bar P\) is a Galois point and they find the characterization of the Galois group \(G_{\bar P}\) of the projection from \(\bar P\) when \(\deg({\pi_P})\) is an odd prime and \(\deg(\pi_{\bar P}) = 2\deg({\pi_P})\).