Galois points for a plane curve and its dual curve, II

Let C⊂P2C⊂P2 be a plane curve of degree at least three. A point P   in projective plane is said to be Galois if the function field extension induced by the projection πP:C⇢P1πP:C⇢P1 from P is Galois. Further we say that a Galois point is extendable if any birational transformation induced by the Galois group can be extended to a linear transformation of the projective plane. This article is the second part of [2], where we showed that the Galois group at an extendable Galois point P   has a natural action on the dual curve C⁎⊂P2⁎C⁎⊂P2⁎ which preserves the fibers of the projection πP‾ from a certain point P‾∈P2⁎. In this article we improve this result, and we investigate the Galois group of πP‾. In particular, we study both when P‾ is a Galois point, and when deg⁡(πP)deg⁡(πP) is prime and deg⁡(πP‾)=2deg⁡(πP). As an application, we determine the number of points at which the Galois groups are certain fixed groups for the dual curve of a cubic curve.