Function Field Theory of Plane Curves by Dual Curves

We study the structure of function fields of plane curves following our method developed previously (K. Miura and H. Yoshihara, 2000, J. Algebra226, 283–294). Let K be the function field of a smooth plane curve C of degree d ( ≥ 4) and let KP be a maximal rational subfield of K for P 2. We study the field extension K/KP from a geometrical viewpoint. Especially, we give a sufficient condition that the Galois group of the Galois closure of K/KP becomes a full symmetric group.