Galois morphism computing the gonality of a nonsingular projective curve on a Hirzebruch surface

Let Π:Xe→P1 (e≥0) be the rational ruled complex surface defined by OP1⊕OP1(−e) on P1, i.e., the eth Hirzebruch surface. Let C be a nonsingular projective curve on Xe, and π:C→P1 the restriction of Π to C. We assume that C is not rational nor elliptic nor hyperelliptic. Then, we consider the question: when is the function field extension C(C)/C(P1) induced by π Galois? We determine the defining equation of C and the Galois group when the function field extension is Galois. We also prove the following theorem: if C is not isomorphic to a nonsingular plane curve, then every automorphism of C can be extended to an automorphism of Xe.