Hyperelliptic d-osculating covers and rational surfaces

1.1. Let P1 and (X, q) denote, respectively, the projective line and a fixed ellip-tic curve marked at its origin, both defined over an algebraically closed field K of arbitrary characteristic p 6 = 2. We will study all finite separable marked mor-phisms pi: (Γ, p) → (X, q), called hereafter hyperelliptic covers, such that Γ i