Intrinsicness of the Newton polygon for smooth curves on P × P

Let C be a smooth projective curve in P1 × P1 of genus g 6= 4, and assume that it is birationally equivalent to a curve defined by a Laurent polynomial that is non-degenerate with respect to its Newton polygon ∆. Then we show that the convex hull ∆(1) of the interior lattice points of ∆ is a standard rectangle, up to a unimodular transformation. Our main auxiliary result, which we believe to be interesting in its own right, is that the first scrollar Betti numbers of ∆-non-degenerate curves are encoded in the combinatorics of ∆(1), if ∆ satisfies some mild conditions. MSC2010: Primary 14H45, Secondary 14J25, 14M25