Intrinsicness of the Newton polygon for smooth curves on $\mathbb{P}^1\times \mathbb{P}^1$

Abstract

Let $C$ be a smooth projective curve in $\mathbb{P}^1\times \mathbb{P}^1$ of genus $g\neq 4$, 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 $Δ$. Then we show that the convex hull $Δ^{(1)}$ of the interior lattice points of $Δ$ 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 $Δ$-non-degenerate curves are encoded in the combinatorics of $Δ^{(1)}$, if $Δ$ satisfies some mild conditions.