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

TL;DR

This paper proves that for certain smooth curves on  imes , the Newton polygon's interior lattice points form a standard rectangle, revealing a deep link between algebraic geometry and combinatorics.

Contribution

It establishes that the interior lattice points of the Newton polygon are always a standard rectangle for these curves, and connects scrollar Betti numbers to the polygon's combinatorics.

Findings

Interior lattice points form a standard rectangle up to unimodular transformation.

Scrollar Betti numbers are determined by the combinatorics of the Newton polygon.

Results hold for genus not equal to 4 and under mild conditions on the polygon.

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 $\Delta$. Then we show that the convex hull $\Delta^{(1)}$ of the interior lattice points of $\Delta$ 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 $\Delta$-non-degenerate curves are encoded in the combinatorics of $\Delta^{(1)}$, if $\Delta$ satisfies some mild conditions.