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 $g\ne 4$, 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.

The current version, together with 1410.1698, 1402.4652, 1410.1692 and 1402.4651, extends and corrects the previous version of this submission (titled "On the intrinsicness of the Newton polygon"), which has now become obsolete

References in corpus (1)

• Lattice points in Minkowski sums