A combinatorial interpretation for Schreyer's tetragonal invariants

TL;DR

This paper links Schreyer's tetragonal invariants to the Newton polygon of Laurent polynomials, providing geometric interpretations and intrinsicness results for small lattice width in tetragonal curves.

Contribution

It offers a new combinatorial interpretation of Schreyer's invariants for tetragonal curves on toric surfaces, connecting algebraic invariants with Newton polygons.

Findings

Integers $b_1$ and $b_2$ relate to Newton polygons of Laurent polynomials.

Established intrinsicness of Newton polygons for small lattice width.

Connected algebraic invariants with combinatorial geometry.

Abstract

Schreyer has proved that the graded Betti numbers of a canonical tetragonal curve are determined by two integers $b_1$ and $b_2$, associated to the curve through a certain geometric construction. In this article we prove that in the case of a smooth projective tetragonal curve on a toric surface, these integers have easy interpretations in terms of the Newton polygon of its defining Laurent polynomial. We can use this to prove an intrinsicness result on Newton polygons of small lattice width.