A generalisation of the toric resolution of curves

TL;DR

This paper extends the toric resolution technique to construct explicit models of smooth curve completions over perfect fields, using combinatorial algorithms based on Newton polygons.

Contribution

It generalizes the toric resolution of curves to a broader class of smooth projective curves with an explicit, combinatorial construction.

Findings

Provides an explicit model over $k$ for the smooth completion of $C_0$

Develops a combinatorial algorithm using Newton polygons

Applicable to any smooth projective curve

Abstract

Let $k$ be a perfect field and let $C_0:f=0$ be a smooth curve in the torus $\mathbb{G}_{m,k}^2$. Let $\mathbb{T}_\Delta$ be the toric variety associated to the Newton polygon of $f$. Extending the toric resolution of $C_0$ on $\mathbb{T}_\Delta$, we construct an explicit model over $k$ of the smooth completion of $C_0$. Such a model exists for any smooth projective curve and can be described via a combinatorial algorithm using an iterative construction of Newton polygons.