Models of curves and Newton polytopes

Abstract

The purpose of this thesis is to construct explicit regular models of curves, both over fields and over discrete valuation rings. Given a perfect field k and a smooth plane curve Y/k, we know there exists a unique non-singular projective curve C containing Y. The problem is to find C explicitly. Under certain conditions, a method called toric resolution describes such a curve from a certain elementary combinatorial object attached to Y. Unfortunately, this approach does not always work. We extend this classical construction to any curve, preserving its computational and combinatorial nature.

Let K be the field of fraction of some discrete valuation ring O and C/K a hyperelliptic curve of genus g. A regular model of C over O is a regular proper flat 2-dimensional scheme X/O with generic fibre isomorphic to C. A classical question in arithmetic geometry is how to construct such a model. An answer is known when g is at most 2, thanks to algorithms developed by Tate and Liu (in residue characteristic not 2). However, there was no general algorithm for an unbounded g.
In this thesis, we explicitly construct a regular model of C over O with normal crossings for hyperelliptic curves of arbitrary genus, when the residue characteristic of K is not 2 (and some cases when it is 2).
The description relies on a new notion we introduce: the MacLane cluster picture.

Date of Award	27 Sept 2022
Original language	English
Awarding Institution	University of Bristol
Supervisor	Tim Dokchitser (Supervisor)