Minimal Weierstrass models and regular models of hyperelliptic curves

TL;DR

This paper investigates minimal Weierstrass models of hyperelliptic curves over discrete valuation fields, exploring their properties, implications for stable reduction, and applications to Jacobian invariants.

Contribution

It provides new insights into the structure of minimal Weierstrass models, characterizes stable reduction for genus 2 curves, and offers methods to compute Jacobian invariants.

Findings

Characterization of stable reduction for genus 2 hyperelliptic curves.

Properties of minimal regular and canonical models when multiple minimal Weierstrass models exist.

Methods to compute Euler factors and volume forms of Jacobians using specific minimal models.

Abstract

Let $C$ be a hyperelliptic curve of genus $g\ge 2$ over a discrete valuation field $K$ with perfect residue field. We study the minimal Weierstrass models of $C$. When there is more than one such model, we find interesting properties on the minimal regular model and the canonical model of $C$. For curves of genus $2$, we characterize the existence of the stable reduction in terms of the minimal Weierstrass models. When there is more than one such model, we can compute the Euler factor of $\mathrm{Jac}(C)$ and a volume form of the N\'eron model of $\mathrm{Jac}(C)$, using two specific minimal Weierstrass models.