Minimal Weierstrass models and regular models of hyperelliptic curves

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 Néron model of $\mathrm{Jac}(C)$, using two specific minimal Weierstrass models.

💡 Research Summary

The paper investigates hyperelliptic curves (C) of genus (g\ge2) over a discrete valuation field (K) with perfect residue field (k). Its central theme is the study of minimal Weierstrass models (MWMs) of (C) and the consequences when more than one such model exists.

A Weierstrass model is defined as the normalization of a smooth (\mathbb{P}^1)-model of the quotient (C/\langle\sigma\rangle) (where (\sigma) is the hyperelliptic involution) and is given by a single affine equation
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