Clusters and semistable models of hyperelliptic curves in the wild case

TL;DR

This paper develops a method to compute semistable models of hyperelliptic curves over discrete valuation fields, especially addressing the complex case when the residue characteristic is 2, by analyzing root clusters of the defining polynomial.

Contribution

It introduces a practical approach to determine the relatively stable model of hyperelliptic curves in the wild case, with explicit techniques for the challenging residue characteristic 2 scenario.

Findings

Components correspond to root clusters in residue characteristic not 2.

In characteristic 2, components relate to clusters with even number of roots.

A polynomial F(T) helps identify components not linked to even-cardinality clusters.

Abstract

Given a Galois cover $Y \to X$ of smooth projective geometrically connected curves over a complete discrete valuation field $K$ with algebraically closed residue field, we define a semistable model of $Y$ over the ring of integers of a finite extension of $K$, which we call the relatively stable model $\mathcal{Y}^{\mathrm{rst}}$ of $Y$, and we discuss its properties. We focus on the case when $Y : y^2 = f(x)$ is a hyperelliptic curve, viewed as a degree-$2$ cover of the projective line $X := \mathbb{P}_K^1$, and demonstrate a practical way to compute the relatively stable model. In the case of residue characteristic $p \neq 2$, the components of the special fiber $(\mathcal{Y}^{\mathrm{rst}})_s$ correspond precisely to the non-singleton clusters of roots of the defining polynomial $f$, i.e. the subsets of roots of $f$ which are closer to each other than to the other roots of $f$ with respect to the induced discrete valuation on the splitting field; this relationship, however, is far less straightforward in the $p=2$ case, which is our main focus (the techniques we introduce nevertheless also allow us to recover the simpler, already-known results in the $p\neq 2$ case). We show that, when $p = 2$, for each cluster containing an even number of roots of $f$, there are $0$, $1$, or $2$ components of $(\mathcal{Y}^{\mathrm{rst}})_s$ corresponding to it, and we determine a direct method of finding and describing them. We also define a polynomial $F(T) \in K[T]$ whose roots allow us to find the components of $(\mathcal{Y}^{\mathrm{rst}})_s$ which are not connected to even-cardinality clusters.