Clusters and semistable models of hyperelliptic curves in the wild case

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.