Clusters, toric ranks, and 2-ranks of hyperelliptic curves in the wild case

Abstract Given a Galois cover $$Y \rightarrow X$$ Y → 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 $$\mathcal {Y}^{\textrm{rst}}$$ Y rst of Y , and and we discuss its properties, focusing on the case when $$Y: y^2 = f(x)$$ Y : y 2 = f ( x ) is a hyperelliptic curve viewed as a degree-2 cover of the projective line $$X{:}{=} \mathbb {P}_K^1$$ X : = P K 1 . Over residue characteristic different from 2, it follows from known results that the toric rank (i.e. the number of loops in the graph of components) of the special fiber of $$\mathcal {Y}^{\textrm{rst}}$$ Y rst can be computed directly from the knowledge of the even-cardinality clusters of roots of the defining polynomial f . We instead consider the “wild” case of residue characteristic 2 and demonstrate an analog to this result, showing that each even-cardinality cluster of roots of f gives rise to a loop in the graph of components of the special fiber of $$\mathcal {Y}^{\textrm{rst}}$$ Y rst if and only if the depth of the cluster exceeds some threshold, and we provide a computational description of and bounds for that threshold. As a bonus, our framework also allows us to provide a formula for the 2-rank of the special fiber of $$\mathcal {Y}^{\textrm{rst}}$$ Y rst .