Models of Hyperelliptic Curves with Tame Potentially Semistable Reduction

Let $C$ be a hyperelliptic curve $y^2 = f(x)$ over a discretely valued field $K$. The $p$-adic distances between the roots of $f(x)$ can be described by a completely combinatorial object known as the cluster picture. We show that the cluster picture of $C$, along with the leading coefficient of $f$ and the action of $\mathrm{Gal}(\overline{K}/K)$ on the roots of $f$, completely determines the combinatorics of the special fibre of the minimal strict normal crossings model of $C$. In particular, we give an explicit description of the special fibre in terms of this data.

Let K be a field complete with respect to a discrete valuation v K , with algebraically closed residue field k of characteristic p > 2. In this paper we study hyperelliptic curves C/K with genus g = g(C) 1 , where C is given by Weierstrass equation y 2 = f (x). We write R for the set of roots of f (x) in the algebraic closure K of K and c f for the leading coefficient of f , so and |R| ∈ {2g +1, 2g +2}. Following [DDMM18] we associate to C a cluster picture, defined by the combinatorics of the root configuration of f . Cluster pictures are a relatively new innovation which have already proved invaluable in studying the arithmetic of hyperelliptic curves: for example, in calculating semistable models, conductors, minimal discriminants and Galois representations in [DDMM18], Tamagawa numbers in [Bet18], root numbers in [Bis19] and differentials in [Kun19].
Using cluster pictures we will calculate a combinatorial description of the minimal strict normal crossings (SNC) model X of C/K: a model whose singularities on the special fibre X k are normal crossings (i.e. locally they look like the union of two axes), and where blowing down any exceptional component of X k would result in a worse singularity. Such models can be used to calculate arithmetic invariants, to study the Galois representation, and to deduce the existence of K-rational points of C. For the case of elliptic curves, Tate's algorithm is sufficient to calculate the minimal SNC model of a given curve. In [DDMM18] the authors calculate the SNC model when C has semistable reduction, and in [Dok18] when C has a particularly nice cluster picture. 2 We extend these results to the more general case where C has tame potentially semistable reduction 1 Unless explicitly mentioned otherwise, we assume g ≥ 2 throughout the paper. 2 In fact, the methods of [Dok18] work for a much larger class of smooth projective curves, but we restrict our attention to its applications for hyperelliptic curves. over K -that is, when there exists some finite extension L/K such that C has semistable reduction over L, and [L : K] is coprime to p. It is important to note that our theorems do not apply in the case where a wild extension is required for semistability. However this condition is not too strong since for large enough p, every curve of genus g has tame potentially semistable reduction. Most of the information we require to deduce the special fibre of X is contained in the cluster picture of C.
Definition 1.1. A cluster is a non-empty subset s ⊆ R of the form s = D ∩ R for some disc D = z + π n K O K , where z ∈ K, n ∈ Q and π K is a uniformiser of K. If s is a cluster and |s| > 1, we say that s is a proper cluster. For a proper cluster s we define its depth d s to be It is the minimal n for which such a cluster exists. We write d s = as bs with a s , b s coprime.The cluster picture Σ C/K of C is the collection of all clusters of the roots of f . When there is no risk of confusion, we may simplify this to Σ C .
The cluster picture Σ C/K comes with a natural action of G K = Gal(K/K). The cluster picture, along with the valuation of the leading coefficient and this action is all we need to calculate a combinatorial description of the minimal SNC model of C.
Theorem 1.2. Let K be a complete discretely valued field with algebraically closed residue field of characteristic p > 2. Let C : y 2 = f (x) be a hyperelliptic curve over K with tame potentially semistable reduction. Then the dual graph, with genus and multiplicity, of the special fibre of the minimal SNC model of C/K is completely determined by Σ C/K (with depths), the valuation of the leading coefficient v K (c f ) of f , and the action of G K . Remark 1.3. In [Bis19] the author classifies the possible cluster pictures which can arise from hyperelliptic curves with tame potentially semistable reduction. He also shows that the inertia action is determined by the cluster picture (with depths). Given time and determination, this fact, along with Theorem 1.2, allows us to classify the minimal SNC models which can arise from such hyperelliptic curves of a given genus. We do so for elliptic curves in Appendix A.
A maximal subcluster s ′ of a cluster s is called a child of s, denoted s ′ < s, and s is the parent of s ′ , denoted P (s ′ ). We say that s is odd (resp. even) if |s| is odd (resp. even) Furthermore, s is a twin if |s| = 2, and s isübereven if s has only even children. A cluster s = R is principal if |s| ≥ 3. The cluster R is not principal if it has a child of size 2g(C), or if R is even and has exactly two children; otherwise R is principal. The remaining theorems given in the introduction will assume that R is principal. Full theorems including the case when R is not principal are given in Section 7.
Every Galois orbit of principal clusters X contributes components to the special fibre X k . More precisely: orbits of principal,übereven clusters contribute either one or two components and orbits of principal non-übereven clusters contribute one component. We call these components central components, and they are linked by either one or two chains of rational curves which we call linking chains. The central components of two orbits X and X ′ are linked by a chain (or chains) of rational curves if and only if there exits some s ∈ X and s ′ ∈ X ′ such that s ′ < s. Orbits of twins gives rise to a chain of rational curves which intersects the component(s) arising from their parent's orbit. Some central components are also intersected by other chains of rational curves: loops, tails and crossed tails. Loops are chains from a component to itself; tails are chains which intersect the rest of the special fibre in only one place; crossed tails are similar to tails but with two additional components intersecting the final component of the chain, which are called crosses. Figures 1 and 2 give a pictorial description of the different types of chains of rational curves that can occur, where the dashed lines illustrate all the components of X k which are intersected by the chain.
In this paper we explicitly describe the structure, multiplicities and genera of components of X k . We will give a precise statement shortly but first, let us illustrate the main result of the paper via a worked example.
. . .    Example 1.4. Let K = Q ur p , and C/K be the hyperelliptic curve given by The cluster picture of C/K is shown in Figure 3a and the special fibre X k of the minimal SNC model of C/K is shown in Figure 3b. The principal clusters in Σ C/K are s 1 , s 2 , s 3 , s 4 , s 5 , and R, as labeled in Figure 3a. Note that s 3 , s 4 and s 5 are permuted by G K and denote their orbit by X. None of the principal clusters in this example areübereven, so by Theorem 1.5, each orbit of principal clusters gives rise to just one central component -these are drawn in bold and labeled in Figure 3b. The clusters s 1 , s 2 are children of R, so there is a linking chain (or chains) between Γ R and Γ si for i = 1, 2, and between Γ s2 and Γ X . Each of the components Γ s1 , Γ s2 , and Γ X also  have tails intersecting them. We will discuss later how we determine the exact number and length of these linking chains (see Theorem 1.11). We can compare the chains intersecting the central components in X k to the tails appearing in the minimal SNC models of related elliptic curves, shown in Table 3 in Appendix A. The chains intersecting Γ R , along with Γ R itself, look much like a type I 0 elliptic curve. Similarly type III for s 1 , type II for s 2 , and type I * 0 for X (here with multiplicities multiplied by |X| = 3). Here we give an abridged version of the description of the structure of the special fibre, given in full in Theorem 7.12. In stating this theorem we use an invariant ǫ X . This is a subtle invariant of even clusters defined fully in Definition 3.10. However, in practice for X with s ∈ X even ǫ X is given by ǫ X = (−1) |X|(vK (c f )+ r ∈s vK (rs−r)) , where r s is any root of s. Theorem 1.5 (Structure of SNC model). Let K be a complete discretely valued field with algebraically closed residue field of characteristic p > 2. Let C/K be a hyperelliptic curve with tame potentially semistable reduction. Then the special fibre of its minimal SNC model is structured as follows. Every principal Galois orbit of clusters X contributes one component Γ X , unless X is ubereven with ǫ X = 1, in which case X contributes two components Γ + X and Γ − X . These components are linked by chains of rational curves in the following cases (where, for any orbit Y , we write Γ + Y = Γ − Y = Γ Y if Y contributes only one central component): Any chain above where the "To" column has been left blank is a crossed tail. Finally, some central components Γ X are also intersected transversally by tails. These are explicitly described in Theorem 1.11.
The case when R is not principal is described in Theorem 7.12. We do not given explicit equations for the components in the special fibre. However, these can be calculated using the method laid out in this paper if desired (see Remark 7.13).
The linking chains, tails, and the multiplicities and genera of the components in the special fibre are given explicitly in Theorem 1.11 below. In order to describe the chains of rational curves in detail, we introduce the notion of sloped chains of rational curves. We will also need a few other numerical invariants.
Definition 1.6. Let t 1 , t 2 ∈ Q and µ, λ ∈ N with λ minimal be such that E i is a chain of rational curves where E i has multiplicity µd i . Then C is a sloped chain of rational curves with parameters (t 2 , t 1 , µ). If C is a tail, then C is a sloped chain with parameters (⌊t 1 − 1⌋, t 1 , µ), so we usually just write (t 1 , µ) for its parameters.
This allows us to state a more detailed description of the chains of rational curves appearing in X k . To state the conditions required for certain tails to appear, we use several invariants associated to clusters. Notation 1.7. Writes for the set of odd children of s, and s sing for the set of size 1 children of s. Definition 1.8. Let s be a cluster, then the semistable genus of s is given by |s| = 2g ss (s) + 1 or 2g ss (s) + 2, or g ss (s) = 0 if s isübereven. If X is an orbit with s ∈ X the semistable genus of X is defined by g ss (X) = g ss (s). From this we define the genus of an orbit X. If X = {s} is a trivial orbit with d s = as bs , where (a s , b s ) = 1, and g ss (s) > 0 then g(s) is given by Otherwise, g(X) = g(s) = 0 if g ss (s) = 0. For a general orbit X, we define g(X) = g(s) for s ∈ X, when s is instead considered as a cluster in Σ C/KX , where K X is the unique extension of K of degree |X|.
Definition 1.9. Let X be an orbit of clusters with s ∈ X. Define e X to be the minimal degree of extension required to make the clusters in X satisfy the conditions of the Semistability Criterion 3.9. X also has the following invariants where r s is any root of s.
Definition 1.10. A child s ′ < s is stable if it has the same stabiliser as s, and an orbit is stable if all (equivalently any) of its children are stable.
Theorem 1.11. Let K and C/K be as in Theorem 1.5. Let X be a principal orbit of clusters in the cluster picture of a hyperelliptic curve C with tame potentially semistable reduction and with R principal. Then Γ ± X has genus g(X). Furthermore, it has multiplicity |X|e X if X is non-übereven, or if X isübereven with ǫ X = 1; otherwise Γ ± X has multiplicity 2|X|e X if X isübereven with ǫ X = −1. Suppose further that e X > 1, and choose some s ∈ X. Then the central component(s) associated to X are intersected transversely by the following sloped tails with parameters (t 1 , µ) −d X |X| X has no stable child, λ X ∈ Z, and either g ss (X) > 0 or X isübereven X has a stable singleton or g ss (X) = 0, X is notübereven and X has no proper stable odd child The central components are intersected by the following sloped chains of rational curves with parameters (t 2 , t 2 + δ, µ): Note that here the names indicate the components which each chains intersect, as explicitly written in the second table of Theorem 1.5. Finally, the crosses of any crossed tail have multiplicity µ 2 . In practice, when g ss (X) > 0 there is a one-to-one correspondence between the chains intersecting a central component Γ X and the tails of the unique central component Γ X of the minimal SNC model X X of a related curve C X . Choose some s ∈ X. Then the curve C X is a hyperelliptic curve over K X (an extension of K of degree |X|) which has as its roots a centre for each odd child of s (with a correction to the leading coefficient to account for the rest of the cluster picture). This preserves the multiplicities in the corresponding chains (up to some small corrections), and the genus of Γ X is equal to the genus of Γ X (or 0 if g ss (X) = 0). This idea has been briefly explored in Example 1.4, comparing parts of the special fibre to minimal SNC models of certain elliptic curves. We have a closer look at this idea in Example 1.12 below. Since C X will often have a lower genus then C, this allows us to construct the minimal SNC model of C in terms of simpler models. We now give some more examples, the first of which provides the motivation behind Theorem 1.11. Example 1.12. Let C be the hyperelliptic curve given by Weierstrass equation and defined over K = Q ur p . The cluster picture of C/K consists of two proper clusters R and s, pictured in Figure 4a. The special fibre of the minimal SNC model X of C/K shown in Figure  4b. Now, define elliptic curves C 1 and C 2 by Weierstrass equations C 1 : y 2 = f 1 (x) = x 3 − p 2 and , the roots of f 1 (x) contribute the roots in R \ s, and the roots of f 2 (x) contribute the roots in s. The coefficient in the defining equation of C 2 is chosen to somehow "see" the roots of f 1 . It is therefore interesting to compare the minimal SNC models of C i to that of C for i = 1, 2. Notice that C 1 and C 2 are type IV and type III * elliptic curves respectively, as shown in Table 3 in Appendix A. The cluster pictures and special fibres of the minimal SNC models of C 1 and C 2 are shown in Figures 5 and 6 respectively. It is clear that the roots of both f 1 and f 2 are making their own contributions to 2 3 (a) Cluster picture Σ C 1 /K .   X k , as both the special fibres of the minimal SNC models of C i can be seen as "submodels" of X k for i = 1, 2. This shows how R and s each make their own contribution to X k . Since s is an even child of R, and ǫ s = 1, there are two linking chains between their contributions in X k .
Example 1.13. Let K = Q ur p , and C/K be the hyperelliptic curve given by C : The cluster picture of C/K is shown in Figure 7a, and the special fibre of the minimal SNC model of C/K is shown in Figure 7b. The central components of the model, which arise from clusters in Σ C/K , are labeled. Note that R contributes components to the model which look like those appearing the the minimal SNC model of a type I 0 elliptic curve; s 1 those of a type IV * elliptic curve; s 2 those of a type II * elliptic curve; and s 3 those of a type II elliptic curve. The special fibers of the minimal SNC models of these Kodaira types are all shown in Table 3 in the Appendix. This reflects the general phenomenon discussed above that the chains intersecting a central component arising from a cluster s "correspond" to the tails of a hyperelliptic curve constructed from s.
Example 1.14. Let K = Q ur p , and C/K be the hyperelliptic curve given by C : The cluster picture of C/K is shown in Figure 8a. Note that t 1 , t 2 and t 3 are swapped by G K and denote their orbit by X. and the special fibre of the minimal SNC model of C/K is shown in Figure 8b. The central components of the model, which arise from clusters in Σ C/K , are labeled, as is the crossed tail T X arising from the orbit of twins X. The component Γ R and its chains look like a type II elliptic curve. Since s isübereven, we cannot construct a curve C s to compare the contributions of s to. However, observe that s and its children contribute a divisor which looks like the minimal SNC model of a Namikawa-Ueno type III * 2 curve. In our final proof we will use induction on the number of proper clusters and this is a useful example to look back to when we do so.
The paper is structured as follows: in Sections 2 -4, we restate key definitions and theorems from literature, which we will make use of in the remainder of the paper. We start with a brief introduction to cluster pictures in Section 2, before moving onto look at models and the methods used to calculate them in Section 3. Results from [Dok18] concerning the use of Newton polytopes will be discussed in Section 4. In Sections 5 and 6, we calculate the minimal SNC model for two special cases. The first of these special cases, Section 5, is where C has tame potentially good reduction -that is, it has a smooth model over a tame extension of K. This will act as a base case for our eventual proof by induction. The second of these cases, Section 6, examines curves C with a cluster picture which consists of exactly two proper clusters s < R. Curves with such cluster pictures are used to deduce the linking chains between central components in the main theorems. These main theorems are stated and proved in Section 7. In Appendix A we give the minimal SNC models and cluster pictures for the different Kodaira-Néron types of elliptic curves for the convenience of the reader.
1.1. Notation. For the convenience of the reader, the following two tables collate the general notation and terminology which we make use of throughout the paper. Table 1 lists the general notation associated to fields, hyperelliptic curves, and models. Table 2 lists the notation and terminology associated to cluster pictures and Newton polytopes. Whenever a component in a figure is drawn in bold it is a central component. In any figure describing the special fiber of a model numbers indicate multiplicities, except those preceded by g, which indicate the genus of a component. So 2 indicates a rational curve of multiplicity 2 and 2g1 indicates a genus 1 curve of multiplicity 2.
algebraically closed residue field of K K algebraic closure of K C hyperelliptic curve over K given by 10) e X (7.9) δ s (2.7) c s (3.12) g(X) (7.9) Table 2. Notation associated to cluster pictures and Newton polytopes

Background -Cluster Pictures
Let C/K be a hyperelliptic curve given by Weierstrass equation y 2 = f (x), with genus g(C) ≥ 1. The p-adic distances between roots of f (x) contain a large amount of useful information. To visualise these p-adic distances we use cluster pictures, as described in [DDMM18]. In this section we will outline the key definitions we require for this paper concerning cluster pictures; the interested reader can find more in [DDMM18]. Let R be the set of roots of f (x) in the algebraic closure K of K, and c f the leading coefficient of f (x), so and |R| ∈ {2g + 1, 2g + 2}. Sometimes we may write g(C) = g.
Recall the definitions of clusters and cluster pictures given in Definition 1.1. The cluster picture Σ C is a way of visualising which roots of f are p-adically close. In a non-archimedean algebra, two discs either have a non empty intersection or one is contained in the other. So Definition 1.1 gives us that any two clusters are either disjoint or is one contained in the other. Moreover d s ′ > d s if s ′ s. Every root is a cluster -that is, {r} ∈ Σ C for every r ∈ R -and R ∈ Σ C . In order to work with clusters we need a significant amount of terminology from [DDMM18] which we describe here.
Definition 2.1. A cluster s is even (resp. odd ) if |s| is even (resp. odd). Furthermore s is a twin if |s| = 2.
Definition 2.2. Suppose s is a cluster. If s ′ s is a maximal subcluster of s then s ′ is a child of s and s is a parent of s ′ . We write s ′ < s, and P (s ′ ) = s. Denote by s the set of all children of s, and by s the set of all odd children.
Definition 2.3. A cluster s is a cotwin if it has a child of size 2g whose complement is not a twin.
Definition 2.4. A cluster isübereven if it only has even children.
Definition 2.5. A centre z s of a proper cluster s is any element z s ∈ K such that v K (z s − r) ≥ d s for all r ∈ s. Equivalently, z s is a centre of s if s can be written as D ∩ R, where D = z s + π ds O K . Note that any root r ∈ s can be chosen as a centre, and if s = {r} then the only centre is z s = r.
Definition 2.6. If s and s ′ are two clusters we write s ∧ s ′ for the smallest cluster containing both s and s ′ .
Definition 2.7. If s and s ′ are two clusters then the distance between them is δ(s, s ′ ) = d s + d s ′ − 2d s∧s ′ . For a proper cluster s = R define the relative depth to be δ s = δ(s, P (s)) = d s − d P (s) .
Definition 2.8. A cluster s is principal if |s| ≥ 3 except if either s = R is even and has exactly two children, or if s has a child of size 2g.
We will see later that principal clusters form an important class of clusters. Roughly, if C/K is a hyperelliptic curve, every orbit of principal clusters in Σ C/K makes a contribution to the minimal SNC model of C over K.
Definition 2.9. For a cluster s that is not a cotwin we write s * for the smallest cluster containing s, whose parent is notübereven. If no such cluster exists we write s * = R. If s is a cotwin, we write s * for its child of size 2g.
Definition 2.10. For a proper cluster s we write g ss (s) for the semistable genus of s. If s is noẗ ubereven the genus is determined by |s| = 2g ss (s) + 1, or 2g ss (s) + 2.
It is important to note that g ss (R) is not necessarily the same as g(C); in fact, they will only be the same when R has no proper children. If C has semistable reduction over L and s ∈ Σ C/K is principal, the semistable genus of s represents the genus of the contribution of s to the special fibre of the minimal semistable model of C over L.
We also need some new terminology, and the reminder of the definitions in this section are not given in [DDMM18].
Definition 2.11. A cluster picture Σ is nested if for all proper clusters s, s ′ ∈ Σ either s ⊆ s ′ , or s ′ ⊆ s. If C is a hyperelliptic curve, we say C is nested if Σ C is nested.
Since the elements of R lie in K, there is a natural action of G K on R, hence also on Σ C . Since K has algebraically closed residue field, G K = I K where I K is the inertia subgroup of G K . It will be important later to know exactly how G K acts on the clusters of Σ C . The following lemma is useful for this purpose.
Lemma 2.12. Let Σ C be such that K(R)/K is a tame extension, and let s ∈ Σ C be a proper cluster fixed by G K .
(i) There exists a centre z s of s such that z s ∈ K.
(ii) Any child s ′ < s is in an orbit of size b s , except possibly for one child s f , where we can Definition 2.13. Let s ′ < s be clusters in Σ C . Then s ′ is a stable child of s if the stabiliser of s also stabilises s ′ . Otherwise s ′ is an unstable child of s.
Remark 2.14. Let s ∈ Σ C be fixed by G K . If s has depth d s with denominator > 1 then, by Lemma 2.12 ii), s has at most once stable child.
Definition 2.15. If r ∈ s is a root which is not contained in a proper child of s then we call r a singleton of s. Define s sing to be the set of all singletons of s. In other words s sing is the set of all children of size 1 of s.

Background -Models
Let C be a hyperelliptic curve over K. A model X of C is a flat scheme over O K which has generic fibre X K isomorphic to C/K. We will insist that all of our models are proper over O K . Strict normal crossing (SNC) models are models with are regular as schemes and whose special fibre X k is an SNC divisor -that is, a curve over k whose worst singularities are normal crossings. Note that we do not insist that the irreducible components themselves are smooth. For a given curve, there is a unique SNC model X min which is minimal in the sense that any map of SNC models X min → X is an isomorphism ([Liu02, Proposition 9.3.36]).
Another class of models that are of particular interest to us are semistable models. These are SNC models which have a reduced special fibre. Curves which have a semistable model are said to have semistable reduction. The minimal SNC models of such curves can be calculated explicitly from the cluster picture, this is done in [DDMM18].
In this section we collate some facts about models from [Lor90], [CES03], and [DDMM18] for the convenience of the reader. Similar techniques concerning quotients of models are also used in [Hal10].
3.1. Chains of Rational Curves. Chains of rational curves are ubiquitous in our descriptions of SNC models. The following definition explains what we mean by a chain of rational curves and defines the three main types of chains that we are interested in: tails, linking chains and crossed tails.
Definition 3.1. Let X be a SNC model of a hyperelliptic curve defined over K. Suppose E 1 , . . . , E λ are smooth irreducible rational components of where (E · F ) is the usual intersection pairing defined on regular models. If (E λ · X k \ C) = 0 then C is a tail. If (E λ · X k \ C) = 1 then C is a linking chain.
We say a chain of rational curves C = λ i=1 E i is a loop if C is a linking chain such that E 1 and E λ both intersect the same component of X k \ C.
Furthermore, if (E λ ·X k \ C) = 2 then C is a crossed tail if E λ intersects two rational components of X k \ C, say E + λ+1 and E − λ+1 , such that (E ± λ+1 · E λ ) = 1 and (E ± λ+1 · X k \ E λ ) = 0. We call the components E ± λ+1 the crosses. Illustrations of the definitions of tails, linking chains, loops, and crossed tails are shown in Figures 1 and 2 in Section 1.
Blowing down a component E results in a regular model if and only if it is rational and has self intersection −1 (Castelnuovo's Criterion, [Liu02, Theorem 9.3.8]). However, blowing down a general rational component of X k of self intersection −1 will not necessarily produce an SNC model. For example blowing down the component of multiplicity 3 in the minimal SNC model of an elliptic curve of Kodaira type IV (shown in Figure 9 below) is no longer an SNC model. After  blowing down a component of a chain of rational curves of self-intersection −1, the special fibre is still an SNC divisor. Therefore, we will be interested in blowing down all such components. If a chain of rational curves cannot be blown down any further, we call it minimal.
Quotients of Models. This section collates several results from [Lor90] and [CES03] concerning taking quotient of models. Let C be a hyperelliptic curve over K and let L/K be a tame field extension of degree e such that C L = C × K L is semistable over L. Note that the cluster picture of C L /L is the same as the cluster picture of C/K, except all the depths have been multiplied by e. Since k is algebraically closed, the extension L/K is totally tamely ramified, hence L/K is Galois with Gal(L/K) cyclic.
Let Y be the minimal semistable model of C L /O L , so Y k is a reduced, SNC divisor of Y . Any σ ∈ Gal(L/K) induces a unique automorphism of Y of the same degree which makes the following diagram commute [Lor90,p. 136]: Although a slight abuse of notation, we will also refer to this automorphism on Y as σ, and define G = σ : Y → Y where σ generates Gal(L/K). The model Y , as well as the automorphism induced on the special fibre, will be given more explicitly in Section 3.3.
Since Y is projective, the quotient Z = Y /G given by q : Y → Z can be constructed by glueing together the rings of invariants of G-invariant affine open sets of Y . The resulting scheme Z /O K is a model of C/K. Furthermore, since Z is a normal scheme, its singularities are closed points lying on the special fibre Z k . The following proposition, from [Lor90,p. 137], gives the multiplicities of the components of Z k .
Blowing up a singularity on Z k results in a chain of rational curves, as in Definition 3.1. The following theorem tells us that we can blow up the singularities on Z k to obtain an SNC model of C/K, after blowing down all rational components in chains with self intersection −1, from which we can obtain a minimal SNC model. Theorem 3.4. There exists a regular SNC model X /O K and a proper birational map φ : X → Z such that φ is an isomorphism away from the singular locus of Z , and for any singular point z ∈ Z , φ −1 (z) = λ i=1 E i is a minimal chain of rational curves. In order to describe the intersection of this chain with the rest of the special fibre we set δ(z) to be the number of irreducible components of Z on which z lies, that is δ(z) = 1 or 2. The chain φ −1 (z) intersects the rest of the special fibre X k in precisely δ(z) points. If δ(z) = 1 then The singularities on Z k are tame cyclic quotient singularities, and there is a precise description of the chain of rational curves that arises after resolving them. We will prove in Proposition 5.11 that singularities z ∈ Z k which lie on precisely one irreducible component of Z k are tame cyclic quotient singularities. The definition is as follows: Definition 3.5. Let S be a scheme over O K and let s ∈ S be a closed point. The point s is a tame cyclic quotient singularity if there exists -a positive integer m > 1 which is invertible in k, We will call the pair (m, r) the tame cyclic quotient invariants of s. The following theorem, [CES03, Theorem 2.4.1], tells us how to resolve tame cyclic quotient singularities.
Theorem 3.6. Let S be a flat, proper, normal curve over O K with smooth generic fibre. Suppose s ∈ S k is a tame cyclic quotient singularity with tame cyclic quotient invariants (m, r), as in Definition 3.5 above.
Consider the Hirzebruch-Jung continued fraction expansion of m r given by Then the minimal regular resolution of s is a chain of rational curves Remark 3.7. Note that in [CES03] the E i are labeled in the opposite order. Instead we use the same labeling of the components in our chain as in both [Dok18], and [Lor90].
3.3. Semistable Models. In order to take the quotient of Y , a semistable model of C/L, we must first know what Y looks like, as well as what the Galois action is. The point of this section is to describe just that, following [DDMM18].
A critical step in the proof of the main theorem in this paper will be extending the field so that C has semistable reduction. The following theorem, a criterion for C to have semistable reduction in terms of the cluster picture of C, allows us to do just that. First we need the following definition: Theorem 3.9 (The Semistability Criterion). Let C : y 2 = f (x) be a hyperelliptic curve, and let R be the set of roots of f (x) in K. Then C has semistable reduction over L if and only if (i) the extension L(R)/L has ramification degree at most 2, (ii) every proper cluster of Σ C/L is I L invariant, (iii) every principal cluster s ∈ Σ C/L has d s ∈ Z and ν s ∈ 2Z.
Once the field has been extended so that C has semistable reduction, there is a very explicit description of the special fibre of Y in terms of the cluster picture of C in [DDMM18, Theorem 8.5]. For this we need some definitions. To simplify some invariants, we assume that all clusters s ∈ Σ C/K have eδ s > 1 2 , since a cluster s with eδ s = 1/2 introduces singular irreducible components. This will be sufficient for our purposes since these invariants are used to describe the explicit automorphism on Y k and we can always extend our field so that the minimal semistable model has no singular components. Note that the valuation on K is normalised with respect to K, such that the valuation of a uniformiser π L of L is v K (π L ) = 1 e . Definition 3.10. For σ ∈ G K let For a proper cluster s ∈ Σ C define Define θ s = c f r / ∈s (z s − r). If s is either even or a cotwin, we define ǫ s : For all other clusters s set ǫ s (σ) = 0. [DDMM18, Definition 8.2] When we write ǫ s without reference to some σ ∈ Gal(L/K), this means ǫ s (σ) for σ ∈ Gal(L/K) a generator.
Remark 3.11. Our definition of λ s differs slightly to that in [DDMM18]. In [DDMM18] λ s is defined to be νs 2 −d s s ′ <s,δ s ′ > 1 2 ⌊ |s ′ | 2 ⌋, and a second quantityλ s is defined to be νs 2 −d s s ′ <s ⌊ |s ′ | 2 ⌋. This is to account for singular components of the special fibre. Given our assumption that every cluster in Σ C/L has relative depth > 1 2 , when we calculate these for C/L we find that λ s =λ s , so for simplicity we do not write the tilde.
For s ′ < s define red s (s ′ ) = red s (r) for any r ∈ s ′ .
Theorem 3.13. Let C/K be a hyperelliptic curve and L/K an extension over which C is semistable. Let C/L have minimal semistable model Y over O L . Then Y k has an irreducible component Γ s,L for every non-übereven, principal cluster s ∈ Σ C/L , and two components Γ + s,L and Γ − s,L for everyübereven, principal cluster s ∈ Σ C/L . These are linked by chains of rational curves as follows (where we write Γ + s,L = Γ − s,L = Γ s,L when s is not ubereven): is not principal, then we also get the following chains or rational curves.
In addition, the component Γ s associated to a principal cluster s ∈ Σ C/L is a hyperelliptic curve over k given by the equation Remark 3.14. The equations for Γ s,L given here differs slightly to that given in [DDMM18] since the authors there allow for irreducible components to be singular.
For such semistable models, we are also able to explicitly describe the Galois action from Section 3.2, using the invariants given in Definitions 3.8 and 3.10.
Theorem 3.15. Let C be a hyperelliptic curve as above. Let σ ∈ G K . Then, with notation as in Theorem 3.13, so s is principal and non-übereven, σ permutes the components of Y k in the following way: orientation. This induces an action on the remaining components of Y k . In addition, if s is a principal cluster then σ maps Γ s,L to Γ σ(s),L and Remark 3.16. The quantity ǫ s (σ) = −1 if and only if σ swaps the two points at infinity of Γ s,L . It is also worth noting that ǫ s = (−1) ν s * −|s * |d s * since This concludes the required information about semistable models and their relation to cluster pictures.

Background -Models of Curves via Newton polytopes
In this section we describe a method from [Dok18] for calculating a SNC model of a curve C/K which is ∆ v -regular. The notion of ∆ v -regularity, given in [Dok18, Definition 3.9], applies to more general smooth projective curves, so is too broad for our purposes. Instead we restrict our attention to the case where C has a nested cluster picture, and note that this condition implies ∆ v -regularity. The results of this section will be applied directly in Section 6. 4.1. Newton polytopes. Here we briefly collate the key definitions regarding Newton polytopes necessary for this paper. We begin with the definition of a Newton polytope.
The Newton polytopes of C over K and O K respectively are: Above every point P ∈ ∆ there is exactly one point (P, v K (P )) ∈ ∆ v . This defines a piecewise affine function v ∆(C) : ∆(C) → R. When there is no risk of confusion, we may sometimes write ∆ = ∆(C), and ∆ v = ∆ v (C) and the pair ( and L(Z), F (Z), ∆(Z) to include points on the boundary. We use subscripts to restrict to the set of points P with v K (P ) in a given set, for instance: We associate the following to ∆: Definition 4.4. The denominator δ λ , for every v-face or v-edge λ is defined to be the common denominator of v ∆ (P ) for P ∈ λ(Z). For two alternate, but equivalent, definitions see [Dok18, Notation 3.2].
Remark 4.5. We shall see that the denominator of a v-face or v-edge λ, in some sense, tells us the multiplicity of the component or chain of the SNC model arising from λ. Roughly, for a v-face F , δ F is the multiplicity of the component Γ F , and for a v-edge L, δ L is the minimum multiplicity appearing in the chain of rational curves arising from L.
We distinguish between v-edges which lie on precisely one or two v-faces of the Newton polytope, the former giving rise to tails and the latter to linking chains.
With these definitions in mind, let us move onto a description of the SNC model in terms of Newton polytopes.

4.2.
Calculating a Model. Before we begin, we give a few constants related to v-faces and v-edges which will be necessary for our description.
where v i is the unique affine function Z 2 → Q that agrees with v ∆ on F i .
Theorem 4.9. Suppose C : y 2 = f (x) is a nested hyperelliptic curve over K. Then there exists a regular model C ∆ /O K of C/K with strict normal crossings. Its special fibre is as follows: (i) Every v-face F of ∆ gives a complete smooth curve Γ F /k of multiplicity δ F and genus (1) Remark 4.10. In (1), λ = 0 indicates that Γ F1 and Γ F2 intersect |L(Z) Z | − 1 times in the inner case, and that L contributes no tails in the outer case.
Remark 4.11. An explicit equation for Γ F is given in [Dok18, Definition 3.7], where it is denoted by X F . However this is more information than necessary for our situation so we do not give this description here. A description of a similar object X L is also given in [Dok18, Definition 3.7], and in Theorem 3.13 of [Dok18] the number of rational chains that a v-edge L gives rise to is described in terms of |X L (k)|. However, it is straightforward to show that in our case |X L (k)| = |L(Z) Z | − 1, so we omit this description also.
Remark 4.12. To see that the sequences in Theorem 4.9 exist, take all numbers in [s L 2 , s L 1 ] ∩ Q of denominator ≤ max{denom(s L 1 ), denom(s L 2 )} in decreasing order. This is essentially a Farey series, so satisfies the determinant condition in (1). One can then repeatedly remove, in any order, terms of the form where (a + c) and (b + d) are coprime, until no longer possible. This corresponds to blowing down P 1 s of self intersection -1 (see Remark 3.16 in [Dok18]). The resulting minimal sequence is unique (else this would contradict uniqueness of minimal SNC model), and still satisfies the determinant condition. If (s L 2 , s L 1 ) ∩ Z = {N, . . . , N + a} = ∅ this minimal sequence has the form with d 0 , . . . , d h strictly decreasing and d l , . . . , d λ+1 strictly increasing. If (s L 2 , s L 1 ) ∩ Z = ∅ this minimal sequence has the form with d 0 , . . . , d l strictly decreasing, d l+1 , . . . , d λ+1 strictly increasing, and d i > 1 for all 1 ≤ i ≤ λ.
Notice that shifting either s L 1 or s L 2 by an integer does not change the denominators d i , that appear in this sequence. If s 2 > 0 (else shift by an integer), the numbers are the approximants of the Hirzebruch-Jung continued fraction expansion of s L 2 , similarly for The following definition allows us to talk about different parts of chains of rational curves arising from v-edges in the Newton polytope of C.
Definition 4.13. Let t 1 , t 2 ∈ Q and µ ∈ N. Pick m i , d i as in Theorem 4.9; that is, such that If A is non-empty, let a 0 be the minimal element of A and let a 1 the maximal element of A. Suppose C = λ i=1 E i is a chain of rational curves where E i has multiplicity µd i . Then C is a sloped chain of rational curves with parameters (t 2 , t 1 , µ) and we split C into three sections. If A = ∅ we define the following: If instead A = ∅ we define: and there is no level section.
We define the length of each section to be the number of E i contained in it, and each section is allowed to have length 0. For instance, the level section has length 0 if and only if A = ∅, and the downhill section has length 0 if and only if 1 ∈ A.
Remark 4.14. A tail is a sloped chain with level section of length 1 and no uphill section. Therefore any tail can be given by just two parameters, namely t 1 and µ (since t 2 = 1 µ ⌊µt 1 − 1⌋). We will often refer to a tail as a tail with parameters (t 1 , µ). It follows from Remark 4.12 that a tail with parameters (t 1 , µ) has the same multiplicities as the tail obtained by resolving a tame cyclic quotient singularity with tame cyclic quotient invariants 1 µt1 .
Remark 4.15. All of our chains of rational curves, be they tails, linking chains or crossed tails, are sloped chains. For example, a linking chain in a semistable model will consist of only level section. Both tails and crossed tails in a minimal SNC model will have no uphill section.

Curves with Tame Potentially Good Reduction (The Base Case)
In this section we calculate the minimal SNC model of a hyperelliptic curve C/K with genus g = g(C) ≥ 1 which has tame potentially good reduction. That is, there exists a field extension L/K of degree e such that e and p are coprime, and C has a smooth model over O L . In order to calculate this model, we assume that L is the minimal such extension. The minimal SNC model of a hyperelliptic curve has a rather straightforward description: it consists of a central component with some tails (in the sense of Definition 3.1) whose multiplicities can be explicitly described using the results of Section 3.2. The size and depth of the unique proper cluster s, as well as the valuation of the leading coefficient c f will be sufficient to calculate the (dual graph with multiplicity of the) minimal SNC model of C over K: Theorem 5.1. Let C be hyperelliptic curve over K with tame potentially good reduction. Then the special fibre X k of the minimal SNC model X of C/K consists of a component Γ = Γ s,K , the central component, of multiplicity e. Furthermore, if e > 1 then the following tails intersect the central component Γ: Remark 5.2. The genus of the central component can be calculated using Riemann Hurwitz, and we prove an explicit formula for it in Proposition 5.26.
Since C has tame potentially good reduction, by [DDMM18, Theorem 1.8(3)] we can assume (possibly after a Möbius transform) that that the cluster picture of C over K consists of a single proper cluster s. After an appropriate shift of the affine line we can assume that s is centered around 0 and that C is given by one of the following two equations: where the u r ∈ K are units.
We will proceed in the manner of Section 3.2. Let Y be the smooth Weierstrass model of C over L (the hyperelliptic curve over O L given by y 2 = f (x)), and let q : Y → Z be the quotient map induced by the action of Gal(L/K). We will explicitly describe the singular points of Z , show that they are tame cyclic quotient singularities in the sense of Definition 3.5, and give their tame cyclic quotient invariants in Proposition 5.11. Theorem 3.6 then tells us the self intersection numbers of the rational curves in the tails obtained by resolving the tame cyclic quotient singularities. After using intersection theory, this allows us to describe the special fibre of the minimal SNC model X of C/K in full.
5.1. The Automorphism and its Orbits. To describe the singularities on Z k , we must first explicitly describe the Galois automorphism on the unique component Γ s,L = Γ ⊆ Y k of the special fibre of the smooth Weierstrass model of C over L. The following fact from [Lor90, Fact IV p. 139] describes the singularities of Z k in terms of the quotient q : Y → Z .
Proposition 5.3. Let z 1 , . . . , z d be the ramification points of the morphism q : Γ → Z k . Then {z 1 , . . . , z d } is precisely the set of singular points of Z k .
Furthermore, the ramification points of q correspond to points whose preimage is an orbit of size strictly less than e.
Definition 5.4. Let X be an orbit of clusters. If |X| < e, we say that X is a small orbit.
Hence describing the singular points of Z k is equivalent to describing the small orbits of Gal(L/K). In order to list these orbits, note that some of the cluster invariants from Definitions 3.8 and 3.10, and Theorem 3.15 can be simplified in this tame potentially good reduction case. In particular: Lemma 5.5. Let C/K be a hyperelliptic curve with tame potentially good reduction and unique proper cluster s. Then: Proof. This follows directly from the definitions in Definitions 3.8 and 3.10, and Theorem 3.15.
By Theorem 3.15, any σ ∈ Gal(L/K) induces the following automorphism on Γ: Since α s (σ) and γ s (σ) are non-zero and k is algebraically closed, the only points which can lie in orbits of size strictly less than e are points at infinity, or points where x = 0 or y = 0. This gives four cases which we will take care to distinguish between, as it will make it easier to describe the minimal SNC model for a general cluster picture. With this in mind we make the following definitions: Definition 5.6. We split the small orbits that can occur into the following types.
The following lemmas describe in which situations we see these small orbits. We will assume e > 1 since no small orbits occur when e = 1. Proof. Let u = 1/x, v = y/x g+1 denote the coordinates at infinity. The curve C has a single point at infinity (u, v) = (0, 0) if deg(f ) is odd, and two points at infinity (u, v) = (0, ± √ c f ) if deg(f ) is even. In the latter case, the action of the automorphism shown in Equation (4) gives that the action at infinity is given by σ : (0, c f ) → (0, χ(σ) eλs c f ). Therefore, when deg(f ) is even, the points at infinity are swapped if and only if χ(σ) eλs = −1 for some σ ∈ Gal(L/K). This is the case if and only if v K (c f ) is odd. In this case, the orbit at infinity has size 2 and is only a small orbit if e > 2. Proof. If f (0) = 0 then {(0, 0)} ∈ Γ is the unique (0, 0)-orbit. If f (0) = 0 then (0, ± c f ) ∈ Γ, and these points are swapped by some element of the Galois group (see Equation (4)) if and only if λ s ∈ Z. If λ s ∈ Z then the orbit has size 2 hence it is only a small orbit if e > 2. Proof. By Theorem 3.9, e is the minimal integer such that ed s ∈ Z and eν s ∈ 2Z. Since ed s ∈ Z, we can deduce that b s | e. Furthermore, since 2b s ν s ∈ 2Z, e = b s or e = 2b s . It is straightforward to check that the other conditions of Theorem 3.9 are satisfied over a field extension of degree e. Proof. This follows since the non-zero points with y = 0 are of the form (ζ i bs , 0) for ζ bs a primitive b th s root of unity. Note that (y = 0)-orbits always have size b s so if e = b s then the (y = 0)-orbits are not small orbits.
These lemmas allow us to fully describe how many singularities Z k has. The following proposition tells us that they are tame cyclic quotient singularities in the sense of Definition 3.5. Theorem 3.6 then allows us to resolve these singularities.
Proposition 5.11. Let z ∈ Z k be a singularity which is the image of a Galois orbit Y ⊆ Y k . Then z is a tame cyclic quotient singularity. In addition, with notation as in Definition 3.5, m r = e r where 1 ≤ r < e and r mod e is given in the following table: Recall that for z to be a tame cyclic quotient singularity, there must exist m > 1 invertible in k, a unit r ∈ (Z/mZ) × and integers m 1 > 0 and m 2 ≥ 0 such that m 1 ≡ m 2 mod m, and such that O Z ,z is equal to the subalgebra of µ m -invariants in k t 1 , t 2 /(t m1 1 t m2 2 − π K ) under the action t 1 → ζ m , t 2 → ζ r m . We will show that m = e |Y | = |Stab(Y )|, m 1 = e, m 2 = 0 and will explicitly calculate r.
Let Y ⊆ Y k be a small orbit and let Q ∈ Y . Then O Z ,z is the subalgebra of µ m -invariants of O Y ,Q under the action of Stab(Y ), where m = |Stab(Y)|. This follows from the definition of Z as the quotient of Y under the action of Gal(L/K), which for a generator σ ∈ Gal(L/K) sends To prove that z is a tame cyclic quotient singularity we must calculate O Y ,Q .
First, suppose Y is a (y = 0) or a (0, 0)-orbit, and write Q = (x Q , 0). Then O Y ,Q is generated by π L , x − x Q and y. However, since x − x Q = uy 2 for a unit u ∈ O Y ,Q , O Y ,Q is generated by π L and y. Therefore, O Y ,Q ∼ = k π L , y /(π e L − π K ), and O Z ,z is the subalgebra of µ m -invariants of this under the action π L → ζ m π L , y → ζ eλs m y where ζ m = χ(σ) |Y | generates Stab(Y ) (as Gal(L/K) is cyclic). Let r be such that 0 < r < m and r ≡ eλ s |Y | mod m. Then to prove z is a tame cyclic quotient singularity all that is left to show is that r is a unit in (Z/mZ) × and that e ≡ 0 mod m. The second is clear, and for the first note that since ζ r m also generates Stab(Y ), it must be a primitive m th root of unity hence r must be a unit.
If Y is an (x = 0)-orbit, then Q = (0, ±1). By a similar argument to above, O Y ,Q ∼ = k π L , x /(π e L −π K ) and O Z ,z is the subalgebra of µ m invariants under the action π L → ζ m π L , x → ζ r m x, where m = e |Y | and r is such that 0 < r < m and r ≡ ed s |Y | mod m. If Y is an ∞ orbit, then we can calculate m, r, m 1 and m 2 explicitly by going to the chart at infinity.
Lemma 5.9 allows us to do even better for the (y = 0)-orbits.
Corollary 5.12. If Y is a (y = 0)-orbit which gives rise to a tame cyclic quotient singularity z ∈ Z k , then the tame cyclic quotient invariants (m, r) of z are such that m r = 2.
Proof. The orbit Y is a (y = 0)-orbit hence has size b s . Lemma 5.9 tells us that, |Y | < e if and only if e = 2b s . In this case eλ s |Y | = 2b s · νs 2 · b s = b 2 s ν s . Since b s = e 2 and b s ν s is an odd integer, this gives eλ s |Y | ≡ e 2 mod e, hence m r = 2.
Now that we know that the singularities of Z k are tame cyclic quotient singularities, and that (by Theorem 3.4) the resolution of each of these is a chain of rational curves, let us develop some language to distinguish between their resolutions. 5.2. Tails. Resolving singularities in Section 5.1 will result in tails: chains of rational curves intersecting the central component once and intersecting the rest of the special fibre nowhere else. It will be helpful to distinguish between the different tails that can arise based on the type of orbit they arise from.
Definition 5.13. Define the following tails depending on the type of singularity of Z k they arise from: • ∞-tail : arising from the blow up of a singularity of Z k which arose from an ∞-orbit, • (y = 0)-tail : arising from the blow up of a singularity of Z k which arose from an orbit of non-zero roots, • (x = 0)-tail : arising from the blow up of a singularity of Z k which arose from an orbit on the points (0, ± √ c f ), • (0, 0)-tail : arising from the blow up of a singularity of Z k which arose from the point (0, 0).
Remark 5.14. The tails defined in Definition 5.13 are the only tails that can possibly occur in X k . This is because any tail must arise from a singularity of Z k which lies on just one component, namely a singularity which arises from one of the small orbits discussed in Section 5.1.
We now move onto the proof of Theorem 5.1.
Proof of Theorem 5.1. The central component Γ is the image of the unique component of Y k under q. Since blowing up points on Γ does not affect its multiplicity, this has multiplicity e, by Proposition 3.3. The description of the tails follows from Lemmas 5.7, 5.8, and 5.10, since the tails are in a bijective correspondence with the orbits of points of Y k of size strictly less than e. We must check that Γ really appears in the minimal SNC model. Suppose Γ is exceptional. Then g(Γ) = 0, hence Riemann-Hurwitz says Therefore there must be at least three ramification points, so Γ intersects at least three tails.
Remark 5.15. The method for calculating the multiplicities of the rational curves in these tails is described in Theorem 3.6 using the tame cyclic quotient invariants given in Proposition 5.11. An even more explicit calculation of g(Γ) in terms of the Newton polytope is given in Proposition 5.26.
5.3. Relation to Newton polytopes. Up to this point, this section has described the minimal SNC model of a hyperelliptic curve C/K with tame potentially good reduction using the methods from Section 3.2. However, such a hyperelliptic curve has a nested cluster picture so we can also calculate the minimal SNC model using Newton polytopes and the techniques described in Section 4. By the uniqueness of the minimal SNC model, these two methods will give the same result: for the reader's sanity, in this section we will show that this is indeed the case. Recall that without loss of generality we can assume that C/K with tame potentially good reduction is given by one of the following two equations: The Newton polytope of C is shown in Figure 10a if b s | |s|, and in Figure 10b if b s ∤ |s|. In each case there is exactly one v-face of ∆ v (C), which we shall label F . Therefore, by Theorem 4.9, the minimal SNC model consists of a central component Γ s = Γ F , and possibly tails arising from the three outer v-edges of F . Figure 10. Newton polytope ∆ v (C) of a hyperelliptic curve C with tame potential good reduction.
Lemma 5.17. The multiplicity of Γ s = Γ F is δ F ; that is δ F = e.
Proof. We will first show that e | δ F , and then that δ F | e. Note that, in both Newton polytopes in Figure 10, the valuation map is given by the affine function v ∆ (x, y) = ν s − d s x − νs 2 y. Since e is such that ed s ∈ Z and eν s ∈ 2Z, we have ev ∆ (x, y) = eν s − ed s x − e νs 2 y ∈ Z. As δ F is the common denominator of all v ∆ (x, y) for x, y ∈ ∆, this gives that δ F | e.
Note that By minimality of e, this implies e | δ F .
Proof. We will first check that this v-edge gives the correct number of ∞-tails, and then calculate the slope to check that the multiplicities of the components are the same.
Let us call this v-edge L. By Theorem 4.9 then L contributes |L(Z) Z |−1 tails to the SNC model. Since the points (0, 2), (|s|, 0) ∈ L(Z) Z , it contributes two tail if and only if P = ( |s| 2 , 1) ∈ L(Z) Z . If s is odd then P ∈ L ∩ Z 2 , hence L contributes one tail. If s is even then . Therefore L contributes one tail if s is even and v K (c f ) is odd, and two tails if s and v K (c f ) are even. This agrees with Theorem 5.1.
A quick calculation tells us that δ L = 2 if and only if s is even and v K (c f ) ∈ 2Z, and that δ L = 1 otherwise. Therefore, δ L = |Y |, where Y is the orbit at infinity. The unique surjective affine function which is zero on L and non-negative on F is L * F (x, y) = 2|s| − 2x − |s|y if s is odd, and L * F (x, y) = |s|−x− 1 2 |s|y if s is even. Therefore, s L 1 = (g +1)d s −λ s if s is odd, and s L 1 = −d s |Y | if s is even. Since the multiplicities of the components of a tail are the Hirzebruch-Jung approximants of the slopes, we are done after comparing the slopes to the table in Proposition 5.3.
If e = 2 (when s is even and v K (c f ) is odd) has s 1 L ∈ Z, so the associated tail is empty, which agrees with the table in Theorem 5.1.
Lemma 5.19. In both cases, when 0 ∈ R and when 0 / ∈ R, the (y = 0)-tails arise from the outer v-edge of ∆ v (C) on the x-axis.
Proof. Denote this v-edge by L. As above, we check that |L(Z) Z | = |s sing |/b s and that s L 1 = −λ s b s . Then, by comparing to the tables in Proposition 5.3 and Theorem 5.1, we are done.
First let us calculate |L(Z) Z |. The valuation on the x-axis is given by v ∆ (x, 0) = ν s − d s x. Since ν s ∈ Z, we have that v ∆ (x, 0) ∈ Z if and only if b s | x. From this we see that |L(Z) Z | = |s sing |/b s . Now, δ L = b s , the size of any (y = 0)-orbit, and the unique surjective affine function which is zero on L and non-negative on F is L * (F ) (x, y) = y. Therefore, Observe that this gives rise to a non-empty tail if and only if s L 1 ∈ Z, which occurs if and only if e = 2b s .
Proof. This is proved by a similar method to that in the proofs of Lemmas 5.18 and 5.19.

The Curve
Cs. To finish off this section, we drop the requirement for C/K to have tame potentially good reduction. We will describe a hyperelliptic curve with potentially good reduction which we associate to a principal cluster s ∈ Σ C with g ss (s) > 0. This new curve, which we will denote by Cs, will be invaluable in describing the components of the minimal SNC model of C/K which are associated to s ∈ Σ C . For s ∈ Σ C/K with g ss (s) > 0, the cluster picture Σs of Cs/K will be such that the singletons in Σs correspond to odd children of s and the even children of s are in effect discarded. The leading coefficient of Cs/K is chosen so that everything behaves well, and allows us to make the comparisons we wish between the minimal SNC model of C/K and the minimal SNC model of Cs/K. We describe this formally now.
Definition 5.21. Let C/K be a hyperelliptic curve, not necessarily with tame potentially good reduction. Let s ∈ Σ C/K be a principal cluster with g ss (s) > 0 such that s is fixed by G K . Suppose furthermore that σ(z s ′ ) = z σ(s ′ ) for any σ ∈ G K , s ′ ∈ Σ C/K . We define another hyperelliptic curve Cs/K by Remark 5.22. The cluster picture Σs /K of Cs/K has a unique cluster, namely the cluster consisting of all roots, which we have labeled Rs above.
Remark 5.23. Let Y be the minimal semistable model of C over O L , for some L/K such that C/L is semistable. Let s be a principal cluster with g ss (s) > 0. If we reduce Cs mod m, we obtain Γ s,L , the component of Y k corresponding to s (see Theorem 3.13 for the equation of Γ s,L ). In addition, c fs has been carefully chosen so that d s = ds, ν s = νs and λ s = λs. In particular, the automorphisms induced by Galois on Γ s,L and Γs ,L are the same.
Definition 5.24. For a principal, Galois-invariant cluster s, define e s to be the minimum integer such that e s d s ∈ Z and e s ν s ∈ 2Z. Furthermore, if g ss (s) > 0 define g(s) to be the genus of Γs and if g ss (s) = 0 define g(s) = 0. We call g(s) the genus of s.
We give an explicit formula for g(s) shortly.
Remark 5.25. By the Semistability Criterion in Theorem 3.9, if s is notübereven then e s is the minimum integer such that Cs has semistable reduction over a field extension L/K of degree e s . In particular, the central component Γs of Xs ,k has multiplicity e s and genus g(s). If e s = 1 then g ss (s) = g(s), but the converse is not necessarily true.
Proposition 5.26. If g ss (s) > 0, the genus g(s) is given by Proof. By Theorem 4.9, we know g(s) is given by |F (Z) Z |. This is the number of interior points with integer valuation of the unique face F of the Newton polytope of C s . By examining Figure  10, we see that all interior points are of the form (x, 1) with 1 ≤ x ≤ g ss (s). For such points, v ∆ (x, 1) = λ s − d s x. Therefore, When λ s ∈ Z this is therefore equal to When λ s ∈ Z, this is equal to When λ s ∈ Z and b s is odd this set is always empty, and when λ s ∈ Z and b s is even it has size gss(s) bs + 1 2 .
We finish off this section with a lemma which will be helpful in the future.
Lemma 5.27. Let C be a hyperelliptic curve and let s ∈ Σ C/K be a principal cluster which is fixed by Galois. Let L be an extension such that C is semistable over L, and let σ generate Gal(L/K). Then σ| Γs,L : Γ s,L → Γ s,L has degree e s .
Proof. The map σ| Γs,L is given by (x, y) → (χ(σ) eds x, χ(σ) eλs y). The result follows as e s , by definition, is the minimal integer such that e s d s , e s λ s ∈ Z.

Calculating Linking Chains
The minimal SNC model of a general hyperelliptic curve C/K can roughly be described as follows. Each principal cluster of Σ C has one or two central components, and some tails associated to it. These central components are linked by chains of rational curves. Section 5 will allow us to describe these central components and tails, while this section will be used to describe these linking chains. This includes describing any loops, as these are just linking chains which go to and from the same central component. In this section we will also see the simplest example of the general philosophy that the components of the special fibre of the minimal SNC model of C/K associated to a principal cluster s "look like" the special fibre of the minimal SNC model of Cs/K.
Throughout the rest of this section we will take C/K to be a hyperelliptic curve such that such that Σ C/K consists of exactly two proper clusters: a proper cluster s and a unique proper child s ′ < s. This is illustrated in Figure 11. Note that d s ′ > d s and |s| > |s ′ |. If C is such that s is even and |s| = |s ′ | + 1 then C/K has potentially good reduction, this case is covered in Section 5. To avoid this case we will assume that if s is even then |s| ≥ |s ′ | + 2. Figure 11. Cluster picture with a parent s and a unique proper child s ′ with no proper children of its own.
Since hyperelliptic curves of this type are nested we can directly apply the methods from [Dok18] given in Theorem 4.9. Before we apply Theorem 4.9, we need to understand the Newton polytope of C/K. 6.1. The Newton polytope. Without loss of generality, we can assume that the defining equation of C/K will be either or where the u r are units. If C has defining equation (5), then ν s ′ = v K (c f ) + (|s| − |s ′ |)d s + |s ′ |d s ′ , and the Newton polytope ∆ v (C) of C will be as shown in Figure 12a. If instead C has defining equation (6), the Newton polytope will be as shown in Figure 12b.
Lemma 6.1. Let C have Newton polytope as shown in Figure 12a. Then there is an isomorphism ψ : F 1 → ∆ v (Cs), from the closure of the v-face marked F 1 to the Newton polytope of Cs (whose only v-face we label Fs), shown in Figure 13. In particular ψ preserves valuations and δ F1 = δ Fs . In this sense we say that F 1 corresponds to the cluster s. Similarly the v-face F 2 in Figure 12a corresponds to s ′ . Figure 12. Newton polytope ∆ v (C) of C.
Proof of Lemma 6.1. Let us compare the v-face F 1 in Figure 12a to the Newton polytope, ∆ v (Cs), of Cs. This is given in Figure 13a if s ′ is even, and given in Figure 13b if s ′ is odd. If s ′ is even we can define It is easy to show that this is an isomorphism, and that the valuations are preserved. Similarly if s ′ is odd we can define which is also an isomorphism that preserves the valuations. In particular, in both cases we have δ F1 = δ Fs , and if v 1 is the unique affine function agreeing with Similarly, we can see that the v-face F 2 in Figure 12a corresponds to s ′ by considering the Newton polytope ∆ v (Cs′ ) of Cs′ . This is shown in Figure 14. We see that the map is an isomorphism that preserves the valuations, that is v 2 (x, y) = v ∆(Cs ′ ) (x, y), and δ F2 = δ Fs ′ , where v 2 is the unique affine function agreeing with v ∆(C) on F 2 . We can make a similar comparison of the v-faces of the Newton polytope in Figure 12b.
Lemma 6.2. Let C have Newton polytope as shown in Figure 12b. Then the v-face marked F 1 on the Newton polytope in Figure 12b corresponds to the cluster s. That is there is a valuation preserving isomorphism between F 1 and ∆ v (Cs), and δ F1 = δ Fs , where Fs is the unique v-face of ∆ v (Cs). Similarly the v-face marked F 2 on the Newton polytope in Figure 12b corresponds to the cluster s ′ .
Proof. Again, we can see that the v-face marked F 1 on the Newton polytope in Figure 12b corresponds to the cluster s by looking at the Newton polytope of Cs. This is shown in Figure 13a if s ′ is even, and in Figure 13b if s ′ is odd. Take ψ to be exactly as in the proof of Lemma 6.1 in both the s ′ even and s ′ odd cases. This gives us an isomorphism between F 1 and ∆ v (Cs ′ ) which preserves the valuations. We can also see that δ F1 = δ Fs . Similarly we can see that the v-face marked F 2 on the Newton polytope in Figure 12b corresponds to the cluster s ′ by looking at the Newton polytope of Cs′ . This is shown in Figure 15. The affine map F 2 → ∆ v (C s ′ ) : (x, y) → (x, y) is an isomorphism which preserves the valuations, and we can see that Fs ′ Figure 15. Newton polytope ∆ v (Cs′ ) of Cs′ , where C has defining equation (6).
6.2. Structure of the SNC Model. The following theorem describes the structure of the special fibre of the minimal SNC model for hyperelliptic curves whose cluster picture looks like Figure 11.
Theorem 6.3. Let C/K be a hyperelliptic curve with cluster picture as in Figure 11. If s is principal, then the special fibre of the minimal SNC model has a component Γ s,K arising from s with multiplicity e s and genus g(s). If s ′ is principal then there is a component Γ s ′ ,K arising from s ′ of multiplicity e s ′ and genus g(s ′ ). These are linked by sloped chain(s) of rational curves with parameters (t 2 , t 2 + δ, µ), which are described in the following table: The chains where the "To" column has been left empty are crossed tails with crosses of multiplicity 1. If s is principal and e s > 1 then Γ s has the following tails with parameters (t 1 , µ): −d s 1 s even and ǫ s = 1 T ∞ 1 −d s 2 s even, ǫ s = −1 and e s > 2 If s ′ is principal and e s ′ > 1 then Γ s ′ has the following tails with parameters (t 1 , µ): Remark 6.4. Note that for this particular type of hyperelliptic curve, s will be principal unless it is a cotwin (i.e. if |s ′ | = 2g(C)), and s ′ will be principal unless it is a twin. Since we have assumed that g ≥ 2, these cases cannot coincide. Note also that neither s nor s ′ can beübereven in this case.
Remark 6.5. Suppose s is principal. In X k we can see most of the components of X s,k . The central component Γ s will have the same multiplicity and genus as Γ s , and will have almost the same tails. The only difference being that one or two of the tails (the (0, 0)-tail in the case s ′ is odd and the (x = 0)-tail(s) otherwise) will instead form either part of a linking chain between Γ s and Γ s ′ (in the case s ′ principal); or a loop or a crossed tail associated to s ′ (in the case where s ′ is a twin). We will say that the downhill section of the linking chain corresponds to this tail. If the linking chain, loop or crossed tail in X k has a non-trivial level section, then all the components of the tails in X s,k appear in the linking chain(s) in X k . If the level section has length zero then some of the lower multiplicity components do not appear -we expand on this in Section 6.3. Similarly, if s ′ is principal, we see most of the components of X s ′ ,k in X k . In this case, Γ s ′ has the same tails as Γ s ′ except that the infinity tail(s) of the latter are absorbed into the linking chain(s) L s,s ′ (or the loop or crossed tail arising from s if it is a cotwin). In this case, we say that the uphill section of the linking chain corresponds to the infinity tail in X s ′ ,k .
We shall see that this is a phenomenon which generalises to the main theorems in Section 7.
Remark 6.6. The length of the level section of a linking chain, loop or crossed tail C ⊆ X k (that is, the number of P 1 s with multiplicity µ) is equal to |(µt 2 , µ(t 2 + δ)) ∩ Z|. Let Y be the minimal regular model of C over L, q : Y → Z be the quotient by Gal(L/K) and φ : X → Z the resolution of singularities. Then any irreducible component E in the level section of C is not an exceptional divisor -that is to say, it is the image of µ components of Y k which are permuted by Gal(L/K). This can be seen by looking at the explicit automorphisms on the components of Y given in [DDMM18, Theorem 6.2].
We now illustrate Theorem 6.3, and the Remarks following it, with an example.
Example 6.7. Consider the hyperelliptic curve C : y 2 = (x 2 − p)(x 3 − p 5 ) over K = Q ur p . The special fibre of the minimal SNC model of C/K can be seen in Figure 16. The central components Γ s , and Γ s ′ are labeled and shown in bold.
If we consider the curves Cs and Cs ′ and the special fibres of their minimal SNC models we find that they are as pictured in Figure 17 below. We can see that all the components in both    Figure 16. This provides a visualisation of what we mean when we say the tails of Γ s correspond to those of Γs, the tails of Γ s ′ correspond to those of Γs ′ , and that some of these tails form part of the linking chains of the special fibre of the minimal SNC model of C.
The proof of Theorem 6.3 will be presented as a series of lemmas, starting with a lemma describing the central components Γ s and Γ s ′ .
Lemma 6.8. If s is principal then the special fibre has an irreducible component Γ s = Γ F1 of multiplicity e s and genus g(s). If s ′ is principal then there is a component Γ s ′ = Γ F2 of multiplicity e s ′ and genus g(s ′ ).
Remark 6.10. We proved Lemma 6.8 by appealing to Newton polytopes, but we could also have proceeded as follows: let L be the minimal field extension such that C has semistable reduction over L. Note that s has an associated component Γ s,L in the semistable model Y of C over O L , and the image of this component under the quotient map gives Γ s , after resolving the singularities. The multiplicity of Γ s then follows from Proposition 3.3, and the genus from Riemann-Hurwitz. Similarly, Γ s ′ is the image of Γ s ′ ,L when s ′ is principal.
The next two lemmas tell us that Γ s and Γ s ′ have the tails we expect.
Proof. This is a consequence of our discussion above, relating F 1 to the Newton polytope of C s . The conditions in Theorem 6.3 for the tails to occur follow since ǫ s = (−1) vK (c f ) .
Lemma 6.12. If s ′ is principal and e s ′ > 1, the following tails of Γ s ′ arise from outer v-edges of the v-face F 2 in Figure 12, with conditions as in Theorem 6.3: Proof. This is a consequence of our discussion above, relating F 2 to the Newton polytope of C s ′ . The conditions in Theorem 6.3 for these tails to occur follow since ǫ s ′ = (−1) ν s ′ −|s ′ |d s ′ .
In order to find the lengths of the level sections of the linking chains, we must calculate the slopes of the unique inner v-edge L, adjacent to both v-faces F 1 and F 2 in Figure 12.
Lemma 6.13. If s ′ is odd then s L 1 = −λ s and s L 2 = −λ s − δ s ′ 2 . If s ′ is even then s L 1 = −δ L d s and s L 2 = −δ L d s ′ . Proof. Suppose s ′ is odd. Then the only points in L(Z) are the endpoints (0, 2) and (|s ′ |, 0), so δ L = 1. The unique function L * F1 : Z 2 → Z such that L * F1 L = 0 and L * F1 F1 ≥ 0 is given by To calculate s L 1 and s L 2 we need P 0 and P 1 such that L * F1 (P 1 ) = 1 and L * F1 (P 0 ) = 0. We will take P 0 = (|s ′ |, 0) and P 1 = ( |s ′ |+1 2 , 1). The unique affine function which agrees with v ∆ on F 1 is defined by Therefore Similarly, the unique affine function which agrees with v ∆(C) on F 2 is defined by So, If s ′ is even, very similar calculations can be made. In this case P 0 = (0, 2), and P 1 = (1, 2). Then, with v 1 and v 2 as before, we can calculate the slopes and find that s L 1 = −δ L d s , and s L 2 = −δ L d s ′ .
We can now prove Theorem 6.3.
Proof of Theorem 6.3. Recall that e s is the minimum integer such that e s d s ∈ Z, and e s ν s ∈ 2Z.
If e s = 1 then d s , λ s ∈ Z, hence the slopes of the outer v-edges of F 1 are integers and Γ s has no tails. If e s > 1 then Lemma 6.11 describes the tails of Γ s . Similarly if e s ′ = 1 then Γ s ′ has no tails and if e s ′ > 1 then Lemma 6.12 describes the tails of Γ s ′ . The statement about the parameters of the tails and the linking chain follows from Remark 4.12 and the calculation of the slopes in Lemma 6.13. The multiplicity of the level section is δ L where L is the inner v-edge between F 1 and F 2 . The two cases left to worry about are when s ′ is a twin or when s is a cotwin. We will only argue the case where s ′ is a twin, as the case where s is a cotwin is proved similarly. Recall from Remark 3.16 that Suppose that ǫ s ′ = 1. Since v ∆ (0, 2) = 0 ∈ 2Z and |s ′ | = 2 we have that ( |s ′ | 2 , 1) = (1, 1) ∈ Z 2 , and v ∆ (1, 1) ∈ Z. So, |L(Z) Z | = 3 and by Theorem 4.9 there are two linking chains from Γ s to the component Γ F2 arising from the v-face F 2 of ∆ v (C) in Figure 12. The component Γ F2 is exceptional by [Dok18, Proposition 5.2] and the linking chains between Γ s and Γ F2 are minimal. After blowing down Γ F2 , this results in a loop from Γ s to itself.
Suppose instead that, ǫ s ′ = −1. Then there is a single chain of rational curves from Γ s to Γ F2 , and Γ F2 has two rational other rational curves intersecting it transversely (which arise from the v-edge connecting (0, 0) and (0, 2)). Therefore, Γ F2 is not exceptional and must appear in the minimal SNC model. This means, if we consider Γ F2 as a component of the level section, that this chain of rational curves is a crossed tail. 6.3. Small Distances. Let s 1 and s 2 be the principal clusters such that there is a linking chain C ⊆ X k from Γ s1 to Γ s2 . If C has level section of length greater than 0, it is straightforward to compare the multiplicities of C to those of the corresponding tails (see Remark 6.5). All of the multiplicities of the corresponding tails appear in the uphill and downhill sections of C. However, if the level section is empty and the downhill section of C corresponds to a tail, say T 1 , then not all of the multiplicites of T 1 ⊆ X s1,k appear in the downhill section of C. Similarly if the uphill section corresponds to a tail, say T 2 ⊆ X s2,k . We shall show that in this case, T 1 and T 2 "meet" at a component of second least common multiplicity. In other words, if we consider a chain of rational curves C ′ such that C ′ has level section of length 1, and whose downhill and uphill sections correspond to T 1 and T 2 respectively, then we "cut out" a section of C ′ to obtain C. Let us clarify this with an example before we proceed to a prove a more precise statement.
Example 6.14. Consider the hyperelliptic curves given by y 2 = (x 4 − p)(x 5 − p 2+10m ) over K = Q ur p for m ∈ Z ≥0 , with cluster pictures shown in Figure 18. The level section of the linking chain between Γ s and Γ s ′ has length m. Figure 19 shows the special fibres of the minimal SNC models for both when m = 1, and the small distance case (when m = 0). Here we can see that when m = 1 the uphill and downhill sections of the linking chain have a common multiplicity greater than 1, namely 3, and that to obtain the m = 0 case we remove the dashed section of the linking chain and glue back along the multiplicity 3 components.   Figure 19. Example of "cutting out" a section of linking chain to obtain the small distance case.
Let us solidify this with a precise statement.
Roughly, Theorem 6.15, states that when there is no level section, rather than seeing all of the multiplicities of the tails which the uphill and downhill sections correspond to, the two tails "meet" at the component of minimal shared multiplicity greater than µ. Before we prove this theorem, let us prove a couple of lemmas.
Lemma 6.17. Let q 1 , q 2 ∈ Q be such that [q 1 , q 2 ] ∩ Z = ∅. Then there is a unique fraction with minimal denominator in the set [q 1 , q 2 ] ∩ Q, when written with coprime numerator and denominator.
Proof. Suppose not, and suppose r 1 , r 2 ∈ [q 1 , q 2 ] ∩ Q can be written r i = mi d with m i , d coprime and d the minimal denominator of elements in the set [q 1 , q 2 ] ∩ Q. We will show that there exists a rational number r lying between r 1 and r 2 of denominator < d.
Write −1) , and consider the set S = [m 1 (d − 1), m 2 (d − 1)] ∩ Z. Since m 2 > m 1 and m 1 , m 2 ∈ Z, |S| ≥ d and there must exist a multiple of d in S. That is, there exists m ∈ Z such that md ∈ S. Since m i and d are coprime, we have m 1 < md < m 2 . Therefore, which contradicts the minimality of d.
Lemma 6.18. With notation as in Theorem 6.15, there exists some l j < λ j , for j = 1, 2, such that µ (1) l2 . Proof. Write s i = µt i . Recall that we assumed that, [s 2 , s 1 ] ⊂ (0, 1), so [s 2 , s 1 ] ∩ Z = ∅. Let m d be the unique fraction of minimal denominator in [s 2 , s 1 ], which exists by Lemma 6.17. Then if is the reduced sequence giving rise to the linking chain C, as in Remark 4.12, where (m i , d i ) = 1, d 0 > · · · > d l and d l < · · · < d λ+1 for some 1 ≤ l ≤ λ, we must have that d l = d. Consider the following two reduced sequences: These give rise to the multiplicites µ for 1 ≤ i ≤ λ j , j = 1, 2 of the tails T j . We will show that there exist 0 ≤ l 1 < λ 1 + 1 and 0 ≤ l 2 < λ 2 + 1 with d l2 . We will first prove that d (1) l1 = d for some l 1 ∈ Z. Since [s 2 , s 1 ] ⊂ (0, 1), we have that s 2 > ⌊s 1 ⌋ = 0. So, some fraction of denominator d, say m d , appears in the full sequence of fractions in [⌊s 1 ⌋, s 1 ] ∩ Q of denominator less than or equal to max{d 0 , d λ+1 }. To obtain a reduced sequence, we remove all terms of the form as in Remark 4.12. We can only remove m d if there exists some q ∈ Q with denom(q) < d and s 1 > q > m d . No such q can exists since d is the minimal denominator of any element of [s 2 , s 1 ]∩Q. Therefore, m d cannot be removed in the reduction proccess and so must apppear in the reduced sequence. Therefore there exists 0 ≤ l 1 < λ 1 + 1 such that d (1) l1 = d. Proving that there exits 0 ≤ l 2 < λ 2 + 1 such that d l2 is done similarly.
We can now prove Theorem 6.15.
Proof of Theorem 6.15. The fractions m0 d0 , m1 d1 , . . . , m l d l in the reduced sequence depend only on the elements of [s 1 , m l d l ] of denominator less than or equal to max(d 0 , d λ+1 ), as do the fractions We are left to show that l 1 , l 2 are maximal such that µ Suppose there is some r 1 , r 2 such that λ i > r i > l i and µ l1 . In addition to this, d ∈ [0, s 2 ). Let q ′ be the unique rational with least denominator d ′ in [q 1 , q 2 ]. By uniqueness, d ′ < d (1) r1 < d. Therefore, q ′ ∈ (s 1 , q 2 ) or (q 1 , s 2 ). Suppose for now that q ′ ∈ (s 1 , q 2 ), and consider again the reduced sequence However 1 − q 2 cannot appear in this reduced sequence since a fraction with smaller denominator, 1 − q ′ , appears to the left of it in the non-reduced sequence. So, at some step in the reduction process 1 − q 2 would have been removed. Therefore, q ′ ∈ (s 1 , q 2 ). Similarly, one can show that q ′ ∈ (q 1 , s 2 ). This is a contradiction. So no such r 1 and r 2 exist.

Main Theorems
The previous two sections have looked at the minimal SNC models of very specific cases of hyperelliptic curves. In this section, we state our main theorems in full generality. Theorem 7.12 gives the structure of the special fibre of the minimal SNC model of C/K, and Theorems 7.17 and 7.18 give more explicit descriptions of the multiplicities and genera of the components appearing in the special fibre. 7.1. Orbits. Before we can state and prove the main results of this paper, we need to extend some of the definitions of Section 2. Since the definitions in Section 2 come from [DDMM18], where the authors deal only with the semistable case, they do not deal with orbits of clusters. So, here we make some new definitions which extend the preexisting ones to orbits.
Definition 7.1. Let X be a Galois orbit of clusters. Then X isübereven if for all s ∈ X, s is ubereven. Define an orbit X to be odd, even, and principal similarly.
Definition 7.2. Let X be an orbit of clusters. Define K X /K to be the field extension of K of degree |X|.
Remark 7.3. By Lemma 2.12, K X /K is the minimal field extension over which for any s ∈ X, σ ∈ Gal(K/K X ) we have σ(s) = s.
Definition 7.4. Let X be a Galois orbit of clusters, and choose some s ∈ X. Then we define Furthermore, we define ǫ X = ǫ |X| s . Remark 7.5. Note that the invariants defined in Definition 7.4 are well defined, i.e they do not depend on the choice of s ∈ X.
Definition 7.6. An orbit X ′ is a child of X, written X ′ < X, if for every s ′ ∈ X ′ there exists some s ∈ X such that s ′ < s. Define δ X ′ = δ s ′ for some s ′ ∈ X ′ .
Definition 7.7. Let X be a principal orbit of clusters with g ss (X) > 0 and choose some s ∈ X. Then C X is defined to be the curve Cs over K X . We denote the minimal SNC model of C X /K X by X X /O KX , and the central component by Γ X /k.
Remark 7.8. The curve C X depends on a choice of s ∈ X, but the combinatorial description of the special fibre of the minimal SNC model will not. Since this is what we need C X for, we do not need to worry about this.
Definition 7.9. Let X be a principal orbit of clusters. Define e X to be the minimal integer such that e X |X|d s ∈ Z and e X |X|ν s ∈ 2Z for all s ∈ X. Define g(X) = g(s) for s ∈ X over K X , where g(s) is as defined in Definition 5.24.
Remark 7.10. Analogously to Section 5.4, the curve C X /K X is semistable over an extension of K X of degree e X and the quotient map Γ s,L → Γ s,KX has degree e X for s ∈ X.
7.2. The Special Fibre of the Minimal SNC Model. We state here the first of our main theorems. Roughly this tells us that the cluster picture, the leading coefficient of f , and the action of G K on the cluster picture is enough to calculate the structure of the minimal SNC model, along with the multiplicities and genera of the components.
Theorem 7.11. Let K be a complete discretely valued field with algebraically closed residue field of characteristic p > 2. Let C : y 2 = f (x) be a hyperelliptic curve over K with tame potentially semistable reduction and cluster picture Σ C/K . Then the dual graph, with genus and multiplicity, of the special fibre of the minimal SNC model of C/K is completely determined by Σ C/K (with depths), the valuation of the leading coefficient v K (c f ) of f , and the action of G K .
The proof of this will follow from the theorems proved in the rest of this section, and we make this more precise later. First we split Theorem 7.11 into several smaller theorems. The first tells us which components appear in the special fibre of the minimal SNC model. Roughly, there is a central component for every orbit of principal, nonübereven clusters, one or two central components for every orbit of principalübereven clusters, and a chain of rational curves associated to each orbit of twins. These central components are linked by chains of rational curves, and certain central components will also have tails intersecting them. The following theorem gives us the structure of the special fibre but is missing important details such as multiplicities, genera and lengths of these chains. These remaining details will be discussed in a later theorem.
Theorem 7.12 (Structure of SNC model). Let K be a complete discretely valued field with algebraically closed residue field of characteristic p > 2. Let C/K be a hyperelliptic curve with tame potentially semistable reduction. Then the special fibre of its minimal SNC model is structured as follows. Every principal Galois orbit of clusters X contributes one central component Γ X , unless X isübereven with ǫ X = 1, in which case X contributes two central components Γ + X and Γ − X . These central components are linked by chains of rational curves, or are intersected transversely by a crossed tail in the following ways (where, for any orbit Y , we write Note that any chain where the "To" column has been left blank is a crossed tail. If R is not principal then we also get the following chains of rational curves: R = s 1 ⊔ s 2 , with X = {s 1 , s 2 } a principal, even orbit, ǫ si = 1 T X Γ X -R = s 1 ⊔ s 2 , with X = {s 1 , s 2 } a principal, even orbit, ǫ si = −1 L t Γ − s Γ + s R = s ⊔ t, s principal and even, t a twin, ǫ t = 1 T t Γ s -R = s ⊔ t, s principal and even, t a twin, ǫ t = −1 Finally, a central component Γ X is intersected transversally by some tails if and only if e X > 1. These are explicitly described in Theorem 7.18.
Remark 7.13. At no point do we give explicit equations for the central components Γ ± X . However, these can be calculated using the method laid out in this paper. In particular, one can take the explicit equations given in Theorem 3.13 for the components Γ ± s,L in the semistable model of C/L and apply [DD18, Theorem 1.1] with the automorphisms given in Theorem 3.15.
Before we prove this, let us prove a couple of lemmas. Recall that L is a field over which C has semistable reduction and that Γ s,L is the component associated to a cluster s in the special fibre of the minimal semistable model Y of C over L.
Lemma 7.14. Let s be a principal cluster with g ss (s) = 0.
(i) If s = R and s is notübereven (resp.übereven) then Γ s,L (resp. each of Γ + s,L and Γ − s,L ) intersects at least two other components. (ii) If s = R and s is notübereven (resp.übereven) then Γ s,L (resp. each of Γ + s,L and Γ − s,L ) intersects at least three other components.
Proof. (i) Let s = R and suppose s is notübereven. Since g ss (s) = 0, s can have at most two odd children and in particular at most two singletons. Since, g(C) ≥ 2, we have |s| ≥ 5. If |s| is odd then s must have an even child s ′ and, by Theorem 3.13, Γ s,L is intersects by the two linking chains to Γ s ′ ,L . Note that, since s is principal, s cannot be the union of two odd clusters. So, if |s| is even then s has an even child and we are done by Theorem 3.13.
If s = R isübereven then s has only even children. In particular, it must have at least two even children s 1 and s 2 . Therefore, each of Γ ± s,L intersects L ± s1 and L ± s2 (the linking chains to the children).
(ii) Let s = R and suppose s is notübereven. Since s is principal, we know |s| ≥ 3. Therefore, s must have at least one proper child s ′ . Suppose that P (s) is principal. If s ′ < s is even then Γ s,L intersects the linking chain to Γ P (s),L and the two linking chains to Γ s ′ ,L . Otherwise s must be the union of two odd clusters, hence s is even. In this case there are two linking chains to Γ P (s),L and one to Γ s,L . A similar argument works if s isübereven. If P (s) = R = s ⊔ s 2 is not principal, the argument is also similar, except linking chains to Γ P (s),L are replaced by linking chains to Γ s2,L .
Proposition 7.15. Let Y be the semistable model of C/L and Z the imagine under the quotient map. Let X be the SNC model obtained by resolving the singularities of Z such that all rational chains are minimal. Let X be a principal orbit of clusters. Let Γ X,K ∈ X k be the image of Γ s,L for some s ∈ X under the quotient by Gal(L/K). Then if g(Γ X,K ) = 0 and (Γ X,K · Γ X,K ) = −1, Γ X,K intersects at least three other components of the special fibre (i.e. blowing down Γ X,K would not result in an SNC model).
Proof. If |X| > 1, there is a non trivial field extension of K to K X . Over K X , each s ′ ∈ X is fixed by Gal(K/K X ). The Galois group Gal(K X /K) then induces anétale morphism s ′ ∈X Γ s ′ ,KX → Γ X,K . Therefore, g(Γ X,K ) = g(Γ s ′ ,KX ), (Γ X,K · Γ X,K ) = (Γ s ′ ,KX · Γ s ′ ,KX ), and Γ X,K and Γ s ′ ,K intersect the same number of other components. So, it is enough to prove this proposition when |X| = 1, and from now on let X = {s}. The Riemann-Hurwitz formula gives us that where q : Γ s,L → Γ s,K is the quotient by Gal(L/K). If g(Γ s,L ) > 0, this implies that P ∈Γs,K e s |q −1 (P )| − 1 ≥ 2e s .
So, if g(Γ s,L ) > 0, there must be at least three points P ∈ Γ s,K with |q −1 (P )| < e s . These ramification points are singular points by Proposition 5.3. So, after blowing up these singular points, we see that Γ s,K intersects at least three other components of X k . It remains to deal with the case when g(Γ s,L ) = 0. If e s = 1, Lemma 7.14 implies that Γ s,K intersects two or more other components. In this case Γ s,K will have multiplicity e s = 1. This tells us that (Γ s,K · Γ s,K ) < −1, so Γ s,K is not exceptional.
Suppose instead that e s > 1. We will show that the component Γ s,K intersects at least three components. There are two ramification points P 0 and P ∞ of the morphism q : Γ s,L → Γ s,K , the images of 0 and ∞ respectively in Γ s,L . Both P 0 and P ∞ are singularities. If q −1 (P 0 ) is an intersection point of Γ s,L with another component Γ then P 0 will be the intersection point of Γ s,K and q(Γ) 3 . Otherwise, blowing up P 0 introduces a component intersecting Γ s,K . Similarly for P ∞ . If s = R then q −1 (P ∞ ) will never be an intersection point by [DDMM18,Proposition 5.20]. Since Γ s,L has two intersection points with other components Q 1 and Q 2 , either q(Q 1 ) = q(Q 2 ), or q(Q 1 ) = q(Q 2 ) = P 0 (since |q −1 (P 0 )| = 1). If q(Q 1 ) = q(Q 2 ) then these are both intersection points with other components, hence Γ s,K intersects at least 3 components at P ∞ , q(Q 1 ) and q(Q 2 ) which are all distinct. If q(Q 1 ) = q(Q 2 ) = P 0 then P ∞ , q(Q 1 ) and P 0 are distinct intersection points with other components. A similar argument works if s = R.
We are now able to prove our structure theorem (Theorem 7.12).
Proof of Theorem 7.12. First let us find which central components appear. Over L, by Theorem 3.13, we know there is a component for every principal, non-übereven cluster, and we know the action of Gal(L/K) on these central components is the same as the action on the clusters. After taking the quotient by Gal(L/K), we get a component for every orbit of principal, nonübereven clusters. Similarly over L, by Theorem 3.13, we know there are two components for everyübereven cluster s. These are swapped by Galois if and only if ǫ s = −1. After taking the quotient this gives us two components for anübereven orbit X if ǫ X = 1 and a single component if ǫ X = −1. We call these components the central components. Showing which linking chains which appear is done similarly, using the information given in Theorem 3.13.
To ensure these central components do in fact appear in the minimal SNC model, we must check that they cannot be blown down. Any central component Γ X,K ∈ X k is the image of Γ s,L ∈ Y k for some s ∈ X. A central component Γ X,K can only be blown down if g(Γ X,K ) = 0, and (Γ X,K · Γ X,K ) = −1. However, by Proposition 7.15, any central component Γ X,K with g(Γ X,K ) = 0 and (Γ X,K · Γ X,K ) = −1 intersects at least three other components of the special fibre. Therefore, if Γ X,K were to be blown down, X k would no longer be an SNC divisor. So Γ X,K must appear in the special fibre of the minimal SNC model.
Remark 7.16. Note that a linking chain can have length 0 -this indicates an intersection between central components (in the case X ′ < X both principal) or a singular central component (in the case where X is principal and X ′ < X is an orbit of twins). 7.3. A More Explicit Description. Theorem 7.12 describes the structure of the special fibre, but says nothing about the multiplicity or genera of the components. The following two theorems fill in these details. The first focuses on the central components, and the second describes the chains of rational curves present in the special fibre.
Theorem 7.17 (Central Components). Let K and C/K be as in Theorem 7.12. Let X be a principal orbit of clusters in Σ C/K . If X is notübereven then Γ X has multiplicity |X|e X and genus g(X). If X isübereven with ǫ X = 1 then Γ + X and Γ − X have multiplicity |X|e X and genus 0, and if ǫ X = −1 then Γ X has multiplicity 2|X|e X and genus 0.
Proof. Let X be a principal, non-übereven orbit, and choose some s ∈ X. Recall that K X is the minimal field extension of K such that the clusters of X are fixed by Gal(K/K X ), and L is the minimal field extension of K such that C is semistable over L. The image Γ s,KX of Γ s,L after taking the quotient by Gal(L/K X ) has multiplicity e X , since the action on Γ s,L has multiplicity e X (by Lemma 5.27). There are |X| such components, which are permuted by Gal(K X /K) in the minimal SNC model of C/K X . So, Γ X has multiplicity |X|e X by [Lor90,Fact IV]. The multiplicities of components corresponding toübereven clusters follows similarly, being careful to account for whether Γ + s,L and Γ − s,L are swapped by Gal(L/K) in the semistable model (which happens precisely when ǫ s = −1).
To find the genus of the central components, note that if g(Γ s,L ) = 0 then g(Γ X,K ) = 0. So let us assume that g(Γ s,L ) > 0. In this case, as mentioned in Remark 5.23, Γ s,L is isomorphic to the special fibre of the smooth model of C s over L. Furthermore, the action on Γ s,L is the same as the action on Γ s,L . Hence, the genus of Γ s,KX is g(X), and also the genus of Γ X,K .
Theorem 7.18 (Description of Chains). Let K and C/K be as in Theorem 7.12. Let X be a principal orbit of clusters with e X > 1. Choose some s ∈ X of depth d s with denominator b s . Then the central component(s) associated to X are intersected transversely by the following sloped tails with parameters (t 1 , µ) (writing −λ X |X|b X |s sing | ≥ 2, and e X > b X /|X| T x=0 Γ X 1 −d X 2|X| X has no stable child, λ X ∈ Z, e X > 2 and either g(X) > 0 or X isübereven −d X |X| X has no stable child, λ X ∈ Z, and either g(X) > 0 or X isübereven T (0,0) Γ X 1 −λ X |X| X has a stable singleton or g(X) = 0, X is notübereven and X has no proper stable odd child The central components are intersected by the following sloped chains of rational curves with parameters (t 2 , t 2 + δ, µ): If R is not principal we get additional sloped chains with parameters (t 2 , t 2 + δ, µ) as follows: δ(s 1 , s 2 ) 2 R = s 1 ⊔ s 2 , X = {s 1 , s 2 } principal, even orbit, and ǫ si = 1 L − X d s1 δ(s 1 , s 2 ) 2 R = s 1 ⊔ s 2 , X = {s 1 , s 2 } principal, even orbit, and ǫ si = 1 T X d s1 δ(s 1 , s 2 ) + 1 µ 4 R = s 1 ⊔ s 2 , X = {s 1 , s 2 } principal, even orbit, and Finally, the crosses of any crossed tail have multiplicity µ 2 . Remark 7.19. If there is any confusion over which central components linking chains or tails intersect, the reader is urged to refer back to the tables in Theorem 7.12. We have omitted this information from these tables due to spatial concerns.
Remark 7.20. Let X be a principal orbit of clusters in Σ C/K . As in Remark 6.5, we make a comparison between the rational chains intersecting a central component, Γ X ∈ X k to the tails in the special fibre of the minimal SNC model X X . This comparison makes sense when g(Γ s,L ) > 0 for some s ∈ X. The central component Γ X ∈ X k will have the same genus as the central component Γ X ∈ X X,k and multiplicity multiplied by |X|. It will have the same tails (with all multiplicities multiplied by |X|) except these tails will make up part of the linking chains intersecting Γ X in the following cases: • If X = R and P (X) is principal, an ∞-tail in X X,k will form the uphill section of one of the linking chains L ± P (X),X , • If X < R and R is not principal, then any ∞-tail in X X,k will form the uphill section of a chain: the linking chain between Γ s1 and Γ s2 if R = s 1 ⊔ s 2 and X = {s 1 }; the crossed tail if R = s 1 ⊔ s 2 and X = {s 1 , s 2 }; and the loop or crossed tail arising from R if R is a cotwin, • a (y = 0)-tail will form the downhill section of a linking chain L X,X ′ if there exists some X ′ < X, a non-trivial orbit of odd, principal children, • a (x = 0)-tail will form the downhill section of a linking chain L ± X,X ′ if there exists some {s ′ } = X ′ < X, a stable even child, • a (0, 0)-tail will form the downhill section of a linking chain L X,X ′ if there exists some {s ′ } = X ′ < X, a stable odd child. where again, all multiplcities are multiplied by |X|.
7.4. Proof of the Main Theorem. To prove Theorem 7.18, we will proceed by induction on two things: the number of proper clusters in Σ C/K , and the degree e = [L : K] of the minimal extension L/K such that C/L is semistable. The base cases for these are when Σ C/K consists of a single proper cluster (which is covered in Section 5, in particular Theorem 5.1 and Proposition 5.11), and when C has semistable reduction over K i.e. e = 1 (which is covered in Section 3.3). For our inductive hypothesis, suppose that for any hyperelliptic curve where the number of proper clusters in its cluster picture is strictly less than that of C/K, or the degree of an extension needed such that it is semistable is strictly less than that of C, we can completely determine the special fiber of its minimal SNC model. 7.4.1. R Principal. We will start by assuming that the top cluster R is principal, and that it has a Galois invariant proper child s. We will calculate the tails of Γ ± R,K and, in the case where s principal Γ ± s,K . We will also calculate the linking chain(s) (or the chain arising from s if s is a twin) between them. This will be done by comparing the linking chain(s) to those in the special fibre of the minimal SNC model of another hyperelliptic curve over K, which we will call C new . We will write C new : y 2 = f new (x), and denote the set of roots of f new over K by R new . The curve C new /K is chosen so that Σ C new /K has a unique proper cluster s new = R new , enabling us to apply the results of Section 6. We will then use induction to deduce the components of the model arising from the subclusters of s. Finally, we will remove the assumption that s is Galois invariant to complete the proof when R is principal. Let us start with lemmas describing the tails of Γ R,K and Γ s,K .
Lemma 7.21. Let R be principal and suppose that e R > 1. The tails of the central component(s) associated to R are as described in Theorem 7.18.
Proof. First suppose that R is notübereven. Let Y be the semistable model of C/L and consider Γ R,L ⊆ Y . The stabiliser of R has order e R . Under the quotient map, a Galois orbit T of points of Γ R,L gives rise to a singularity on Γ R,K lying on precisely one component of X K if and only if |T | < e R and the points of T lie on Γ R,L and no other components of Y k .
Suppose that g(Γ R,L ) = 0. There are only two orbits with size less than e R , which after an appropriate shift we can assume are at 0 and ∞. The point at ∞ certainly lies on no other component of Y k by [DDMM18, Proposition 5.20], so Γ R,K will always have ∞-tails. By [DDMM18,Proposition 5.20], the point 0 lies on no other component of Y k if and only if R has no stable proper odd child. This is because if s < R is a stable odd child then L R,s intersects Γ R,L at 0, however no other linking chain to a child will ever intersect Γ R,L at 0. Therefore Γ R,K will have a (0, 0)-tail if and only if it has no stable proper odd child. The description of the tails follows.
Suppose instead that g(Γ R,L ) > 0. The orbits of points on Γ R,L of size less than e R are the same as the small orbits Γ R,L , which are described in Lemmas 5.7 -5.10. To complete the description, we must calculate when these small orbits are intersection points with other components. We do this using the explicit description of the components of Y k given in [DDMM18,Proposition 5.20]. From this, we can deduce that the points at ∞ never lie on a component other than Γ R,L , (y = 0)-orbits are intersection points if and only if s has a non-trivial orbit of proper odd children, (x = 0)-orbits are intersection points if and only if s has a stable even child, and the (0, 0)-orbit is an intersection point if and only if R has a proper stable odd child. Now suppose R isübereven. Then each Γ ± R,L has two orbits of size less than e R , which lie at their respective points at 0 and ∞. The points at ∞ do not lie on any other components of Y k and the points at 0 lie on no other component of Y k if and only if R has no stable child. Therefore, Γ ± R,K has a (x = 0)-tail if and only if R does not have a stable child. The description of the tails follows.
Lemma 7.22. Let s < R be a principal, Galois invariant cluster with e s > 1. Then the tails intersecting the central component(s) assosciated to s are as described in Theorem 7.18.
Proof. The proof is similar to that of the previous lemma, noting that all of the orbits at infinity are the intersection points of Γ ± s,L and the linking chain between Γ ± R,L and Γ ± s,L .
Following is a technical lemma allowing us to compare the chain(s) appearing between Γ R,K and Γ s,K to those of a simpler curve C new .
Lemma 7.23. Let s 1 , s 2 be two Galois invariant principal clusters (resp. a principal cluster and a twin) such that either s 2 < s 1 , or R = s 1 ⊔ s 2 is not principal. Then any linking chain between Γ ± s1,K and Γ ± s2,K (resp. the chain of rational curves arising from s 2 intersecting Γ ± s1,K ) is determined entirely by λ si mod Z, the parity of |s 2 |, d si , and when R is not principal d R .
Proof. Assume that both s i are principal, Galois invariant clusters. From Section 3.2, we know that a linking chain between Γ ± s1,K and Γ ± s2,K is completely determined by the length and number of linking chains between Γ ± s1,L and Γ ± s2,L , the order of the action of Gal(L/K) on any individual component of a linking chain between Γ ± s1,L and Γ ± s2,L , and the nature of the singularities at the intersection points of components after taking the quotient. Recall from Theorem 3.13 that there is one linking chain, say C, between Γ ± s1,L and Γ ± s2,L if s 2 is odd and two linking chains, say C + and C − , if s 2 is even. We will write C = C + = C − if s 2 is odd. Theorem 3.13 tell us that the length of C ± is determined by δ(s 1 , s 2 ), which is given in terms of d s1 and d s2 (and d R in the case where R = s 1 ⊔ s 2 is not principal).
Let P be an intersection point of two components E 1 , E 2 ∈ {Γ s1,L , Γ s2,L , C ± }. Let σ Ei be the induced G K action on E i , for a generator σ of Gal(L/K). Suppose σ a E1 , and σ b E2 , generate the stabilisers of P in E 1 and E 2 respectively. Then q(P ) is a tame cyclic quotient singularity with parameters n = gcd(o(σ a E1 ), o(σ b E2 )), m 1 = o(σ a E1 )/n, m 2 = o(σ b E2 )/n, and r = d −a E1 d b E2 /n 2 s 2 even, λ −a E1 λ b E2 /n 2 s 2 odd, where o(τ ) is the order of τ ∈ Gal(L/K). In other words, the tame cyclic quotient singularity is determined entirely by the automorphisms on the E i and the parity of s 2 . Therefore, since the automorphisms on E i are determined entirely by the invariants in the statement of the theorem (by [DDMM18, Theorem 6.2]), we are done.
The case where s 2 is a twin follows similarly.
For the following lemma we first need some notation. Recall that a child of s ∈ Σ C/K is stable if has the same stabiliser as s. Let s nf denote the set of stable children of s, and s f denote the set of unstable children of s.
Lemma 7.24. Let C/K be a hyperelliptic curve with R principal, and let s < R be a Galois invariant proper child. We can construct a hyperelliptic curve, C new , such that the cluster picture Σ C new of C new consists of two proper clusters s new < R new , where |s| ≡ |s new | mod 2, d R = d R new , d s = d s new and λ R − λ R new , λ s − λ s new ∈ Z.
Proof. Let C new be the hyperelliptic curve over K defined by It remains to check that λ R −λ R new , λ s −λ s new ∈ Z. Let us begin with the first. By construction, s new is odd if and only if s is. Therefore, if | R \ s| ≥ 2 it follows that λ R new = λ R . Else, If d R ∈ Z, then clearly λ R new − λ R ∈ Z. Otherwise, d R ∈ Z. By Lemma 2.12, the children of R must lie in orbits of size b R > 1. Therefore, any such orbit must be an orbit of even children of R, since s is fixed and there is at most one child not equal to s. Hence, | R \ R|d R ∈ Z, and so λ R new − λ R ∈ Z. It can be checked similarly that λ s new − λ s ∈ Z.
By the above lemmas and Theorem 6.3, we have proved the statements in Theorem 7.12 about the linking chain(s) between Γ ± s,K and Γ ± R,K where s < R is a Galois invariant proper child. We now turn our focus to the components of X k which arise from s and its subclusters. In order to do this, we construct yet another new hyperelliptic curve, which we shall call C ′ , given by Note that C ′ is also semistable over L, and let Y ′ be the semistable model of C ′ over L. Comparing the cluster pictures of C ′ and C, we see that the cluster picture Σ C ′ appears within the cluster picture Σ C of C. This is illustrated in Figure 20. In particular, s and all of its subclusters in Σ C are drawn in solid black in Figure 20a. These are exactly the clusters that make up Σ C ′ , also shown in solid black.
. . .  The leading coefficient of C ′ has been chosen so that the corresponding clusters in Σ C and Σ C ′ have the same cluster invariants. Therefore, there is a closed immersion Y ′ k → Y k which commutes with the action of G K . The existence of this immersion is illustrated in Figure 21. We can see this by calculating the explicit equations of the components of Y ′ given in Theorem 3.13, and the explicit Galois action on these components given in Theorem 3.15. Therefore, this immersion also commutes with the quotient by Gal(L/K).  After taking this quotient by Gal(L/K), and performing any appropriate blow ups and blow downs, we obtain a closed immersion X ′ k \ T ∞ → X k , where X ′ is the minimal SNC model of C ′ /K and T ∞ is the set of infinity tails of X ′ k . We remove the infinity tails since in the small distance case (see Section 6.3) the whole tails do not appear in X k . By our inductive hypothesis (since the number of proper clusters in Σ C ′ is strictly less than that in Σ C ), we can calculate X ′ k . This gives us a full description of the components of X k which arise from the subclusters of s.
Finally let us remove the assumption that s is G K invariant. Let X < R be a non-trivial orbit of children. Extend K by degree |X| to the field K X , the minimal extension such that each cluster in X is fixed by Gal(K/K X ). By our inductive hypothesis (since C/K X needs an extension of degree strictly less than C/K does in order to have semistable reduction), we can calculate the minimal SNC model of C over K X , which we denote X X . Since each cluster of X is fixed by Gal(L/K X ), there is a divisor D s corresponding to every cluster s ∈ X and all of the subclusters of s. Let D X = s∈X D s be the union of these divisors. Since Gal(K X /K) simply permutes these divisors, the quotient by Gal(K X /K) is anétale morphism, and the image of D X consists of precisely the same components as D s for some s ∈ X, but with all the multiplcities multiplied by |X|. See Figure 22 for an illustration. This concludes the proof when R is principal. 7.4.2. R not principal. Now suppose that R is not principal. If R is a cotwin, then the contribution to the cluster picture from R can be deduced using Remark 6.5 and Lemmas 7.23 and 7.24. The contribution of s < R, the child of size 2g, can be calculated by induction using a curve C ′ as in (7) above.
If R is not principal and not a cotwin then R is even and the union of two proper children. In this case, we will write R = s 1 ⊔ s 2 . Here the s i are either fixed or swapped by G K . We will deal with the case when s i are swapped at the end of this section, so for now suppose that both s i are fixed by G K . The first of these lemmas shows that there is a Möbius transform taking a certain class of curves with R not principal to the curves we studied in Section 6.
Lemma 7.25. Let C/K by a hyperelliptic curve with cluster picture Σ C/K , and set of roots R.
(i) Let s ∈ Σ C/K be a cluster with centre z s . Write every root r ∈ s as r = z s + r h , where v K (r h ) ≥ d s . Then there exists at most one r ∈ s such that v K (r h ) > d s .
The next lemma is analogous to Lemma 7.24, it gives us the existence of some new curve, which we will again call C new , to which we can apply Lemma 7.25. This will allow us to calculate the linking chain(s) between Γ ± s1 and Γ ± s2 , by using Lemma 7.23. The cluster picture of C/K is shown in Figure 24a. The special fibre of the minimal SNC model of C/K is shown in Figure 24b. The central components of the model, which arise from which clusters in Σ C/K , are labeled. This is a hyperelliptic curve of Namikawa-Ueno type IV − III − 0 as shown in [NU73,p. 167]. Observe that R = s 1 ⊔ s 2 is not principal so gives rise to a linking chain between Γ s1 and Γ s2 . Note that the special fibre here is the same as in Example 6.7, and there is in fact a Möbius transform between the two Weierstrass equations.  Appendix A.

Kodaira-Néron Classification for Elliptic Curves
For the convenience of the reader, we have included a table showing the cluster pictures and special fibre X k of the minimal SNC model X for the different Kodaira-Néron types of elliptic curves with tame potentially semistable reduction (for which it is sufficient to take p ≥ 5). Our table differs slightly from the table found in [Sil94,p 365], where instead the special fibers of the minimal regular models for the different types of elliptic curves are shown. This makes a difference for type II, III or IV elliptic curves, whereas for all the other types the minimal regular model is SNC.
Type Cluster Picture X k Type Cluster Picture X k Table 3. Kodaira-Néron types of elliptic curves with tame potentially semistable reduction over K, where K has algebraically closed residue field k of characteristic p = 2. The given cluster picture is the cluster picture of the minimal Weierstrass model of such an elliptic curve.