Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta

Let K be an algebraically closed, complete non-Archimedean field. The purpose of this paper is to carefully study the extent to which finite morphisms of algebraic K-curves are controlled by certain combinatorial objects, called skeleta. A skeleton is a metric graph embedded in the Berkovich analytification of X. A skeleton has the natural structure of a metrized complex of curves. We prove that a finite morphism of K-curves gives rise to a finite harmonic morphism of a suitable choice of skeleta. We use this to give analytic proofs of stronger ‘skeletonized’ versions of some foundational results of Liu-Lorenzini, Coleman, and Liu on simultaneous semistable reduction of curves. We then consider the inverse problem of lifting finite harmonic morphisms of metrized complexes to morphisms of curves over K. We prove that every tamely ramified finite harmonic morphism of Λ-metrized complexes of k-curves lifts to a finite morphism of K-curves. If in addition the ramification points are marked, we obtain a complete classification of all such lifts along with their automorphisms. This generalizes and provides new analytic proofs of earlier results of Saïdi and Wewers. As an application, we discuss the relationship between harmonic morphisms of metric graphs and induced maps between component groups of Néron models, providing a negative answer to a question of Ribet motivated by number theory. This article is the first in a series of two. The second article contains several applications of our lifting results to questions about lifting morphisms of tropical curves.


Introduction
This article is the first in a series of two. The second, entitled Lifting harmonic morphisms II: tropical curves and metrized complexes, will be cited as [3]; references of the form "Theorem II.1.1" will refer to Theorem 1.1 in [3].
Throughout this paper, unless explicitly stated otherwise, K denotes a complete algebraically closed non-Archimedean field with nontrivial valuation val : K → R ∪ {∞}. Its valuation ring is denoted R, its maximal ideal is m R , and the residue field is k = R/m R . We denote the value group of K by = val(K × ) ⊂ R.
The purpose of this paper is to carefully study the extent to which finite morphisms of algebraic K-curves are controlled by certain combinatorial objects, called skeleta. Let X be a smooth, proper, connected K-curve. Roughly speaking, a triangulation (X, V ∪ D) of X (with respect to a finite set of punctures D ⊂ X(K)) is a finite set V of points in the Berkovich analytification X an of X whose removal partitions X an into open balls and finitely many open annuli (with the punctures belonging to distinct open balls). Triangulations of (X, D) are naturally in one-to-one correspondence with semistable models X of (X, D): see Section 5. A triangulation (X, V ∪ D) gives rise to a skeleton (X, V ∪ D) of X. The skeleton of a triangulated punctured curve is the dual graph of the special fiber X k of the corresponding semistable model, equipped with a canonical metric, along with completed infinite rays in the directions of the punctures. There are many skeleta in X an , although if X \ D is hyperbolic (i.e., has negative Euler characteristic χ(X, D) = 2 − 2g(X) − #D), then there exists a unique minimal skeleton. A skeleton of (X, D) is by definition a subset ⊂ X an of the form (X, V ∪ D) for some triangulation of X.

Skeletal simultaneous semistable reduction
First we will prove that finite morphisms of curves induce morphisms of skeleta: Theorem 1.2. Let ϕ : X → X be a finite morphism of smooth, proper, connected K-curves and let D ⊂ X(K) be a finite set. There exists a skeleton ⊂ X such that := ϕ −1 ( ) is a skeleton of X . For any such , the map ϕ : → is a finite harmonic morphism of metric graphs.
See Corollaries 4.18 and 4.28. Harmonic morphisms of graphs are defined in Section 2. Due to the close relationship between semistable models and skeleta, Theorem 1.2 should be interpreted as a very strong simultaneous semistable reduction theorem. In fact, we will show how to formally derive from Theorem 1.2 the simultaneous semistable reduction theorems of Liu-Lorenzini [32], Coleman [23], and Liu [34]. Our version of these results hold over any non-Archimedean field K 0 , not assumed to be discretely valued. As an example, the following weak form of Liu's theorem is a consequence of Theorem 1.2: Corollary 1. Let X, X be smooth, proper, geometrically connected curves over a non-Archimedean field K 0 , and let ϕ : X → X be a finite morphism. Then, there exists a finite, separable extension K 1 of K 0 and semistable models X, X of the curves X K 1 , X K 1 , respectively, such that ϕ K 1 extends to a finite morphism X → X.
We discuss simultaneous semistable reduction theorems in Section 5. We wish to emphasize that Theorem 1.2 does not follow from any classical simultaneous semistable reduction theorem, in that there exist finite morphisms ϕ : X → X of semistable models such that the inverse image of the corresponding skeleton of X is not a skeleton of X . (See, however, [24] where a skeletal version of Liu's theorem is derived from Liu's method of proof in the discretely valued case.) Our proof of Theorem 1.2 is entirely analytic, resting on an analysis of morphisms between open annuli and open balls; in particular, it makes almost no reference to semistable reduction theory and is therefore quite different from Liu and Liu-Lorenzini's approach.
A skeletal simultaneous semistable reduction theorem is important precisely when one wants to obtain a well-behaved morphism of graphs from a finite morphism of curves. This is crucial for obtaining the spectral lower bound on the gonality in [24], for instance.

Lifting harmonic morphisms
Our second goal is in a sense inverse to the first: we wish to lift finite harmonic morphisms of metric graphs to finite morphisms of curves. More precisely, let (X, D) be a punctured K-curve as above, let be a skeleton, and letφ : → be a finite harmonic morphism of metric graphs. It is natural to ask whether there exists a curve X and a finite morphism ϕ : X → X such that ϕ −1 ( ) is a skeleton of X and ϕ −1 ( ) ∼ = as metric graphs over . In general, the answer is 'no': there are subtle 'Hurwitz obstructions' to such a lift, as described below. One solution is to enrich , with the extra structure of metrized complexes of curves. A metrized complex of curves is basically a metric graph and for every (finite) vertex p ∈ , the data of a smooth, proper k-curve C p , along with an identification of the edges adjacent to p with distinct k-points of C p . There is a notion of a finite harmonic morphism of metrized complexes of curves, which consists of a finite harmonic morphism ϕ : → of underlying metric graphs, and for every vertex p ∈ with p = ϕ(p ), a finite morphism ϕ p : C p → C p , satisfying various compatibility conditions. Now, let = (X, V ∪ D) for a triangulated punctured curve (X, V ∪ D). We will show that is naturally a metrized complex of curves, and that finite harmonic morphisms → of metrized complexes do lift to finite morphisms of curves, under a mild tameness hypothesis. Moreover, we have very precise control over the set of such lifts entirely in terms of the morphism of metrized complexes. The main lifting theorem, stated somewhat imprecisely, is as follows: Theorem 1.3 (Lifting theorem). Let (X, V ∪ D) be a triangulated punctured K-curve, let be its skeleton, and let → be a tame covering of metrized complexes of curves. Then, there exists a curve X and a finite morphism ϕ : X → X, branched only over D, such that ϕ −1 ( ) is a skeleton of (X , ϕ −1 (D)) and ϕ −1 ( ) ∼ = as metrized complexes of curves over . There are finitely many such lifts X up to X-isomorphism. There are explicit descriptions of the set of lifts and the automorphism group of each lift (as a cover of X) in terms of the morphism → .
See Theorems 7.4, 7.7, and 7.10 for the precise statements. Tame coverings are defined in Definition 2.21. The hardest part of the proof of Theorem 1.3 is a local lifting result, Theorem 6.18, in which the target curve X is replaced by a neighborhood of a vertex p ∈ . Theorem 6.18 eventually reduces to classical results about the tamely ramified étale fundamental group.
It is worth mentioning that given a metrized complex of curves C with edge lengths contained in , it is not hard to construct a triangulated punctured K-curve (X, V ∪ D) such that (X, V ∪ D) ∼ = C. See Theorem 3.24.
Our lifting results extend several theorems in the literature, as well as making them more precise. Saïdi ([39], Théoréme 3.7) proves a version of the lifting theorem for semistable formal curves (without punctures) over a complete discrete valuation ring. His methods also make use of the tamely ramified étale fundamental group and analytic gluing arguments. Wewers [47] works more generally with marked curves over a complete Noetherian local ring using deformation-theoretic arguments, proving that every tamely ramified admissible cover of a marked semistable curve over the residue field of a complete Noetherian local ring lifts. Wewers classifies the possible lifts (fixing, as we do, a lift of the target) in terms of certain non-canonical 'deformation data' depending on compatible choices of formal coordinates at the nodes on both the source and target curves.
Our results have several advantages compared to these previous approaches. For one, we are able to work over non-Noetherian rank-1 valuation rings and descend the lifting theorems a posteriori to an essentially arbitrary non-Archimedean field: see (7.11). The real novelty, however, is the systematic use of metrized complexes of curves in the formulation of the lifting theorems, in particular the classification of the set of lifts X of → . For example, in our situation, Wewers' non-canonical deformation data are replaced by certain canonical gluing data, which means that the automorphism group of the morphism of metrized complexes acts naturally on the set of gluing data (but not on the set of deformation data). This is what allows us to classify lifts up to isomorphism as covers of the target curve, as well as determine the automorphism group of such a lift. Saïdi's lifting theorem is more similar to ours, with generators of inertia groups over nodal points playing the role of gluing data. It is not clear to the authors whether his methods can be extended to give a classification of lifts. The question of lifting (and classification of all possible liftings) in the wildly ramified case is more subtle and cannot be guaranteed in general. See Remark 7.6.

Homomorphisms of component groups
We will discuss in Section 8 the relationship between harmonic morphisms of metric graphs and component groups of Néron models, connecting our lifting results to natural questions in arithmetic algebraic geometry. The idea is as follows. Let X be a smooth, proper, geometrically connected curve over a discretely valued field K 0 , and suppose that X admits a semistable model X over the valuation ring R 0 of K 0 . Let J be the Jacobian of X, and let X be the group of connected components of the special fiber of the Néron model of J.
Given a metric graph with integer edge lengths, there is a naturally associated abelian group Jac reg ( ), called its regularized Jacobian. The association → Jac reg ( ) satisfies Picard and Albanese funtoriality: that is, given a finite harmonic morphism → , there are natural push-forward and pullback homomorphisms Jac reg ( ) → Jac reg ( ) and Jac reg ( ) → Jac reg ( ).
A series of theorems of Raynaud, translated into our language, imply that X ∼ = Jac reg ( X ) canonically, where X is the skeleton induced by the semistable model X. We observe that this isomorphism respects both Picard and Albanese functoriality.
For modular curves such as X 0 (N) with N prime, the minimal regular model has a special fiber consisting of two projective lines intersecting transversely. Maps from such curves to elliptic curves, and their induced maps on component groups of Néron models, play an important role in arithmetic geometry; for example, Ribet's work establishing that the Shimura-Taniyama conjecture implies Fermat's Last Theorem was motivated by the failure of such induced maps on component groups to be surjective in general. (The cokernel of such maps controls congruences between modular forms.) Ribet noticed that if one considers instead the case where the target curve has genus at least 2, then in all examples he knew of, the induced map on component groups was surjective, and in personal communication with the second author, he asked whether this was a general property of curves whose reduction looks like that of X 0 (N). As a concrete application of Theorem 1.3, the next proposition provides a negative answer to this question. Proposition 1.5. There exists a finite morphism f : X → X of semistable curves over a discretely valued field K 0 with g(X) ≥ 2 such that: • the special fiber of the minimal regular model of X consists of two projective lines intersecting transversely; • the induced map f * : X → X on component groups of Néron models of Jacobians is not surjective.
In order to prove Proposition 1.5, we proceed as follows. As component groups can be calculated in terms of metric graphs, one first finds a finite harmonic morphism of metric graphs ϕ : → such that has two vertices and at least three edges, all of which have length one, and such that ϕ * : Jac reg ( ) → Jac reg ( ) is not surjective. One then enriches ϕ to a morphism of metrized complexes of curves, with rational residue curves, and uses Theorem 3.24 and Theorem 1.3 (along with a descent argument) to lift this to a finite morphism of curves X → X. Such a morphism satisfies the conditions of Proposition 1.5.

Applications to tropical lifting theorems
Tropical geometry has many applications to algebraic geometry, in particular enumerative algebraic geometry. The basic strategy is to associate a combinatorial 'tropical' object to an algebraic object, and prove that counting the former is somehow equivalent to counting the latter. An important ingredient in any such argument is a tropical lifting theorem, which is a precise description of the algebraic lifts of a given tropical object.
Many of the intended applications of our lifting results are contained in the second paper [3], in which we prove a number of tropical lifting theorems. The skeleton of a curve, viewed as a metric graph, is a tropical object associated to the curve; our lifting results can therefore be interpreted as tropical lifting theorems for morphisms of metric graphs, in the following sense. Recall that ⊂ R is the value group of K. A -metric graph is a metric graph whose edge lengths are contained in . We say that a finite harmonic morphismφ : → of -metric graphs is liftable provided that there exists a finite morphism of K-curves ϕ : X → X, a set of punctures D ⊂ X(K), and a skeleton of (X, D) such that ∼ = as metric graphs, ϕ −1 ( ) is a skeleton of (X , ϕ −1 (D)), and ∼ = as metric graphs over ∼ = . Theorem 1.3 (along with Theorem 3.24) shows that the obstruction to lifting a morphism of metric graphsφ : → to a morphism of curves is the same as the obstruction to enriching ϕ with the structure of a morphism of metrized complexes of curves. The following more precise statement is an immediate consequence of Proposition 7.15 and Theorem 3.24: Corollary 1.6. Assume char(k) = 0. Letφ : → be a finite harmonic morphism of -metric graphs. Suppose that ϕ can be enriched to a morphism of metrized complexes of curves. Then, ϕ is liftable.
We remark that we actually take advantage of the stronger form of Theorem 1.3, in particular the calculation of the automorphism group of a lift, when lifting group actions on metrized complexes in [3].
The problem of enriching a morphism of metric graphs to a morphism of metrized complexes is essentially equivalent to the problem of assigning k-curves C p , C p to vertices p ∈ , p ∈ and finding morphisms of k-curves C p → C p with prescribed ramification profiles. The latter is a question about whether certain Hurwitz numbers are nonzero. See Proposition II.3.3. Allowing arbitrary choices of residue curves C p , C p , this can always be done: Theorem. Suppose that char(k) = 0. Any finite harmonic morphism ϕ : → of -metric graphs is liftable to a morphism of curves.
See Theorem II. 3.11. Imposing requirements on the lifted curves X, X give rise to variants of this tropical lifting problem. The most basic such requirement is to specify the genera of the curves. An augmented metric graph is a metric graph along with an integer g(p) ∈ Z ≥0 for every finite vertex p of , called the genus. The genus of an augmented graph is defined to be where h 1 ( ) is the first Betti number. Any metrized complex of curves has an underlying augmented metric graph, where g(p) := g(C p ), the genus of the residue curve. If is a skeleton of a curve X, viewed as an augmented metric graph, then we have g( ) = g(X).
A finite harmonic morphism of augmented metric graphs → is liftable if the morphism of underlying metric graphs is liftable to a finite morphism of curves X → X such that the isomorphism of (resp. ) with a skeleton of X (resp. X ) respects the augmented metric graph structure. In this case, g(X) = g( ) and g(X ) = g( ). By our lifting theorem, → is liftable to a morphism of curves if and only if it can be enhanced to a morphism of metrized complexes such that g(C p ) = g(p) for every finite vertex. Now the obstruction to lifting is nontrivial: there exist finite harmonic morphisms of metric graphs which cannot be enhanced to a morphism of metrized complexes. This corresponds to the emptiness of certain Hurwitz spaces over k. See Section II.5 for the following striking example of this phenomenon: Theorem. There exists an augmented metric graph admitting a finite morphism of degree d > 0 to an augmented metric graph T of genus zero but which is not isomorphic to a skeleton of a d-gonal curve X.
In [3], we also give applications of the lifting theorem to the following kinds of tropical lifting problems, among others: • When T is an augmented metric graph of genus zero, we study a variant of the lifting problem in which the genus of the source curve is prescribed, but the degree of the morphism is not. We use this to prove that linear equivalence of divisors on a tropical curve C coincides with the equivalence relation generated by declaring that the fibers of every finite harmonic morphism from C to the tropical projective line are equivalent.
• We study liftability of metrized complexes equipped with a finite group action, and as an application classify all (augmented) metric graphs arising as the skeleton of a hyperelliptic curve. As mentioned above, this study actually takes advantage of the more precise form of Theorem 1.3, not just the existence of a lift.

Organization of the paper
In Section 2, we recall some basic definitions of graphs with additional structures, including metrized complexes of curves, and define harmonic morphisms between them. We provide somewhat more detail than is strictly necessary for the exposition in this paper but will prove necessary in [3].
In Section 3, we review the structure theory of analytic curves and define the skeleton of a curve. We show that the skeleton is naturally a metrized complex of curves, and we prove that any metrized complex arises in this way. In Section 4, we develop 'relative' versions of the results of Section 3. We prove that a finite morphism of curves induces a finite morphism between suitable choices of skeleta, and that the map on skeleta is a finite harmonic morphism of metrized complexes of curves. In Section 5, we show how the previous two sections can be used to derive various simultaneous semistable reduction theorems.
In Section 6, we prove a local lifting theorem, which essentially says that a tame covering of metrized complexes of curves can be lifted in a unique way to a tame covering of analytic K-curves in a neighborhood of a vertex. This is the technical heart of the lifting theorem. In Section 7, we globalize the considerations of the previous section, giving the classification of lifts of a tame covering of metrized complexes of curves to a tame covering of triangulated punctured K-curves. Section 8 contains an example application, in which we construct a morphism of curves over a discretely valued field such that the covariantly associated morphism of component groups of Jacobians has a prescribed behavior. We use this to answer a question of Ribet.

Related work
Another framework for the foundations of tropical geometry has been proposed by 31] and Mikhalkin [35,36], in which tropical objects are associated to real one-parameter families of complex varieties. We refer to [30,31] for some conjectures relating this framework to Berkovich spaces. In this setting, the notion of metrized complex of curves is similar to the notion of phase-tropical curves, and Proposition 7.15 is a consequence of Riemann's Existence Theorem. We refer the interested reader to the forthcoming paper (Mikhalkin: Phase-tropical curves I. Realizability and enumeration, forthcoming) for more details (see also [17] where this statement is implicit, as well as Corollary II.3.8).
Harmonic morphisms of finite graphs were introduced by Urakawa in [45] and further explored in [8]. Harmonic morphisms of metric graphs have been introduced independently by several different people. Except for the integrality condition on the slopes, they appear already in Anand's paper [4]. The definition we use is the same as the one given in [21,35].
Finite harmonic morphisms of metrized complexes can be regarded as a metrized version of the notion of admissible cover due to Harris and Mumford [29] (where, in addition, arbitrary ramification above smooth points is allowed). Recall that for two semistable curves Y and Y over a field k, a finite surjective morphism ϕ : Y → Y is an admissible cover if ϕ −1 (Y sing ) = Y sing and for each singular point y of Y , the ramification indices at y along the two branches intersecting at y coincide (In addition, one usually imposes that all the other ramifications of ϕ are simple). An admissible cover naturally gives rise to a finite harmonic morphism of metrized complexes: denoting by C the regularization of Y (the metrized complex associated to Y in which each edge has length one), define C as the metrized complex obtained from Y by letting the length of the edge associated to the double point y be 1/r y (where r y is the ramification index of ϕ at y along either of the two branches). The morphism ϕ of semistable curves naturally extends to a finite harmonic morphism ϕ : C → C: on each edge e of C corresponding to a double point y of Y , the restriction of ϕ to e is linear with slope (or 'expansion factor') r y . Conversely, a finite harmonic morphism of metrized complexes of curves gives rise to an admissible cover of semistable curves (without the supplementary condition on simple ramification) by forgetting the metrics on both sides, remembering only the expansion factor along each edge.
A related but somewhat different Berkovich-theoretic point of view on simultaneous semistable reduction for curves can be found in Welliaveetil's recent preprint [46]; harmonic morphisms of metrized complexes of curves play an implicit role in his paper.

Metric graphs and metrized complexes of curves
In this section, we recall several definitions of graphs with some additional structures and morphisms between them. A number of these definitions are now standard in tropical geometry; we refer for example to [10,17,35], and [2]. We also provide a number of examples. Some of the definitions in this section are not necessary in the generality or the form in which they are presented for the purposes of this paper, but will be useful in [3].
Throughout this section, we fix a non-trivial subgroup of (R, +).

Metric graphs
Given r ∈ Z ≥1 , we define S r ⊂ C to be a 'star with r branches' , i.e., a topological space homeomorphic to the convex hull in R 2 of (0, 0) and any r points, no two of which lie on a common line through the origin. We also define S 0 = {0}. All graphs considered in this paper are assumed to be connected. A finite topological graph is the topological realization of a finite graph. That is to say, is a compact topological space of dimension at most one (possibly zero) such that for any point p ∈ , there exists a neighborhood U p of p in homeomorphic to some S r ; moreover, there are only finitely many points p with U p homeomorphic to S r with r = 2.
The unique integer r such that U p is homeomorphic to S r is called the valence of p and denoted val(p). A point of valence different from 2 is called an essential vertex of . The set of tangent directions at p is T p ( ) = lim − →Up π 0 (U p \ {p}), where the limit is taken over all neighborhoods of p isomorphic to a star with r branches. The set T p ( ) has val(p) elements.

Definition 2.2.
A metric graph is a finite connected topological graph equipped with a complete inner metric on is some (finite) set of 1-valent vertices of called infinite vertices of . (An inner metric is a metric for which the distance between two points x and y is the minimum of the lengths of all paths between x and y.) Let be a metric graph containing a finite essential vertex. We say that is a -metric graph if the distance between any two finite essential vertices of lies in . A -point of is a point of whose distance to any finite essential vertex of lies in . Let be a metric graph such that all essential vertices are infinite, so is homeomorphic to a circle or a completed line. We say that is a -metric graph if it is a completed line or a circle whose circumference is in . Given a distinguished point v ∈ of valency equal to 2, a -point of is a point of whose distance to v lies in .
One can equip with a degenerate metric in which the infinite vertices are at infinite distance from any other point of . When no confusion is possible about the subgroup , we will sometimes write simply metric graph instead of -metric graph. Fix a vertex set V ( ). We denote by E( ) the set of edges of . Since is a metric graph, we can associate to each edge e of its length (e) ∈ ∪ {+∞}. Since the metric on \ V ∞ ( ) is complete, an edge e is infinite if and only if (e) = +∞. The notion of vertices and edges of depends, of course, on the choice of a vertex set; we will fix such a choice without comment whenever there is no danger of confusion. Definition 2.4. Fix vertex sets V ( ) and V ( ) for two -metric graphs and , respectively, and let ϕ : → be a continuous map.
• The map ϕ is called a (V ( ), V ( ))-morphism of -metric graphs if we have and let e ∈ E( ) be the edge in the direction of v . We define the directional derivative of ϕ in the direction v to be in the obvious way.
Example 2.5. Figure 1 depicts four examples of morphisms of metric graphs ϕ : → . We use the following conventions in our pictures: black dots represent vertices of and ; we label an edge with its degree if and only if the degree is different from 0 and 1; we do not specify the lengths of edges of and .
The morphisms depicted in Figure 1a-c are finite, while the one depicted in Figure 1d is not.
→ be a morphism of -metric graphs, let p ∈ , and let p = ϕ(p ). The morphism ϕ is harmonic at p provided that, for every tangent direction v ∈ T p ( ), the number is independent of v. The number d p (ϕ) is called the degree of ϕ at p . We set d p (ϕ) = 0 if every edge adjacent to p is contracted, unless = {p}, in which case d p (ϕ) is part of the data of ϕ. We say that ϕ is harmonic if it is surjective and harmonic at all p ∈ ; in this case, since is connected, the number deg(ϕ) = p →p d p (ϕ) is independent of p ∈ and is called the degree of ϕ. Remark 2.7. In the above situation, if all edges adjacent to a vertex p ∈ V ( ) are contracted, then ϕ is automatically harmonic at p . In particular, if consists of a single   Figure 1a is not harmonic, while the ones depicted in Figure 1b-d are (for suitable choices of lengths).

Harmonic morphisms and harmonic functions
Given a metric graph and a non-empty open set U in , a function f : U → R is said to be harmonic on U if f is piecewise affine with integer slopes and for all x ∈ U, the sum of the slopes of f along all outgoing tangent directions at x is equal to 0.
Since we will not use it elsewhere in the paper, we omit the proof of the following result (which is very similar to the proof of [8] Equivalently, a morphism of metric graphs is harmonic if and only if germs of harmonic functions pullback to germs of harmonic functions. For a metric graph , we let Div( ) denote the free abelian group on . Given a harmonic morphism ϕ : → of metric graphs, there are natural pullback and push-forward homomorphisms ϕ * : Div( ) → Div( ) and ϕ * : Div( ) → Div( ) defined by ϕ * (p) = p →p d p (ϕ) p and ϕ * p = ϕ p and extending linearly. It is clear that for D ∈ Div( ), we have deg(ϕ * (D)) = deg(D) and deg(ϕ * (D)) = deg(ϕ) · deg(D).

Augmented metric graphs
Here, we consider -metric graphs together with the data of a non-negative integer at each point. For a finite point p, this integer should be thought as the genus of a 'virtual algebraic curve' lying above p.
13. An augmented -metric graph is a -metric graph along with a function g : → Z ≥0 , called the genus function, such that g(p) = 0 for all points of except possibly for finitely many (finite) -points p ∈ . The essential vertices of are the points p for which val(p) = 2 or g(p) > 0. A vertex set of is a vertex set V ( ) of the underlying metric graph which contains all essential vertices of as an augmented graph.
An augmented metric graph is said to be totally degenerate if the genus function is identically zero.
The genus of is defined to be A harmonic morphism of augmented -metric graphs ϕ : → is a map which is a harmonic morphism between the underlying metric graphs of and .
Note that both summations in the above definition are in fact over essential vertices of . The degree of the canonical divisor of an augmented -metric graph is deg(K ) = 2g( ) − 2. An augmented metric graph of genus 0 will also be called a rational augmented metric graph. Let ϕ : → be a harmonic morphism of augmented -metric graphs.
The ramification divisor of ϕ is the divisor R = R p (p ), where If p is a non-essential vertex of , then one verifies by direct computation that R p = 0 if d p (ϕ) = 0 or if p = ϕ(p ) is a non-essential vertex of . In particular, the sum R p (p ) is in fact finite, so R is well-defined. Note that if p ∈ V ∞ ( ) and d p (ϕ) > 0, then ϕ(p ) is also an infinite vertex and R p ≥ 0, whereas if d p (ϕ) = 0, then R p = −1.
The definition of R v in Equation 2.14.1 is rigged so that we have the following graphtheoretic analogue of the Riemann-Hurwitz formula: (2.14.2) Definition 2.15. Let ϕ : → be a finite harmonic morphism of augmentedmetric graphs. We say that ϕ is étale at a point p ∈ provided that R p = 0. We say that ϕ is generically étale if R is supported on the set of infinite vertices of and that ϕ is étale if R = 0.

Metrized complexes of curves
Recall that k is an algebraically closed field. A metrized complex of curves is, roughly speaking, an augmented metric graph together with a chosen vertex set V ( ) and a marked algebraic k-curve of genus g(p) for each finite vertex p ∈ V ( ). More precisely: Definition 2.17. A -metrized complex of k-curves consists of the following data: • An augmented -metric graph equipped with the choice of a distinguished vertex set V ( ).
• For every finite vertex p of , a smooth, proper, connected k-curve C p of genus g(p).
• An injective function red p : T p ( ) → C p (k), called the reduction map. We call the image of red p the set of marked points on C p . Example 2.18. Figure 2 depicts a particular metrized complex of curves over C. At each finite vertex p there is an associated compact Riemann surface C p , together with a finite set of marked points (in red).

Definition 2.19.
A harmonic morphism of metrized complexes of curves consists of a harmonic (V ( ), V ( ))-morphism ϕ : → of augmented metric graphs, and for every finite vertex p of with d p (ϕ) > 0, a finite morphism of curves ϕ p : C p → C ϕ(p ) , satisfying the following compatibility conditions: Note that we do not require the morphisms ϕ p : C p → C p to be ramified only at the marked points corresponding to the tangent directions at p on the metric graph ; see however Proposition 2.22 which naturally guarantees this property.
A harmonic morphism of metrized complexes of curves is called finite if the underlying harmonic morphism of (augmented) metric graphs is finite. Metrized complexes of curves with harmonic morphisms between them form a category, with finite harmonic morphisms giving rise to a subcategory.
If has at least one edge, then (3) follows from (1) and (2), since the sum of the ramification degrees of a finite morphism of curves along any fiber is equal to the degree of the morphism.
Let ϕ : C → C be a finite morphism of smooth, proper, connected k-curves. Recall that ϕ is tamely ramified if either char(k) = 0 or the ramification degree of ϕ at every closed point of C is prime to the characteristic of k. ramified if and only if ϕ| U : U → U is a tamely ramified cover of U over C relative to D in the sense of ( [40], XIII.2.1.3): see the discussion in §2.0 of loc. cit. 2. A tamely ramified morphism is separable (i.e. generically étale). 3. The Riemann-Hurwitz formula applies to a tamely ramified morphism. Definition 2.21. Let ϕ : C → C be a finite harmonic morphism of metrized complexes of curves. We say that ϕ is a tame harmonic morphism if either char(k) = 0, or char(k) = p > 0 and ϕ p is tamely ramified for all finite vertices p ∈ V ( ). We call ϕ a tame covering if in addition ϕ : → is a generically étale finite morphism of augmented metric graphs.
Let ϕ : C → C be a tame harmonic morphism. Then, each ϕ p : C p → C ϕ(p ) is a separable morphism. Note that if char(k) = p, then d e (ϕ) is prime to p for all edges e of . We have the following kind of converse statement: Moreover, ϕ p is automatically tamely ramified and ϕ is a tame covering.
Proof. Let p = ϕ(p ). Since ϕ is generically étale, we have Since g(p) = g C p , g p = g C p , and d p (ϕ) is the degree of ϕ p : The Riemann-Hurwitz formula as applied to ϕ p says that the right-hand side of the above equation is equal to the degree of the ramification divisor of ϕ p (this holds whenever ϕ p is separable). For all v ∈ T p ( ), the ramification degree e x of ϕ p at x = red p (v ) is d v (ϕ), so we have an equality between the degree of the ramification divisor of ϕ p and x =red p (v ) (e x − 1). This implies both that any ramification point x ∈ C p (k) is the reduction of a tangent direction, and in addition that the contribution of x to the ramification divisor of ϕ p is equal to e x − 1, from which it follows that ϕ p is tamely ramified at x .

Analytic curves and their skeleta
In this section, we recall the definition and basic properties of the skeleton of the analytification of an algebraic curve as outlined in [43] and elaborated in [12]. These foundational results are well known to experts and appear in various guises in the literature. See, for example, Berkovich [14,16], Thuillier [44], and Ducros [25,26]. In this section, we also show that the skeleton is naturally a metrized complex of curves and prove that any metrized complex arises as the skeleton of a curve. Finally, we introduce the important notion of a triangulated punctured curve, which is essentially a punctured curve along with the data of a skeleton with a distinguished set of vertices.
Recall that K is an algebraically closed field which is complete with respect to a nontrivial non-Archimedean valuation val : K → R ∪ {∞}. Its valuation ring is R, its maximal ideal is m R , its residue field is k = R/m R , and its value group is = val(K × ) ⊂ R. The assumption that K is algebraically closed is mostly for simplicity, to avoid complications due to Galois actions on non-split semistable models and nontrivial twists of analytic balls and annuli (see however Section 5 where we explain how to use descent arguments to remove this restriction and get results over more general ground fields). We assume that K is nontrivially valued because the theory is mostly tautological in the trivially valued case.
By an analytic curve, we mean a strictly K-analytic space X of pure dimension 1. For our purposes, the most important example of an analytic curve is the analytification of a smooth, connected, projective algebraic K-curve. If X is an analytic curve, then we define H(X) = X \ X(K) as in ( [12], §5).

Analytic building blocks
We identify the set underlying the analytification of the affine line with the set of multiplicative seminorms · : K[ t] → R ≥0 which extend the absolute value on K. We let val : By a closed unit ball (resp. open unit ball, resp. closed annulus, resp. open annulus, resp. punctured open ball), we will mean a K-analytic space which is isomorphic to the standard closed unit ball (resp. the standard open unit ball, resp. the standard closed annulus of modulus val(a) for some a ∈ R \ {0}, resp. the standard open annulus of modulus val(a) for some a ∈ m R \ {0}, resp. the standard punctured open ball). A generalized open annulus is a K-analytic space which is either an open annulus or a punctured open ball. It is a standard fact that the modulus of a closed (resp. open) annulus is an isomorphism invariant; see for instance [12], Corollary 2.6. We define the modulus of a punctured open ball to be ∞.
Let a ∈ m R (so a = 0 is allowed). There is a natural section σ : Let T be a metric space and let x, y ∈ T. A geodesic segment from x to y is the image of a locally isometric embedding [ a, b] → T with a → x and b → y. We often identify a geodesic segment with its image in T. Recall that an R-tree is a metric space T with the following properties: See [9], Appendix B. It is proved in §1.4 of loc. cit. that H(B(1)) is naturally an Rtree. (Note however that the topology induced by the R-tree structure is finer than the analytic topology.) Since any path-connected subspace of an R-tree is an R-tree as well, if X is a standard open annulus or standard (punctured) open ball, then H(X) is an R-tree. It is proved in [12], Corollary 5.6, that the metric structure on H(X) is an isomorphism invariant, so the same statement applies to open balls and generalized open annuli. For a ∈ m R , the section σ : (0, val(a)) → S(a) + is an isometry.
It also follows from the results in [9], §1.4, that for any x ∈ H(B(1)) and any type-1 point y ∈ B(1), there is a unique continuous injection α :  Proof. Let f ∈ A be nonzero and define F : (0, ∞) → R by F(r) = − log |f • ϕ(σ (r))|, where σ : (0, ∞) → S(0) + ⊂ B(1) + is the canonical section of val. By [12], Proposition 2.10, F is a piecewise affine function which is differentiable away from finitely many points r ∈ ∩ (0, ∞), and for any point r ∈ (0, ∞) at which F is differentiable, the derivative of F is equal to the number of zeros y of f • ϕ with val(y) > r (note that f • ϕ has finitely many zeros since f is algebraic and ϕ has finite fibers). It follows that F( ∩ (0, ∞)) ⊂ . Since f • ϕ has finitely many zeros, the limit lim r→0 F(r) exists and is contained in . We define f 0 = exp(− lim r→0 F(r)). It is easy to see that f → f 0 is a multiplicative seminorm on A extending the absolute value on K, so · 0 is a point in X an . Define ϕ(x) = · 0 . One shows as in the proof of [12], Lemma 3.2, that ϕ is continuous on

Open balls
It remains to show that ϕ(x) = · 0 is a type-2 point of X an . Let f ∈ A be a nonconstant function that has a zero on ϕ(B(1) + ). Since − log f 0 ∈ , there exists α ∈ K × such that αf 0 = 1; replacing f by αf , we may and do assume that f 0 = 1. Since ϕ • f has a zero, by the above we have that F = − log |f • ϕ • σ | is monotonically increasing, so F(r) > 0 for all r ∈ (0, ∞). For any r ∈ (0, ∞), the maximal point of val −1 (r) ⊂ B(1) + is equal to σ (r), so for any y ∈ B(1) + such that val(y) = r, we have F(r) ≤ − log |f • ϕ(y)|. It follows that |f • ϕ(y)| < 1 for all y ∈ B(1) which is a type-2 point of A 1,an . Therefore, ϕ(x) is a type-2 point of X an . Applying

Open annuli
Let a ∈ m R \ {0}. The closure of S(a) + in B(1) consists of S(a) + and the two type-2 points x = σ (0) and y = σ (val(a)), called the ends of S(a) + : again see the proof of [12],

The skeleton of a curve
Let X be a smooth, connected, proper algebraic K-curve, and let D ⊂ X(K) be a finite set of closed points. The set H(X an ) is natually a metric space ( [12], Corollary 5.7), although the metric topology on H(X an ) is much finer than the topology induced by the topology on the K-analytic space X an . The metric on H(X an ) is locally modeled on an R-tree ([12], Proposition 5.9). It is a consequence of the semistable reduction theorem of Deligne-Mumford that there exist semistable vertex sets of (X, D): see ( [12], Theorem 4.22). In the sequel, it will be convenient to consider a curve along with a choice of semistable vertex set, so we give such an object a name. Definition 3.9. A triangulated punctured curve (X, V ∪ D) is a smooth, connected, proper algebraic K-curve X equipped with a finite set D ⊂ X(K) of punctures and a semistable vertex set V of (X, D).

Remark 3.10.
This terminology is loosely based on that used in [26] as well as the forthcoming book of Ducros on analytic spaces and analytic curves. Strictly speaking, what we have defined should be called a semistably triangulated punctured curve, but as these are the only triangulations that we consider, we will not need the added precision.
Let (X, V ∪ D) be a triangulated punctured curve, so (This set is denoted (X, V ∪ D) in [12].) For each i and each α the inclusions H( For each open annulus A i , the map A i → X an extends to a continuous map A i → X an sending the ends of A i to points x i , y i ∈ V . We say that A i is adjacent to x i and y i . The length (e i ) ∈ of the geodesic segment e i = (A i )∪{x i , y i } is then the modulus of A i . We say that V is strongly semistable if x i = y i for each open annulus A i . For each generalized open annulus A i , the map A i → X an extends to a continuous map A i → X an sending the end of A i to a point x i ∈ V and sending the puncture to a point y i ∈ D. We say that A i is adjacent to x i and y i . We define the length of The skeleton = (X, V ∪ D) naturally has the structure of a -metric graph with . , e n } as above. Note that the -points of are exactly the type-2 points of X an contained in . For x ∈ V , the residue field H (x) of the completed residue field H (x) at the type-2 point x is a finitely generated field extension of k of transcendence degree 1; we let C x be the smooth k-curve with function field H (x). For x ∈ V we let g(x) be the genus of C x , and for x ∈ \ V we set g(x) = 0. These extra data give the structure of an augmented -metric graph. (In (3.22), we will see that is in fact naturally a metrized complex of k-curves.) By the genus formula ( [12], (4.1)), the genus of X is equal to the genus of the augmented -metric graph . The open analytic domain X an \ is isomorphic to an infinite disjoint union of open balls ( [12], Lemma 3.4(3)). If B is a connected component of X an \ , then the inclusion B → X an extends to a map B → X an sending the end of B to a point of . We define the retraction τ = τ : X an → by fixing and sending a point x ∈ X an not in to the end of the connected component of x in X an \ . This is a continuous map and in fact Berkovich has proved that it is a strong deformation retraction, cf. [14]. If x ∈ B α is in an open ball connected component of X an \(V ∪ D), then τ (x) ∈ V is the end of B α , and if x ∈ A i , then τ (x) coincides with the image of x under the retraction map τ : Here, we collect some additional facts about skeleta from [12].
1. The skeleton is the set of points in X an that do not admit an open neighborhood isomorphic to B(1) + and disjoint from V ∪ D.
Then, any geodesic segment from x to y in H(X an ) is contained in . 6. Let x, y ∈ X an be points of type 2 or 3, and let [ x, y] be a geodesic segment from x to y in H(X an ). Then, there exists a semistable vertex set V 1 of (X, D) such that The augmented -metric graph structure of a skeleton of (X, D) does not depend on the choice of vertex set for .

Modifying the skeleton
Let X be a smooth, proper, connected K-curve, and let D ⊂ X(K) be a finite set. Let be a skeleton of (X, D), let y ∈ X an \ , let B be the connected component of X an \ containing y, and let x = τ (y) ∈ be the end of B. If y ∈ H(X an ), then the geodesic segment [ x, y] ⊂ B is the unique geodesic segment in H(X an ) connecting x and y. If y ∈ X(K), then we define [ x, y] to be the geodesic segment [ x, y] ⊂ B as in (3.1). The following strengthening of Proposition 3.12(3) will be important in the sequel. Lemma 3.15. Let V be a semistable vertex set of (X, D), let = (X, V ∪D), let W ⊂ X an be a finite set of type-2 points, and let E ⊂ X(K) be a finite set of type-1 points.
1. There exists a minimal semistable vertex set V 1 of (X, D ∪ E) which contains V ∪ W , in the sense that any other such semistable vertex set contains V 1 . 2. Let B be a connected component of X an \ with end x, and let y 1 , . . . , y n be the points of (W ∪ E) ∩ B. Then if n > 0, and (X,

The skeleton
is minimal in the sense that any other skeleton containing and W ∪ E must contain 1 .

Proof.
To prove the first part, we may assume that W = {y} consists of a single type-2 point not contained in and E = ∅, or E = {y} is a single type-1 point not contained in D and W = ∅. In the first case, one sees using Lemma 3.4 that V 1 = V ∪ {y, τ (y)} is the minimal semistable vertex set of (X, D) containing V and y, and in the second case, the other inclusion is proved by induction on n, adding one point at a time as above. The case E = ∅ is similar and is left to the reader. The final assertion follows easily from the first two and Proposition 3.12.
Remark 3.16. The above lemma shows in particular that the metric graph (X,

The minimal skeleton
The and V is stable if there is no x ∈ V of genus zero (resp. one) and valency less than 3 (resp. one) in (X, V ∪ D). It is clear that minimal semistable vertex sets exist. A skeleton of (X, D) is minimal if there is no skeleton of (X, D) properly contained in .
The following consequence of the stable reduction theorem can be found in [12], Theorem 4.22.
Proposition 3.18. Let V be a minimal semistable vertex set of (X, D), and let = (X, V ∪ D).
1. If χ(X \ D) ≤ 0, then is the unique minimal skeleton of (X, D); moreover, is equal to the set of points of X an \ D that do not admit an open neighborhood isomorphic to B(1) + and disjoint from D.

Remark 3.19.
We have χ(X \ D) > 0 if and only if X ∼ = P 1 and D contains at most one point; in this case, any type-2 point of X an serves as a minimal semistable vertex set of (X, D), and there does not exist a unique minimal skeleton. If χ(X \ D) = 0, then either X ∼ = P 1 and D consists of two points, or X is an elliptic curve and D is empty. In the first case, the minimal skeleton of (X, D) is the extended line connecting the points of D and any type-2 point on is a minimal semistable vertex set. If X is an elliptic curve and X an contains a type-2 point x of genus 1, then {x} is both the unique minimal semistable vertex set and the minimal skeleton. Otherwise, X is a Tate curve, is a circle, and any type-2 point of is a minimal semistable vertex set. See [12], Remark 4.24.

Tangent directions and the slope formula
As above, we let X be a smooth, proper, connected algebraic K-curve. A continuous func- To any point, x ∈ X an is associated a set T x of tangent directions at x, defined as the set of germs of geodesic segments in X an beginning at x. If F : X an → R ∪ {±∞} is a piecewise affine function and v ∈ T x , we denote by d v F(x) the outgoing slope of F in the direction v. We say that F is harmonic at a point x ∈ X an provided that there are only [12], (5.11).
Let x ∈ X an be a type-2 point, let H (x) be the completed residue field of X an at x, and let H (x) be its residue field, as in (3.11.1). Then, H (x) is a finitely generated field extension of k of transcendence degree 1, and there is a canonical bijection between the set T x of tangent directions to X an at x and the set DV The following theorem is called the slope formula in [12], Theorem 5.15: Theorem 3.21. Let f ∈ K(X) × be a nonzero rational function on X and let F = − log |f | : X an → R ∪ {±∞}. Let D ⊂ X(K) contain the zeros and poles of f and let be a skeleton of (X, D). Then: F is piecewise affine with integer slopes, and F is affine on each edge of (with respect to a choice of vertex set V ). 3. If x is a type-2 point of X an and v ∈ T x , then

The skeleton as a metrized complex of curves
Recall that is an augmented -metric graph with infinite vertices D. We enrich with the structure of a -metrized complex of k-curves as follows. For x ∈ V , let C x be the smooth, proper, connected k-curve with function field H (x) as in (3.11.1). By definition, C x has genus g(x). We have natural bijections where the first bijection v → ord v is defined in (3.20) and the second associates to a closed point ξ ∈ C x (k) the discrete valuation ord ξ on the function field H (x) of C x . Let v ∈ T x be a tangent direction and define red x (v) to be the point of C x corresponding to the discrete valuation ord v ∈ DV H (x)/k . These data make into a -metrized complex of k-curves.

Lifting metrized complexes of curves
We now prove that every metrized complex of curves over k arises as the skeleton of a smooth, proper, connected K-curve. This fact appears in the literature in various contexts (over discretely valued fields): see for instance [6], Appendix B, and [41], Lemme 6.3. For this reason, we only sketch a proof using our methods.
Proof. Let C be a smooth, proper, connected k-curve. By elementary deformation theory, there is a smooth, proper admissible formal R scheme C with special fiber C. By GAGA, the analytic generic fiber C K is the analytification of a smooth, proper, connected K-curve C (in fact, the formal scheme C is itself algebraizable, see e.g. [25], Théorème 3.7.6). There is a reduction map red : C K → C from the analytic generic fiber of C to (the set underlying) C, under which the inverse image of the generic point of C is a single distinguished point x. The set {x} is a semistable vertex set of C , with associated skeleton also equal to {x}. Moreover, there is a canonical identification of H (x) with the field of rational functions on C by [14], Proposition 2.4.4. See (5.6) for a more detailed discussion of the relationship between semistable vertex sets and admissible formal models.
Let be the metric graph underlying C. Let V be the vertices of , and let E be its edges. By adding valence-2 vertices, we may assume that has no loop edges. Assume for the moment that has no infinite edges. For a vertex x ∈ V , let C x denote the smooth, proper, connected k-curve associated to x and choose an admissible formal curve C x with special fiber isomorphic to C x as above. For clarity, we let red C x denote the reduction map (C x ) K → C x . By [19], Proposition 2.2, for every x ∈ C x (k), the formal fiber red −1 C x ( x) is isomorphic to B(1) + . Let e be an edge of with endpoints x, y, and let x e ∈ C x (k), y e ∈ C y (k) be the reductions of the tangent vectors in the direction of e at x, y, respectively. Remove closed balls from red −1 C x ( x e ) and red −1 C y ( y e ) whose radii are such that the remaining open annuli have modulus equal to (e). We form a new analytic curve X an by gluing these annuli together using some isomorphism of annuli for each edge e. The resulting curve is proper, hence is the analytification of an algebraic curve X. By construction, the image of the set of distinguished points of the curves (C x ) K in X an is a semistable vertex set, and the resulting skeleton (considered as a metric graph) is isomorphic to .
If does have infinite edges, then we apply the above procedure to the finite part of , then puncture the curve X in the formal fibers over the smooth points of the residue curves which correspond to the directions of the infinite tails of .
As an immediate application, we obtain the following property of the 'abstract tropicalization map' from the moduli space of stable marked curves to the moduli space of stable abstract tropical curves: Corollary 3.25. If g and n are nonnegative integers with 2 − 2g − n < 0, the natural map trop : M g,n → M trop g,n (see [12], Remark 4.25) is surjective.
See [1] for an interpretation of the above map as a contraction from M an g,n onto its skeleton (which also implies surjectivity).

Morphisms between curves and their skeleta
This section is a relative version of the previous one, in that we propose to study the behavior of semistable vertex sets and skeleta under finite morphisms of curves. We introduce finite morphisms of triangulated punctured curves, and we prove that any finite morphism of punctured curves can be enriched to a finite morphism of triangulated punctured curves. This powerful result can be used to prove the simultaneous semistable reduction theorems of Coleman, Liu-Lorenzini, and Liu, which we do in Section 5. It is interesting to note that we use only analytic methods on analytic K-curves, making (almost) no explicit reference to semistable models; hence, our proofs of the results of Liu-Lorenzini and Liu are very different from theirs. Using a relative version of the slope formula, we also show that a finite morphism of triangulated punctured curves induces (by restricting to skeleta) a finite harmonic morphism of metrized complexes of curves, which will be crucial in the sequel.

Morphisms between open balls and generalized open annuli
The main results of this section rest on a careful study of the behavior of certain morphisms between open balls and generalized open annuli. Some of the lemmas and intermediate results appear in some form in the literature -see in particular [14] and [15] -although they are hard to find in the form we need them. As the proofs are not difficult, for the convenience of the reader we include complete arguments.
We begin by remarking that if ϕ : X → X is a morphism of smooth strictly K-analytic curves with finite fibers, then ϕ preserves the type of a point, i.e. if x ∈ X has type i then Proof. Part (2) is exactly [12], Proposition 2.5, so suppose that val • ϕ is constant on (S(a)). The map ϕ : S(a ) + → S(a) + ⊂ G an m is given by a unit f on S(a ) + . By [12],    Proof. First suppose that y does not have type 4. Choose isomorphisms B ∼ = B(1) + such that 0 ≥ y and B ∼ = B(1) + such that ϕ(0) = 0. Since y ∈[x , 0] , we may replace [x , y ] by the larger geodesic segment [x , 0] to assume that y = 0. Define F : [12], Proposition 2.10, F is a piecewise affine function, and for any point alized open annulus mapping to S(0) + . By the above, val • ϕ is not constant on . It is clear in this case that the restriction of ϕ to the closure of any connected component of (S(0) + ) \ Z is an embedding with integer expansion factor, so ϕ| [x ,0] is piecewise affine. Note that ϕ| [x ,0] changes expansion factor at most #ϕ −1 (0) times. Now suppose that y has type 4. Suppose that z , w ∈[x , y ] are two distinct points such that ϕ(z ) = ϕ(w ). Assume without loss of generality that z < w . Since w and z have the same image under ϕ, they both have the same type, so w = y because y is the only type-4 point in [x , y ]. Applying the above to the geodesic [x , w ] gives a contradiction. There- , Corollary 1.14).
Assume now that all fibers of ϕ have size bounded by N. The above argument proves that for all z < y , the restriction of ϕ to [x , z ] is piecewise affine, and ϕ| [x ,z ] changes expansion factor at most N times. Therefore, there exists z 0 < y and m ∈ Z >0 such that for all z ∈[z 0 , y ], the restriction of ϕ to [z 0 , z ] is an embedding with expansion factor m. It follows that ϕ| [z 0 ,y ] is an embedding with expansion factor m, so ϕ| [x ,y ] is piecewise affine.
Since ϕ has finite fibers, ϕ : B → B 1 is finite if and only if it is proper by [14], is closed as a subset of the compact space B . One has Int(B /B 1 ) = B by [14], Proposition 3.1.3(i) since B is a boundary-less K-analytic space. Therefore, ϕ is proper.
The fact that ϕ : B → B 1 is order-preserving follows immediately from Lemma 4.5.
where the second equality holds because A is a finite A-algebra. It follows that for we call this number the degree of ϕ.   ((x , z ) Let d be the degree of the finite morphism ϕ, so for every z ∈ B, we have For y ∈ B as in the statement of the Proposition define a function δ : (x, y] → Z by It follows that is a disjoint union of affinoid domains: indeed, by Lemma 4.5, the preimages of z are incomparable (otherwise, the restriction of ϕ to a geodesic segment [z , z ], for z , z ∈ ϕ −1 (z), would not be injective), and so the sets B z are disjoint. Hence, each map B z → B z is finite, and its degree is equal to dim H (z) O ϕ −1 (z),z . Let z 1 , z 2 ∈ (x, y] and assume that z 1 < z 2 , so z 1 has type 2 or 3. For z 1 ∈ ϕ −1 (z 1 ), we have Summing over all z 1 ∈ ϕ −1 (z 1 ) ∩ T , we obtain is an open ball, then we proceed as above to find r + ∈ (0, r] ∩ such that α is an embedding with integer expansion factor when restricted to [0, r + ]. Applying the above argument to the morphism ϕ composed with the automorphism t → a/t of S(a) + (which interchanges the two ends), we find that there exists r − ∈ [0, val(a)) ∩ such that α is an embedding with integer expansion factor when restricted to [r − , val(a)]. Let s ∈ (0, val(a)) ∩ . Replacing A with the annulus val −1 ((0, s)) (resp. val −1 ((s, val(a)))), the above arguments then provide us with s + ∈ (s, val(a)] ∩ (resp. s − ∈[0, s) ∩ ) such that α is an embedding with integer expansion factor when restricted to [s, s + ] (resp. [s − , s]). Hence, we have shown that there exists a neighborhood U s of every point s ∈[0, val(a)] ∩ such that ϕ| U s is an embedding with integer expansion factor.
It remains to treat type-3 points, i.e., points s ∈[0, val(a)] \ . Let s be such a point, and let y = ϕ(s). Then, y has type 3 as well, so by [12], As for the final assertion, choose 0 = r 0 < r 1 < r 2 < · · · < r n = val(a) such that α is an embedding with integer expansion factor on each [r i , r i+1 ]. Let i 0 ∈ {0, 1, . . . , n} be the largest integer such that α(r i 0 ) > α(r i ) (in the canonical partial ordering on B) for all i < i 0 . If i 0 = n, then we are done, so assume that i 0 < n. Let y = α(r i 0 ) ∈ B, and choose an identification B ∼ = B(1) + such that 0 > y. If F = − log |ϕ|, then by [12], Proposition 2.10, at every point r ∈ (0, val(a)), the change in slope of F at r is equal to the negative of the number of zeros of ϕ with valuation r (cf. the proof of Lemma 4.5); in particular, the slope of F can only decrease. By construction, F is monotonically increasing

Morphisms between curves and skeleta
In what follows, we fix smooth, connected, proper algebraic K-curves X, X and a finite morphism ϕ : X → X. Let D ⊂ X(K) and D ⊂ X (K) be finite sets of closed points. The map on analytifications ϕ : X an → X an is finite and open by [14], Lemma 3.2.4.

Proposition 4.11.
Let be a skeleton of (X , D ), and let be a skeleton of (X, D). There exists a skeleton 1 of (X, D ∪ ϕ(D )) containing ∪ ϕ( ), and there is a minimal such 1 with respect to inclusion.
Proof. First, we will prove the Proposition in the case D = D = ∅. Let V be a vertex set for , and let V be a vertex set for containing τ (y ) : y ∈ ϕ −1 . Let A be an open annulus connected component of X an \V , and let e ⊂ be the associated edge. We claim that ϕ(e ) is a geodesic segment between type-2 points of X an . Let C be the connected component of X an \V containing ϕ(A ). Applying the above to each edge e of , we find that there exists a finite set of type-2 points x 1 , y 1 , x 2 , y 2 , . . . , x n , y n ∈ X an such that ϕ( where [x i , y i ] denotes a geodesic segment from x i to y i . Let By Lemma 3.15(2) as applied to W = {x 1 , y 1 , . . . , x n , y n }, we have that 1 is a skeleton of X, so by Proposition 3.12(5), 1 contains ∪ ϕ( ) since 1 contains each geodesic segment [x i , y i ]. Any other skeleton of X containing and all x i , y i also contains the geodesics [x i , τ (x i )] and [y i , τ (y i )], so 1 is the minimal such skeleton. Now, assume that D D = ∅. By Lemma 3.15 (3), there is a minimal skeleton 1 of X containing and ϕ(D ); replacing by 1 and D by D ∪ ϕ(D ), we may assume without loss of generality that ϕ(D ) ⊂ D. We will proceed by induction on the size of D D . As above, we let V be a vertex set for , and we let V be a vertex set for containing τ (y ) : y ∈ ϕ −1 (V ∪ D) . Let V be a semistable vertex set of (X, D). Recall from (3.11.1) that a connected component C of X an \ (V ∪ D) is adjacent to a vertex x ∈ V provided that the closure of C in X an contains x. Proposition 4.12(1) below is exactly [14], Theorem 4.5.3, when D = D = ∅. Proposition 4.12. Suppose that D = ϕ −1 (D) and that one of the following two conditions holds: In the situation of (1), let (resp. ) be the minimal skeleton of (X, D) (resp. (X , D )); and in (2) Proof. First, suppose that (2) holds. Let x ∈ X an \ , and let B be the con- Let V be a semistable vertex set of (X , D ) such that is a disjoint union of open balls, by the above, we have ϕ −1 (V ) ⊂ . Hence, we may enlarge V to contain ϕ −1 (V ) without changing , so we are reduced to (2).
We will require the following lemmas. Lemma 4.14 is similar to [14], Corollary 4.5.4. Proof. Suppose that the genus of X is at least one, so X also has genus at least one. By Lemma 4.6, we only need to show that B is contained in an open ball in X an . Let (resp. ) be the minimal skeleton of X (resp. X ). By In the case X = P 1 , suppose first that B (K) → P 1 (K) is not surjective, so we may assume that ∞ / ∈ B after choosing a suitable coordinate on P 1 (2) of Lemma 4.14 does occur. For instance, let ϕ : P 1 → P 1 be the finite morphism t → t 2 , and let B be the open ball in P 1,an obtained by deleting the closed ball of radius 1/2 around 1 ∈ P 1 (K). If char(k) = 2, then for every point x ∈ P 1 (K), either x ∈ B (K) or −x ∈ B (K), so B → P 1,an is surjective.
Proof. We may assume without loss of generality that Proof. Note that ϕ( ) ⊂ implies ϕ(D ) ⊂ D. Let V (resp. V ) be a vertex set for (resp. ). We claim that x , τ x , which by Lemma 3.15 is the minimal skeleton of (X , ϕ −1 (D)) containing and ϕ −1 (V ). Let B be a connected component of X an \ , let x ∈ be its end, and let x = ϕ(x ) ∈ . Then, B = ϕ(B ) is an open ball and x is its end: if X ∼ = P 1 , then this follows directly from Lemma 4.14; and if X ∼ = P 1 , then B ⊂ X an \ {z}, so ϕ(B) = X an and therefore B is an open ball in this case as well. Hence, x, y by Lemma 4.16, so x , y by Proposition 4.8.
Remark 4.17. When X ∼ = P 1 , the extra hypothesis on in Theorem 4.13 is necessary. Indeed, let ϕ : P 1 → P 1 be a finite morphism and let x ∈ P 1,an be a type-2 point such that ϕ −1 (ϕ(x )) has more than one element. Then, = {ϕ(x )} is a skeleton containing the image of the skeleton = {x }, but ϕ −1 ( ) is not a skeleton.

Tangent directions and morphisms of curves
Our next goal is to prove that a finite morphism of curves induces a finite harmonic morphism of metrized complexes of curves for a suitable choice of skeleta. First, we need to formulate and prove a relative version of Theorem 3.21. Let X, X be smooth, proper, connected K-curves.

Definition 4.21.
A continuous function ϕ : X an → X an is piecewise affine provided that, for every geodesic segment α :[a, b] → X an , the composition ϕ•α is piecewise affine with integer expansion factors in the sense of Definition 4.4.
Let ϕ : X an → X an be a piecewise affine function, let x ∈ X an , and let x = ϕ(x ) ∈ X an . Let v ∈ T x , and let α :[a, b] → X an be a geodesic segment representing v (so a = −∞ if x has type 1). Taking b small enough, we can assume that ϕ • α is an embedding with integer expansion factor m. Let v ∈ T x be the tangent direction represented by ϕ • α. We define this is independent of the choice of α. We call the expansion factor m the outgoing slope of ϕ in the direction v , and we write m = d v ϕ(x ).

Definition 4.22.
A piecewise affine function ϕ : X an → X an is harmonic at a point x ∈ X an provided that, for all has a finite number of nonzero terms and is independent of the choice of v.
Let ϕ : X → X be a finite morphism of smooth, proper, connected K-curves, let x ∈ X an be a type-2 point, and let x = ϕ(x ). Let C x and C x denote the smooth proper connected k-curves with function fields H (x) and H (x ), respectively. We denote by ϕ x the induced morphism C x → C x . We have the following relative version of the slope formula of Theorem 3.21: Theorem 4.23. Let ϕ : X → X be a finite morphism of smooth, proper, connected K-curves.
1. The analytification ϕ : X an → X an is piecewise affine and harmonic. 2. Let x ∈ X an be a type-2 point, let x = ϕ(x ), let v ∈ T x , let v = dϕ(v ) ∈ T x , and let ξ v ∈ C x and ξ v ∈ C x be the closed points associated Equation 3.22.1 to v and v, respectively. Then, ϕ x (ξ v ) = ξ v , and the ramification degree of ϕ x at ξ v is equal to d v ϕ(x ). 3. Let x ∈ X an be a type-1 point, and let v ∈ T x be the unique tangent direction.
Then, d v ϕ(x ) is the ramification degree of ϕ at x .
Proof. First we claim that ϕ is piecewise affine. Let [x , y ] ⊂ X an be a geodesic segment. Suppose first that x and y have type 2 or 3. Then, there exists a skeleton of X containing [x , y ] by [12], Corollaries 5.1 and 5.10. From this, we easily reduce to the case that [x , y ] is an edge of with respect to some vertex set; now, the claim follows from Next, we prove (2). By functoriality of the reduction map, for any rational function By the slope formula (Theorem 3.21), the point ξ v corresponds to the discrete valuation on H (x) given by Since any non-zero element of H (x) is of the form f x for some rational function f on The proof of (3) proceeds in the same way as (2), applying the slope formula to a nonzero rational function f on X with a zero at x = ϕ(x ) and to ϕ * f on X .

Morphisms of curves induce morphisms of metrized complexes
More precisely, this occurs when the morphism of curves respects a choice of triangulations, in the following sense.  Let ϕ : X → X be a finite morphism of smooth, connected, proper K-curves, let D ⊂ X(K) be a finite set, and let D = ϕ −1 (D). There exists a strongly semistable vertex set V of (X, D) such that V = ϕ −1 (V ) is a strongly semistable vertex set of (X , D ) and such that In particular, ϕ extends to a finite morphism of triangulated punctured curves.
Proof. By Corollary 4.18, there is a skeleton of (X, D) such that = ϕ −1 ( ) is a skeleton of (X , D ). Let V 0 be a strongly semistable vertex set of , let V be a strongly semistable vertex set of containing ϕ(V 0 ), and let V = ϕ −1 (V ). Then, V is again a strongly semistable vertex set of .

Remark 4.29.
More generally, suppose that (X, V ∪D) and (X , V ∪D ) are triangulated punctured K-curves and that ϕ : X → X is a finite morphism such that ϕ −1 (D) = D and ϕ −1 (V ) ⊂ V . Then, ϕ −1 ( ) ⊂ by Proposition 4.12 (2). One can show using Theorem 4.23 that the map → X an followed by the retraction X an → is a (not necessarily finite) harmonic morphism of metric graphs → .

Tame coverings of triangulated curves
Many of the results in this paper involve constructing a curve X and a morphism ϕ : X → X inducing a given morphism of skeleta as in Corollary 4.28. For these purposes, it is useful to introduce some mild restrictions on the morphism ϕ.   1. Since D is finite, it follows from (1) and Theorem 4.23(3) that a tame covering of (X, V ∪ D) is a tamely ramified morphism of curves.
2. If has at least one edge, then (2)   Proof. Let (resp. ) be the augmented metric graph underlying (resp. ). By Proposition 2.22, we only have to show that ϕ| is generically étale. Let R = S = x ∈D S x (x ) be the ramification divisor of ϕ : X → X. We will show that R = S. Since ϕ x is generically étale for all x ∈ V , we see from Equation 2.14.1 and the Riemann-Hurwitz formula as applied to ϕ x that R x ≥ 0 for x ∈ V . If x ∈ D , then there is a unique edge e adjacent to x and d e (ϕ) = d x (ϕ), so where the final equality is Theorem 4.23 (3). Since R x ≥ 0 for all x ∈ V , it is enough to show that deg(R) = deg(S). By Equation 2.14.2, we have K = (ϕ| ) * (K ) + R, so counting degrees gives The Lemma now follows from the equalities deg(ϕ) = deg(ϕ| ), g( ) = g(X), g( ) = g(X ), and the Riemann-Hurwitz formula as applied to ϕ : X → X.

Remark 4.34.
We showed in the proof of Lemma 4.33 that the ramification divisor of ϕ| coincides with the ramification divisor of ϕ. Moreover, it follows from Proposition 2.22 that for every x ∈ V , every ramification point x ∈ C x is the reduction of a tangent direction in . In other words, for a tame covering all ramification points of all residue curves are 'visible' in the morphism of underlying metric graphs → .

Moreover, if ϕ is a tame covering and x ∈ X an is a type-2 point not contained in , then ϕ is an isomorphism in a neighborhood of x .
Proof. It is clear that conditions (1) and (2) imply that ϕ is a tame covering, so suppose that ϕ is a tame covering and x ∈ X an is a type-2 point. We have the following cases: • If x ∈ V , then ϕ x is tamely ramified by Lemma 4.33 and Proposition 2.22.
• Suppose that x is contained in the interior of an edge e of . Since ϕ is a tame covering of (X, (V ∪ {ϕ(y )}) ∪ D) by Remark 4.32(3), we are reduced to the case treated above.
• Suppose that x / ∈ but y = τ (x ) is contained in V . Let X (resp. X ) be the semistable formal model of X (resp. X ) corresponding to the semistable vertex set V (resp. V ), and let ϕ denote the unique finite morphism X → X extending ϕ : X → X (see Theorem 5.13). Let x = red(x ) ∈ X (k), and let x = ϕ(x ) and • Suppose that x / ∈ but y = τ (x ) is contained in the interior of an edge of . We claim that ϕ is an isomorphism in a neighborhood of x . Let y = ϕ(y ) ∈ . Then, ϕ is a tame covering of (X, (V ∪ {y}) ∪ D), so replacing V with V ∪ {y} and V with V ∪ ϕ −1 (y) (see Remark 4.32 (3)), we are reduced to the previous case. Remark 4.36. It follows from Proposition 4.35 that if ϕ is a tame covering, then the set-theoretic branch locus of ϕ : X an → X an (i.e. the set of all points in X an with fewer than deg(ϕ) points in its fiber) is contained in . See [27,28] for more on the topic of the Berkovich ramification locus in the case of self-maps of P 1 .

Applications to morphisms of semistable models
In this section, we show how Section 4 formally implies a large part of the results of Liu [34] on simultaneous semistable reduction of morphisms of curves (see also [23]) over an arbitrary valued field. In addition to establishing these results over a more general ground field, we feel that the 'skeletal' point of view on simultaneous semistable reduction is enlightening (see Remark 5.23). To this end, we let K 0 be any field equipped with a nontrivial non-Archimedean valuation val : K 0 → R ∪ {∞}. Its valuation ring will be denoted R 0 , its maximal ideal m R 0 , and its residue field k 0 .
Let X be a smooth, proper, geometrically connected algebraic K 0 -curve. By a (strongly) semistable R 0 -model of X, we mean a flat, integral, proper relative curve X → Spec(R 0 ) whose special fiber X k 0 is a (strongly) semistable curve (i.e. X k 0 is a reduced curve with at worst ordinary double point singularities; it is strongly semistable if its irreducible components are smooth) and whose generic fiber is equipped with an isomorphism X K 0 ∼ = X. By properness of X , any K 0 -point x ∈ X(K 0 ) extends in a unique way to an R 0 -point x ∈ X (R 0 ); the special fiber of this point is the reduction of x and is denoted red(x) ∈ X (k 0 ). Let D ⊂ X(K 0 ) be a finite set. A semistable model X of X is a semistable R 0 -model of (X, D) provided that the points of D reduce to distinct smooth points of X k 0 .
The model X is a stable R 0 -model of (X, D) provided that every rational (resp. genus-1) component of the normalization X k contains at least three points (resp. one point) mapping to a singular point of X k or to the reduction of a point of D.
If K 0 = K is complete and algebraically closed, we define a (strongly) semistable formal R-model of X to be an integral, proper, admissible formal R-curve X whose analytic generic fiber X K is equipped with an isomorphism to X an and whose special fiber X k is a (strongly) semistable curve. There is a natural map of sets red : X an → X k , called the reduction, which extends the reduction map from the previous paragraph to the formal setting; it is surjective and anti-continuous in the sense that the inverse image of a Zariski-open subset of X k is a closed subset of X an (and vice versa). Using the reduction map, for a finite set D ⊂ X(K), we define semistable and stable formal R-models of (X,D) as above.
As we will be passing between formal and algebraic R-models of K-curves, it is worth stating the following lemma; see [12], Remark 4.2(2). Lemma 5.1. Let X be a smooth, proper, connected K-curve. The -adic completion functor defines an equivalence from the category of semistable R-models of X to semistable formal R-models of X.
The inverse of the -adic completion functor of Lemma 5.1 will be called algebraization.

Descent to a general ground field
We will use the following lemmas to descend the geometric theory of Section 4 to the valued field K 0 . Fix an algebraic closure K 0 of K 0 and a valuation val on K 0 extending the given valuation on K 0 . For any field K 1 ⊂ K 0 , we consider K 1 as a valued field with respect to the restriction of val, and we write R 1 for the valuation ring of K 1 . Let K be the completion of K 0 with respect to val. This field is algebraically closed by [20], Proposition 3.4.1/3. Lemma 5.2. Let X 0 be a smooth, proper, geometrically connected curve over K 0 , and let is dense in P 1,an K and the subspace topology on P 1 K (K) ⊂ P 1,an K coincides with the ultrametric topology, this proves the lemma for X 0 = P 1 K 0 . For general X 0 , choose a finite, generically étale morphism ϕ : X 0 → P 1 Lemma 5.4. Let X 0 be a smooth, proper, geometrically connected curve over K 0 , let D ⊂ X 0 (K 0 ) be a finite subset, let X = X 0 ⊗ K 0 K, and let X be a semistable model of (X, D). There exists a finite, separable extension K 1 of K 0 and a semistable model Proof. Let X be the -adic completion of X . For x ∈ X k (k), the subset red −1 ( x) ⊂ X an is open in X an . By Lemma 5.2, there exists a point x ∈ X 0 (K sep 0 ) with red(x) = x, so after passing to a finite, separable extension of K 0 if necessary, we may enlarge D ⊂ X 0 (K 0 ) to assume that X is a stable model of (X, D). On the other hand, by the stable reduction theorem (i.e. the properness of the Deligne-Mumford stack M g,n parameterizing stable marked curves), there exists a stable model X 0 of (X 0 , D 0 ) after potentially passing to a finite, separable extension of K 0 . By uniqueness of stable models (i.e. the separateness of the Deligne-Mumford stack), we have X 0 ⊗ R 0 R ∼ = X . Lemma 5.5. Let X 0 , X 0 be smooth, proper, geometrically connected K 0 -curves, and let ϕ : X 0 → X 0 be a finite morphism. Let X 0 (resp. X 0 ) be a semistable model of X 0 (resp. X 0 ), let X = X 0 ⊗ K 0 K and X = X 0 ⊗ R 0 R (resp. X = X 0 ⊗ K 0 K and X = X 0 ⊗ R 0 R), and suppose that ϕ K : X → X extends to a morphism X → X . Then, ϕ extends to a morphism X 0 → X 0 in a unique way.
Proof. The morphism ϕ gives rise to a section X 0 → X 0 × K 0 X 0 of pr 2 : X 0 × K 0 X 0 → X 0 . Let Z be the schematic closure of the image of X 0 in X 0 × R 0 X 0 . It is enough to show that pr 2 : Z → X 0 is an isomorphism. Since schematic closure respects flat base change, the schematic closure of the image of X in X × R X is equal to Z⊗ R 0 R. By properness of X , the image of X in X × R X is then equal to Z⊗ R 0 R, so pr 2 : Z⊗ R 0 R → X is an isomorphism. Since R 0 → R is faithfully flat, this implies that pr 2 : Z → X 0 is an isomorphism.

The relation between semistable models and skeleta
Let X be a smooth, proper, connected K-curve. The following theorem is due to Berkovich and Bosch-Lütkebohmert; see [12], Theorem 4.6, for more precise references. It follows from Theorem 5.7 that if X is a semistable formal model of X, then is a disjoint union of open balls and finitely many open annuli. The following folklore theorem is proved in [12], Theorem 4.11.
Theorem 5.8. Let X be a smooth, proper, connected algebraic K-curve, and let D ⊂ X(K) be a finite set of closed points. The association X → V (X) sets up a bijection between the set of (strongly) semistable formal models of (X, D) (up to isomorphism) and the set of (strongly) semistable vertex sets of (X, D). Let X be a semistable formal model of (X, D). One can construct the metrized complex of curves (X, V (X) ∪ D) directly from X, as follows. Let V be the set of irreducible components of X k , regarded as a set of vertices of a graph. For x ∈ V , we will write C x to denote the corresponding component of X k . To every double point x ∈ X k , we associate an edge e connecting the irreducible components of X k containing x (this is a loop edge if there is only one such component); if C x is such a component, then we set red x (e) = x. The completed local ring of X at x is isomorphic to R[ [ s, t]] /(st − π) for some π ∈ m R \{0}; we define the length of e to be (e) = val(π). We connect a point x ∈ D to the vertex corresponding to the irreducible component of X k containing red(x). These data define a metrized complex of curves (X, D). By Equation 5.7.1, the graphs underlying (X, D) and (X, V (X) ∪ D) are naturally isomorphic. The edge lengths coincidesee for instance [12], Proposition 4.10 -and the residue curves of the two metrized complexes are naturally isomorphic by [14], Proposition 2.4.4. From this, it is straightforward to verify that (X, D) and (X, V (X) ∪ D) are identified as metrized complexes of curves.
In particular, a semistable formal model X of (X, D) is stable if and only if V (X) is a stable vertex set of (X, D).
As another consequence of Theorem 5.8, we obtain the following compatibility of skeleta and extension of scalars. Proposition 5.10. Let X be a smooth, proper, connected K-curve, let K be a complete and algebraically closed valued field extension of K, and let π : X an K → X an K be the canonical map. If X is a semistable formal model of X, then an isomorphism of augmented metric graphs.
Proof. The reduction map is compatible with extension of scalars, in that the following square commutes with k the residue field of K . The first assertion is immediate because the extension of scalars morphism π : X k → X k is a bijection on generic points. It is also a bijection on singular points, and for any node x ∈ X k the open annulus A = red −1 ( x ) is identified with the extension of scalars of A = red −1 ( π( x )). It is easy to see that π : A → A takes the skeleton of A isomorphically onto the skeleton of A.

Extending morphisms to semistable models: analytic criteria
Let X be a smooth, proper, connected K-curve, and let X be a semistable formal model of X. The inverse image topology on X an is the topology T (X) whose open sets are the sets of the form red −1 (U k ), where U k is a Zariski-open subset of X k . Note that any such set is closed in the natural topology on X an . If U k ⊂ X k is an affine open subset, then U k is the special fiber of a formal affine open U = Spf(A) ⊂ X, the set underlying the generic fiber U K = M (A K ) ⊂ X an is equal to red −1 (U k ), and A is equal to the ring • A K of power-bounded elements in the affinoid algebra A K . This essentially means that X is a formal analytic variety in the sense of [18]; see [12], Proposition 5.12. Let X, X be smooth, proper, connected algebraic curves over K, let ϕ : X → X be a finite morphism, and let X and X be semistable formal models of X and X , respectively. Then, ϕ extends to a morphism X → X if and only if ϕ is continuous with respect to the inverse image topologies T (X) and T (X ), in which case there is exactly one morphism X → X extending ϕ.
Fix smooth, proper, connected algebraic K-curves X, X and a finite morphism ϕ : X → X. The following theorem is well known to experts, although no proof appears in the published literature to the best of our knowledge (see however the book in preparation by Ducros ([25], Section 6.3), where part of this result is discussed).
Theorem 5.13. Let X and X be semistable formal models of X and X , respectively. Then ϕ : X → X extends to a morphism X → X if and only if ϕ −1 (V (X)) ⊂ V (X ), and Remark 5.14. If X and X are semistable formal models of the punctured curves (X, D) and (X , D ), respectively, and ϕ : X → X is a finite morphism with ϕ −1 (D) = D which extends to a morphism X → X, then since ϕ −1 (V (X)) ⊂ V (X ) it follows from Remark 4.29 that there is a natural harmonic morphism of metric graphs (X , V (X ) ∪ D ) → (X, V (X) ∪ D). The morphism X → X is finite if and only if the local degree of this harmonic morphism at every v ∈ V (X ) is positive.
Before giving a proof of Theorem 5.13, we mention the following consequences. Let X 1 and X 2 be semistable formal models of X. We say that X 1 dominates X 2 provided that there exists a (necessarily unique) morphism X 1 → X 2 inducing the identity map on analytic generic fibers. Taking X = X in Theorem 5.13, we obtain the following corollary; this gives a different proof of the second part of ( [12], Theorem 4.11).
Corollary 5.15. Let X 1 and X 2 be semistable formal models of X. Then, In conjunction with Proposition 5.10, we obtain the following corollary. If K is a complete and algebraically closed field extension of K, we say that a semistable formal model X of X K is defined over R provided that it arises as the extension of scalars of a (necessarily unique) semistable formal model of X. Corollary 5.16. Let K be a complete and algebraically closed field extension of K, let X be a semistable formal model of X K , and suppose that there exists a semistable formal model X of X such that X R dominates X . Then, X is defined over R.
Proof. Let π : X an K → X an be the canonical map. By Corollary 5.15, we have V (X R ) ⊃ V (X ), and by Proposition 5.10 the map π defines a bijection V (X R ) Let V = V (X) and V = π(V (X )). We claim that V is a semistable vertex set. Granted this, letting Y be the semistable formal model of X associated to V , It remains to prove the claim that V is a semistable vertex set. The point is that the question of whether or not a subset of V is a semistable vertex set is intrinsic to the augmented -metric graph = (X, V ) ∼ = (X K , V (X R )). We leave the details to the reader.
We will use the following lemmas in the proof of Theorem 5.13. Recall from (3.11.1) that if V is a semistable vertex set of X, then a connected component C of X an \ V is adjacent to a point x ∈ V provided that the closure of C in X an contains x.

an . Then, U is open in the topology T (X) if and only if the following conditions hold:
1. U is closed in the ordinary topology on X an , and 2. for every x ∈ U ∩ V , all but finitely many connected components of X an \ V which are adjacent to x are contained in U.
Proof. Let ζ be a generic point of X k , and let y ∈ X k be a closed point. Let x ∈ V be the unique point of X an reducing to ζ , and let B = red −1 ( y). By the anti-continuity of the reduction map, y is in the closure of ζ if and only if x is adjacent to B. The lemma follows easily from this and the fact that the connected components of X an \ V are exactly the inverse images of the closed points of X k under red. Lemma 5.18. Let V and V be semistable vertex sets of X and X , respectively, and suppose that ϕ −1 (V ) ⊂ V . Let C be a connected component of X an \ ϕ −1 (V ). Then, C has the following form:

Proof. Let
= (X , V ). Suppose that there exists y ∈ C such that τ (y ) / ∈ C . Let B be the connected component of X an \ containing y , so B is an open ball contained in C and τ (y ) is the end of B . It follows that τ (y ) is contained in the closure of C in X an . Since C is a connected component of X an \ ϕ −1 (V ), its closure is contained in ) is open and closed in X an \ ϕ −1 (V ), so B is a connected component of X an \ ϕ −1 (V ) and hence B = C . Now, suppose that τ (C ) ⊂ C . Then, τ (C ) ⊂ C ∩ , so C ⊂ τ −1 (C ∩ ). Let y ∈ τ −1 (C ∩ ), and let B be the connected component of X an \ containing y , as above. Since B = B ∪ {τ (y )} is a connected subset of X an \ ϕ −1 (V ) intersecting C , we have B ⊂ C , so y ∈ C and therefore C = τ −1 (C ∩ ).
Proof. Let V = V (X), and V = V (X ). If there is an extension X → X of ϕ, then the square (5.18.1) commutes. Let x ∈ ϕ −1 (V ), so x = ϕ(x ) reduces to a generic point ζ of X k . Since the reduction ζ of x maps to ζ , the point ζ is generic, so x ∈ V . Therefore, The morphism X → X is finite if and only if every generic point of X k maps to a generic point of X k ; as above, this is equivalent to V = ϕ −1 (V ).
It remains to prove that if ϕ −1 (V ) ⊂ V , then ϕ extends to a morphism X → X. By Proposition 5.12, we must show that ϕ is continuous with respect to the topologies T (X) and T (X ). Let U ⊂ X an be T (X) open, and let U = ϕ −1 (U). Clearly U is closed, so U is closed (with respect to the ordinary topologies). We must show that condition (2) of Lemma 5.17 holds for U . Let x ∈ U ∩ V , and let x = ϕ(x ). If x / ∈ V , then let C ⊂ U be the connected component of X an \ V containing x and let C be the connected component of X an \ ϕ −1 (V ) containing x . Then, C ⊂ U because ϕ(C ) ⊂ C, and C contains every connected component of X an \ V adjacent to x since C is an open neighborhood of x . Now suppose that x ∈ V . Any connected component of X an \ V which is adjacent to x maps into a connected component of X an \ V which is adjacent to x. There are finitely many connected components X an \ V which are adjacent to x and not contained in U by Lemma 5.17. Let C be such a component. Since ϕ is finite, there are only finitely many connected components of ϕ −1 (C); each of these is a connected component of X an \ ϕ −1 (V ). If C is such a connected component, then either C is an open ball connected component of X an \ V or C intersects = (X , V ) by Lemma 5.18. There are finitely many connected components of X an \ V which intersect -these are just the open annulus connected components of X an \ V -so there are only finitely many connected components of X an \ V contained in ϕ −1 (C). Therefore, all but finitely many connected components of X an \V which are adjacent to x map to connected components of X an \V which are contained in U.

Simultaneous semistable reduction theorems
Recall that K 0 is a field equipped with a nontrivial non-Archimedean valuation val : K 0 → R∪{∞}. As above, we fix an algebraic closure K 0 of K 0 and a valuation val on K 0 extending the given valuation on K 0 ; for any field K 1 ⊂ K 0 , we consider K 1 as a valued field with respect to the restriction of val, and we write R 1 for the valuation ring of K 1 . In what follows, X and X are smooth, proper, geometrically connected K 0 -curves and ϕ : X → X is a finite morphism. Proposition 5.20. Let X be a semistable R 0 -model of X. If there exists a semistable R 0 -model X of X such that ϕ : X → X extends to a finite morphism X → X , then there is exactly one such model X up to (unique) isomorphism.
Proof. Suppose first that K = K 0 is complete and algebraically closed. Let X (resp. X ) be the -adic completion of X (resp. X ). In this case, the proposition follows from Theorem 5.8 and the fact that if the morphism X → X of Theorem 5.13 is finite, then V (X ) is uniquely determined by V (X). The general case follows from this case after passing to the completion of the algebraic closure K of K 0 : if X , X are two semistable models of X such that ϕ extends to finite morphisms X → X and X → X , then the isomorphism X R ∼ −→ X R descends to an isomorphism X ∼ −→ X by Lemma 5.5.
Let X 1 , X 2 be semistable R 0 -models of X. Recall that X 1 dominates X 2 if there exists a (necessarily unique) morphism X 1 → X 2 inducing the identity on X.
Proposition 5.21. Let X and X be semistable R 0 -models of X and X , respectively. Let D ⊂ X (K 0 ) be a finite set of points. Then there exists a finite, separable extension K 1 of K 0 and a semistable R 1 -model X of (X K 1 , D ) such that: 1. X dominates X R 1 , 2. for any valued field extension K 1 → K 1 , the morphism ϕ K 1 : where R 1 is the ring of integers in K 1 , and 3. any other semistable R 1 -model of (X K 1 , D ) satisfying the above two properties dominates X .
Proof. Suppose first that K 0 = K is complete and algebraically closed. Let X and X be the -adic completions of X and X , respectively. Let V = V (X) and V = V (X ). By Lemma 3.15, there is a minimal semistable vertex set V of (X , D ) which contains ϕ −1 (V ) ∪ V . Let X be the semistable formal model of X corresponding to V . Then, X dominates X by Corollary 5.15 and ϕ extends to a morphism X → X by Theorem 5.13. Part 3 follows from Corollary 5.15 and the minimality of V . Taking X to be the algebraization of X yields (1) to (3) in this case.
For a general valued field K 0 , suppose that K is the completion of K 0 . By Lemma 5.4, the model of X K constructed above descends to a model X defined over the ring of integers of a finite, separable extension K 1 of K 0 . Properties (1)-(3) follow from Lemma 5.5 and the corresponding properties of X R . Now, we address the behavior of this construction with respect to base change. Using Lemma 5.5, we immediately reduce to the case of an extension K → K of complete and algebraically closed valued fields. Let R be the ring of integers of K . Let X (resp. X ) be the minimal R-model of X dominating X (resp. R -model of X K dominating X R ) mapping to X (resp. X R ). By (3) as applied to X , we have that X R dominates X . By Corollary 5.16, X is defined over R, so by (3) as applied to X , we have X R = X . Theorem 5.22 (Liu). Let X (resp. X ) be a semistable R 0 -model of X (resp. X ). Let D ⊂ X(K 0 ) and D ⊂ X (K 0 ) be finite sets, and suppose that ϕ(D ) ⊂ D. Then, there exists a finite, separable extension K 1 of K 0 and semistable R 1 -models X 1 , X 1 of (X K 1 , D), (X K 1 , D ), respectively, such that 1. X 1 dominates X R 1 and X 1 dominates X R 1 , 2. the morphism ϕ K 1 : X K 1 → X K 1 extends to a finite morphism X 1 → X 1 , and 3. if X 2 , X 2 are semistable formal models of (X K 1 , D), (X K 1 , D ), respectively, satisfying (1) and (2) above, then X 2 dominates X 1 and X 2 dominates X 1 .
Moreover, the formation of X 1 → X 1 commutes with arbitrary valued field extensions The morphism X 1 → X 1 is called the stable marked hull of X X in [34].
Proof. First, assume that K 0 = K is complete and algebraically closed; let X, X be the -adic completions of X , X , respectively. Let V = V (X) and V = V (X ). By Theorem 5.8, Theorem 5.13, and Corollary 5.15, we may equivalently formulate the existence and uniqueness of X 1 → X 1 in terms of semistable vertex sets, as follows. We must prove that there exists a semistable vertex set V 1 of (X, D) such that is a semistable vertex set of (X , D ), and 3. V 1 is minimal in the sense that if V 2 is another semistable vertex set of (X, D) satisfying (1) and (2) above, then V 2 ⊃ V 1 .
First, we will prove the existence of V 1 satisfying (1) and (2). By Lemma 3.15, we may enlarge V to assume that V is a semistable vertex set of (X, D). Let = (X, V ∪ D) and = (X , V ). By Corollary 4.18, there exists a skeleton 1 of (X, D) such that 1 ⊃ ∪ ϕ( ) and such that 1 = ϕ −1 ( 1 ) is a skeleton of (X , ϕ −1 (D)). Let V 1 be a vertex set for 1 . Since any finite subset of type-2 points of 1 which contains V 1 is again a vertex set for 1 by Proposition 3.12(4), we may and do assume that V ⊂ V 1 . Let V 1 ⊂ 1 be the union of a vertex set for 1 with V ∪ ϕ(V 1 ). Then, V 1 is a semistable vertex set of (X, D), and ϕ −1 (V 1 ) ⊂ 1 is a finite set of type-2 points containing V 1 , thus is a semistable vertex set of (X , ϕ −1 (D)) (hence of (X , D ) as well).
To prove that there exists a minimal such V 1 , we make the following recursive construction. Let V (0) = V , let V (0) = V , and for each n ≥ 1 let V (n) be the minimal semistable vertex set of (X, D) containing V (n − 1) ∪ ϕ(V (n − 1)), and let V (n) be the minimal semistable vertex set of (X , D ) containing ϕ −1 (V (n)). These sets exist by Lemma 3.15. By induction, it is clear that if V 2 is any semistable vertex set of (X, D) satisfying (1) and (2) above, then V (n) ⊂ V 2 for each n. Since V 2 is a finite set, for some n we have V (n) = V (n + 1), which is to say that ϕ(V (n)) ⊂ V (n); since V (n) ⊃ ϕ −1 (V (n)), we have that V (n) = ϕ −1 (V (n)) is a semistable vertex set of (X , D ). Then, V 1 = V (n) is the minimal semistable vertex set satisfying (1) and (2) above.
The case of a general ground field reduces to the geometric case handled above exactly as in the proof of Proposition 5.21, as does the statement about the behavior of the stable marked hull with respect to valued field extensions.
Remark 5.23 (The skeletal viewpoint on Liu's theorem). The statement of Theorem 5.22 is strongly analogous to Corollary 4.18, which is indeed the main ingredient in the proof. The difference is that whereas finite morphisms of semistable models correspond to pairs V , V of semistable vertex sets such that ϕ −1 (V ) = V , for finite morphisms of triangulated punctured curves one requires in addition that ϕ −1 ( ) = .
The former condition (of Liu's theorem) does not imply the latter: for instance, let X be the Tate curve y 2 = x 3 + x 2 + , let X = P 1 , and let ϕ : X → X be the cover (x, y) → x. This extends to a finite morphism of semistable models given by the same equations; however, the associated skeleton of X (resp. of P 1 ) is a circle (resp. a point), so ϕ −1 ( ) . In general, if ϕ −1 (V ) = V and ϕ −1 (D) = D , then by Remark 4.19 the image of the skeleton (X , V ∪ D ) is equal to (X, V ∪ D) union a finite number of geodesic segments T 1 , . . . , T r , and by Theorem 4.13 the saturation ϕ −1 (ϕ( (X , V ∪ D )) is a skeleton of X .
Theorem 5.25 below follows the same philosophy in deriving a simultaneous semistable reduction theorem of Liu-Lorenzini from Proposition 4.12.
Remark 5.24. Liu in fact works over an arbitrary Dedekind scheme (a connected Noetherian regular scheme of dimension 1), which includes discrete valuation rings but not more general valuation rings. In his statement of Theorem 5.22, the given models X , X are allowed to be any integral, projective Spec(R 0 ) schemes with generic fibers X and X , respectively. Although we restrict to semistable models X , X in this paper, using a more general notion of a triangulation our methods can be extended to treat relatively normal models X , X .
The following simultaneous stable reduction theorem can be found in [32]. Theorem 5.25 is to Proposition 4.12 as Theorem 5.22 is to Corollary 4.18 Theorem 5.25 (Liu-Lorenzini). Suppose that (X, D) and (X , D ) are both stable and that ϕ −1 (D) = D . Assume that (X, D) and (X , D ) admit stable models X and X , respectively, defined over R 0 . Then, ϕ : X → X extends to a (not necessarily finite) morphism X → X .
Proof. By Lemma 5.5, we may assume that K 0 = K is complete and algebraically closed. Let V (resp. V ) be the minimal semistable vertex set of (X, D) (resp. (X , D )). By Theorem 5.13, we must show that ϕ −1 (V ) ⊂ V . Let = (X, V ) and = (X , V ).
Let V 1 be a vertex set for with respect to which has no loop edges. Recall that V = x ∈ V 1 : g(x) ≥ 1 orx has valency at least 3 by Proposition 3.18 (2).
this is a vertex set for since ϕ −1 ( ) ⊂ by Proposition 4.12 (2). If x ∈ V has genus at least 1, then any x ∈ ϕ −1 (x) has genus at least 1; so, ϕ −1 (x) ⊂ V . Let x ∈ V have genus 0, so x has valency at least 3 in . Let x. By the above, the same is true of x , so x has valency at least 3 in , so x ∈ V , as desired.

A local lifting theorem
We now begin with the classification of lifts of harmonic morphisms of metrized complexes of curves to finite morphisms of algebraic curves. We begin by working in the neighborhood of a vertex of a metrized complex.
Let X be a smooth, proper, connected curve over K, and let x ∈ X an be a type-2 point. Let V ⊂ X an be a strongly semistable vertex set of X containing x, let = (X, V ), and let τ : X an → be the retraction. Let e 1 , . . . , e r be the edges of adjacent to x and let  Conversely, let X be a smooth, proper, connected formal curve over Spf(R). If x ∈ X an is the point reducing to the generic point of X k , then (X K , x) is a proper star-shaped curve. 1. When Y is proper, so Y ∼ = X K for a smooth, proper, connected formal curve over Spf(R); then, C y ∼ = X k and the reduction map Y → C y coincides with the canonical reduction map X K → X k . 2. Let D ⊂ Y (K) be a compatible divisor, and let 0 = (Y , {y} ∪ D). Every edge e of 0 is contained in a unique connected component of Y \ {y}, and red(e) ∈ C y (k) is the closed point corresponding to the tangent direction represented by e.

Tame coverings
We now study a class of morphisms of star-shaped curves analogous to (4.30). We begin with the following technical result. where α ∈ R × and |g| sup < 1. Since δ is not divisible by char(k) if char(k) > 0, the Taylor expansion for δ √ 1 + g has coefficients contained in R, hence converges to a δth root of 1 + g(u) on A. Choosing a δth root of α as well and letting u B = u δ √ α δ √ 1 + g, we have that u B is a parameter on B such that ϕ * (t) = u δ B . Hence, for each such B, there are exactly δ choices of a parameter u B on B such that ϕ * (t) = u δ B ; choosing a compatible set of such parameters for all B yields a parameter on A satisfying (1).
Part (2) follows immediately from (1). A morphism from a K-analytic space X to B(1) + is given by a unique analytic function f on X such that |f (x)| < 1 for all x ∈ X, so ( Proof. In the situation of (1), let x = red(B). Since ϕ y is not branched over x, there are δ distinct points of C y mapping to x; hence, there are δ connected components B 1 , . . . , B δ of Y \ {y } mapping onto B. The restriction of ϕ to each B i is finite of degree 1.

A local lifting theorem
Let (Y , y, D) be a punctured star-shaped curve with skeleton 0 = (Y , {y} ∪ D) and residue curve C y . Let C be a smooth, proper, connected k-curve, and let ϕ : C → C y be a finite, tamely ramified morphism branched only over the points of C y corresponding to tangent directions at y represented by edges in 0 . A lifting of C' to a punctured starshaped curve over (Y,y,D) is the data of a punctured star-shaped curve (Y , y , D ), a tame covering ϕ : (Y , y , D ) → (Y , y, D), and an isomorphism of the residue curve C y with C which identifies ϕ with the morphism ϕ y : C y → C y induced by ϕ. An isomorphism between two liftings (Y , y ) and (Y , y ) is a Y -isomorphism Y → Y such that the induced morphism C y ∼ −→ C y respects the identifications C y ∼ = C and C y ∼ = C . Before giving the proof of this theorem, we need a technical lemma. Let X be a finitely presented, flat, separated R scheme, and let X be its -adic completion; this is homomorphism π t 1 (U)→ π t 1 (U k ), the generic fiber U is connected and ϕ K : U → U corresponds to the inverse image H≤ π t 1 (U) of H. Let X be the smooth compactification of U ; by abuse of notation, we also let ϕ K : X → X denote the morphism extending ϕ K : U → U. The ramification profile of ϕ K over a point x ∈ D is determined by the cycle decomposition of the action of the inertia element σ x on the coset space π t 1 (U)/H. Since π t 1 (U)/H ∼ −→ π t 1 (U k )/ H and σ x specializes to σ red(x) , it follows that the ramification profiles of ϕ K : X → X above x and of ϕ : C → X k = C y above red(x) are identical. Applying the Riemann-Hurwitz formula to ϕ K and to ϕ shows that the genus of X equals the genus of C .
Letting U and U denote the -adic completions of U and U , respectively, we have a Cartesian square by Lemma 6.19. In other words, U K is the inverse image of U K under ϕ K : U an → U an . Let y ∈ U K be the unique point reducing to the generic point of U k = U . By functoriality of the reduction map, this is the unique point of U K mapping to y since the map U k → U k is dominant. Therefore, ϕ −1 K (y) = {y }. Moreover, since the genus of the point y is equal to the genus of C (the smooth compactification of U k ), which coincides with the genus of X by the genus formula ( [12], (4.1)), we see that {y } is the minimal skeleton of X . In particular, X has a smooth formal model X with special fiber C = C y , and X → X extends to a finite morphism X → X lifting ϕ : C → C y . Let ϕ : X → X be the algebraization of this morphism. The restriction of ϕ to the inverse image of U is the unique étale morphism extending U → U k , so X → X is in fact a compactification of the morphism ϕ : U → U above.
Let D = ϕ −1 K (D) ⊂ X , and let Y = X an . In order to show that (Y , y , D ) is a punctured star-shaped curve, we must argue that D is a compatible divisor in Y , i.e. that distinct points of D reduce to distinct points of C y = C . Suppose not, so there are two points x 1 , x 2 ∈ D with the same reduction x ∈ D . Let x = red( x ) ∈ D k . Since distinct points of D have distinct reductions in D k , by functoriality of the reduction map, we have ϕ K (  Y , y, D), still in the case when Y is proper. Given such a lift, running parts of the above argument backward, one sees that Y ∼ = X an where X is the smooth compactification of the generic fiber of the unique étale extension U → U of U → U k . As all of these constructions are unique up to unique isomorphism, the same is true of Y . Now, suppose that (Y , y) is not proper. Let Y → X be a compactification (6) of Y ; we will identify Y with its image in X. Let D 1 be the union of D with a choice of one K-point from every connected component of X \ Y . Let ϕ : (X , y , D 1 ) → (X, y, D 1 ) be a lifting of C to a punctured star-shaped curve over (X, y, D 1 ), and let Then, (Y , y , ϕ −1 (D)) is a lifting of C to a punctured star-shaped curve over (Y , y, D).
Let ψ : Y → Y be an automorphism of Y as a lifting of C . Since ψ induces the identity map C y → C y on residue curves, ψ takes each connected component of Y \ {y } to itself. Recall that X \ Y consists of a disjoint union of closed balls around the points of where ζ ∈ R × is a δth root of unity. Therefore, ψ| A extends uniquely to a B-morphism ψ| B : B → B fixing 0. Gluing these morphisms together, we obtain an X-morphism ψ : X → X extending ψ : Y → Y . By construction, this is an automorphism of X as a lifting of C to X, which is thus the identity. Therefore, ψ : Y → Y is the identity.
Let ϕ : (Y , y , D ) → (Y , y, D) be another lifting of C to a punctured star-shaped curve over (Y , y, D). By Corollary 6.16, there exists a compactification Y → X of Y and an extension of ϕ to a tame covering ϕ : (X , y , ϕ −1 (D 1 )) → (X, y, D 1 ). This is another lifting of C to a punctured star-shaped curve over (X, y, D 1 ), so there is an isomorphism ψ : (X , y ) ∼ −→ (X , y ) of liftings. Since Y = ϕ −1 (Y ) and Y = ϕ −1 (Y ), the isomorphism ψ restricts to an isomorphism (Y , y ) ∼ −→ (Y , y ) of liftings. Corollary 6.21. With the notation in Theorem 6.18, let (Y , y , D ) → (Y , y, D) be a lifting of C to a punctured star-shaped curve over (Y , y, D). Then, the natural homomorphism y, D) is a second lifting of C to a punctured star-shaped curve over (Y , y, D), then the natural map is bijective.

Classification of liftings of harmonic morphisms of metrized complexes
Fix a triangulated punctured curve (X, V ∪ D) with skeleton = (X, V ∪ D), and let τ : X an → be the canonical retraction. Throughout this section, we assume that has no loop edges. Let ϕ : → be a tame covering of metrized complexes of curves. A lifting of to a tame covering of (X, V ∪ D) is a tame covering of triangulated punctured (4.30)) equipped with a -isomorphism ϕ −1 ( ) ∼ = of metrized complexes of curves. Since V = ϕ −1 (V ) and D = ϕ −1 (D), we will often denote a lifting simply by X . Let ϕ : X → X and ϕ : X → X be two liftings of , and let ψ : X ∼ −→ X be an X-isomorphism of curves. Then, ψ restricts to a -automorphism ψ| : We will consider liftings up to X-isomorphism preserving , i.e. such that ψ is the identity, and we will also consider liftings up to isomorphism as curves over X.
For every finite vertex x ∈ V , let (x) be the connected component of , as in Example 6.14. Then, (Y (x), x, D(x)) is a punctured star-shaped curve. By Proposition 2.22, for every finite vertex x ∈ V ( ) lying above x, the morphism ϕ x : C x → C x is finite, tamely ramified, and branched only over the points of C x corresponding to tangent directions at x represented by edges of (x).
) be the unique lifting of C x to a punctured star-shaped curve over (Y (x), x, D(x)) provided by Theorem 6.18, and let in such a way that for every edge e of (x ), the point red x (e ) ∈ C x (k) is identified with the pointred(e ) defined in (6). (6). .
Let E ± f ( ) denote the set of oriented finite edges of , and for e ∈ E ± f ( ) let e denote the same edge with the opposite orientation. Let G( , X) denote the set of tuples such that e = −1 e , where e = − → x y . We call G( , X) the set of gluing data for a lifting of to a tame covering of (X, V ∪ D), and we emphasize that G( , X) is nonempty.
Let α ∈ Aut ( ), so α is a degree-1 finite harmonic morphism ∼ −→ preserving → . Let x ∈ V f ( ), and let x = α(x ) and x = ϕ(x ) = ϕ(x ). Part of the data of of punctured star-shaped curves inducing the isomorphism C x ∼ −→ C x on residue curves. Let e ∈ E f ( ) be an edge adjacent to x , let e = α(e ), and let e = ϕ(e ) = ϕ(e ). Then, α x restricts to an isomorphism Define the conjugation action of Aut ( ) on G( , X) by the rule 1. There is a canonical bijection between the set of gluing data G( , X) and the set of liftings of to a tame covering of (X, V ∪ D), up to X-isomorphism preserving . In particular, there exists a lifting of . Any such lifting has no nontrivial automorphisms which preserve . Proof. Given ( e ) ∈ G( , X), one can glue the local lifts {Y (x )} x ∈V f ( ) via the isomorphisms Equation 7.2.1 to obtain an analytic space which one easily verifies is smooth and proper, hence arises as the analytification of an algebraic curve X . Moreover, the morphisms Y (x ) → Y (x) glue to give a morphism X an → X an , which is the analytification of a morphism ϕ : X → X. By construction, if V = ϕ −1 (V ) and D = ϕ −1 (D), then (X , V ∪ D ) → (X, V ∪ D) is a lifting of to a tame covering of (X, V ∪ D). Now, let ϕ : (X , V ∪ D ) → (X, V ∪ D) be any lifting of to a tame covering of (X, V ∪ D). As explained in Example 6.14, for every x ∈ V , the inverse image ϕ −1 (Y (x)) is a disjoint union of tame covers of the punctured star-shaped curve (Y (x), x, D(x)), one for each x ∈ ϕ −1 (x). By Theorem 6.18, we have canonical identification is the retraction. Hence, X an is obtained by pasting the local liftings Y (x ) via a choice of isomorphisms ( e ) ∈ G( , X).
Let ψ : X ∼ −→ X be an X-automorphism preserving . Since ψ is the identity on the set ⊂ X an , for x ∈ V , we have ψ(Y (x )) = Y (x ). We also have that for x ∈ V the map of residue curves ψ x : C x → C x is the identity, so by Corollary 6.21, ψ restricts to the identity morphism Y (x ) → Y (x ). Since X an = x ∈V Y (x ), we have that ψ is the identity. It follows from this that the tuple ( e ) can be recovered from the class of X modulo X-isomorphisms preserving , so different gluing data give rise to curves which are not equivalent under such isomorphisms. This proves (1). Let ϕ : (X , V ∪ D ) → (X, V ∪ D) and ϕ : (X , V ∪ D ) → (X, V ∪ D) be the liftings of associated to the tuples of gluing data e , e ∈ G , X , respectively. Suppose that there exists α ∈ Aut ( ) such that α· e = e . Then, for all e = − → (α(x )) glue to give an X-isomorphism α : X ∼ −→ X . Conversely, the restriction of an X-isomorphism α : X ∼ −→ X to is a -automorphism of . It is easy to see that these are inverse constructions. Taking X = X we have an injective homomorphism Aut X (X ) → Aut ( ); it follows formally from the above considerations that its image is the stabilizer of e . Remark 7.5. Let X be a lifting of to a tame covering of (X, V ∪ D). It follows from Theorem 7.4(1) that the natural homomorphism is injective. It is not in general surjective: see Example 7.8 (where all the stabilizer groups of elements of G( , X) are proper subgroups of Aut ( )). Remark 7.6. It is worth mentioning that the question of lifting (and classification of all possible liftings) in the wildly ramified case is more subtle and cannot be guaranteed in general. Some lifting results in the wildly ramified case are known: e.g., Liu proves in [33], Proposition 5.4, that any finite surjective generically étale admissible cover of proper semistable curves over an algebraically closed field k lifts to a finite morphism of smooth proper curves over K. However, the same statement for metrized complexes cannot be true in general: consider the metrized complex C consisting of a single finite vertex v and an infinite vertex u attached to v by an infinite edge e with C v ∼ = P 1 k and red v (e) = ∞. Consider the degree p generically étale morphism of metrized complexes C → C which restricts to a degree p Artin-Schreier cover ϕ : P 1 k → P 1 k , étale over A 1 k → A 1 k and ramified at ∞ (with ramification index p) and which is linear of slope p on the infinite edge e. This map cannot be lifted to a degree p cover P 1 K → P 1 K in characteristic zero; otherwise, we would obtain a (connected) étale degree p cover of A 1 K which is impossible since A 1 K is simply connected.
We now state a variant of Theorem 7.4 in which we allow loop edges. Proof. The only issue is that in Theorem 7.7, we merely require that V is a semistable vertex set whereas; whereas in Theorem 7.4, we require it to be strongly semistable. However, we can modify by inserting a (valence 2) vertex v along each loop edge, with C v ∼ = P 1 k , and with the marked points on C v being 0 and ∞. Call the resulting metrized complex , and let V 0 denote the set of vertices which have been added to the vertex set V for . We construct a new metrized complex from by adding V 0 : to the vertex set V for and letting C v ∼ = P 1 k for v ∈ V 0 , with the marked points on C v being 0 and ∞. We can extend the tame covering ϕ : → to a tame covering → by letting the map from C v to C ϕ(v ) , for v ∈ V 0 , be z → z d , where d is the local degree of ϕ along the loop edge corresponding to v . By Theorem 7.4, there is a tame corresponds to V 0 (resp. V 0 ). Removing the vertices in V 0 and V 0 gives a tame covering Example 7.8. In this example, we suppose that char(k) = 2. Let E be a Tate curve over K, let be its (set-theoretic) skeleton, and let τ : E an → be the canonical retraction. Let U : G an m /q Z ∼ −→ E an be the Tate uniformization of E an . The two-torsion subgroup of E is U ±1, ± √ q ; choose a square root of q, let y = τ (U(1)) and z = τ (U √ q ), and let V = y, z ⊂ . This is a semistable vertex set of E and = (E, V ) is the circle with circumference val(q); the points y and z are antipodal on . Let e 1 , e 2 be the edges of ; orient e 1 so that y is the source vertex and e 2 so that z is the source vertex. The residue curves C y and C z are both isomorphic to P 1 k ; fix isomorphisms C y ∼ = P 1 k and C z ∼ = P 1 k such that the tangent direction at a vertex in the direction of the outgoing (resp. incoming) edge corresponds to ∞ (resp. 0).
Let be a circle of circumference 1 2 val(q), let V = {y , z } be a pair of antipodal points on , and let e 1 = − → y z and e 2 = − → z y be the two edges of , with the indicated orientations. Enrich with the structure of a metrized complex of curves by setting C y = C z = P 1 k and letting the tangent direction at a vertex in the direction of the outgoing (resp. incoming) edge correspond to ∞ (resp. 0). Define a morphism ϕ : → of metric graphs by ϕ(y ) = y, ϕ z = z, ϕ e 1 = e 1 , ϕ e 2 = e 2 , with both degrees equal to 2. See Figure 3. Topologically, → is a homeomorphism. Let ϕ y : C y → C y and ϕ z : C z → C z both be the morphism t → t 2 : P 1 k → P 1 k . This makes ϕ into a degree-2 tame covering of metrized complexes of curves.
For x ∈ V , the analytic space Y (x) is an open annulus of logarithmic modulus val(q); fix an isomorphism Y (x) ∼ = S(q) + for each x. For x ∈ V , the analytic space Y (x ) is an open annulus of logarithmic modulus 1 2 val(q); fix isomorphisms Y (x ) ∼ = S √ q + such that the morphisms Y (x ) → Y (ϕ(x )) are given by t → t 2 : S √ q + → S(q) + . For e ∈ E( ), we have d ϕ (e ) = 2 by definition. Hence, G( , E) has four elements, which we label (±1, ±1). By Theorem 7.4(1), there are four corresponding classes of liftings of to a tame covering of (E, V ) up to isomorphism preserving . Any -automorphism of is the identity on the underlying topological space. Hence, an element of Aut ( ) is a pair of automorphisms (ψ y , ψ z ) ∈ Aut C y (C y ) × Aut C z (C z ) such that ψ y , ψ z fix the points 0, ∞ ∈ P 1 k . Thus, ψ y , ψ z = ±1, so Aut ( ) = {±1} × {±1}. For x = y , z , the automorphism −1 : C x ∼ −→ C x lifts to the automorphism −1 : Therefore, the conjugation action of Aut ( ) on G( , E) is given as follows: By Theorem 7.4(2), there are two isomorphism classes of lifts of to a tame covering of (E, V ), and each such lift has two automorphisms. These liftings can be described concretely as follows. Fix a square root of q, let E ± be the algebraization of the analytic elliptic curve G an m / ± √ q Z , and let ψ ± : E ± → E be the morphism t → t 2 on uniformizations. Let ± = ψ −1 ± ( ). Then, ± is isomorphic to as a tame covering of and E ± is a lifting of to a tame covering of (E, D). The elliptic curves E ± are not isomorphic (as K schemes) because they have different q invariants, so they represent the two isomorphism classes of liftings of . The nontrivial automorphism of E ± is given by translating by the image of −1 ∈ G an m (K) (this is not a homomorphism). In fact, since ψ ± : E ± → E is an étale Galois cover of degree 2, this is the only nontrivial automorphism of E ± as an E-curve; so the homomorphism is injective but not surjective (its image is {±(1, 1)}).
Consider now the automorphism group Aut 0 ( ) ⊂ Aut ( ) consisting of all degree-1 finite harmonic morphisms α : → respecting ϕ : → and inducing the identity on the metric graph underlying . The restriction of the conjugacy action of Aut ( ) on G( , X) to the subgroup Aut 0 ( ) admits a simplified description that we describe now. Combining this with arguments similar to those in the proof of Theorem 7.4 yields a classification of the set of liftings of up to isomorphism as liftings of the metric graph underlying ; see Theorem 7.10.
First, for each x ∈ V ( ) with image x = ϕ(x ), let Aut 0 C x (C x ) be the subgroup of Aut C x (C x ) that fixes every point of C x of the form red(e ) for some edge e of adjacent to x . Then, Denote by E 1 the finite abelian group (ϕ)).
The discussion preceding Theorem 7.4 shows that the set of gluing data G( , X) is canonically a principal homogeneous space under E 1 .
The subgroup Aut 0 C x (C x ) ⊂ Aut C x (C x ) corresponds to the subgroup Aut 0 Y (x) (Y (x )) of automorphisms in Aut Y (x) (Y (x )) which act trivially on the skeleton (x ). Restriction of a Y (x)-automorphism of Y (x ) to τ −1 x (e • ) defines a homomorphism ρ x ,e : Aut 0 d e (ϕ)).
Fix an orientation of each finite edge of . For x ∈ V f ( ) with image x = ϕ(x ), let ρ x : Aut 0 C x (C x ) → E 1 be the homomorphism whose e coordinate is 1. G( , X) is canonically a principal homogeneous space under E 1 , and the conjugacy action of Aut 0 ( ) on G( , X) is given by the action of the subgroup ρ Aut 0 ( ) = ρ(E 0 ) ⊆ E 1 on G( , X). 2. The set of liftings of up to isomorphism as liftings of the metric graph underlying is a principal homogeneous space under H 1 (E • ), and the group of automorphisms of a given lifting as a lifting of the metric graph underlying is isomorphic to H 0 (E • ).

Descent to a general ground field
Let K 0 be a subfield of K, let X 0 be a smooth, projective, geometrically connected K 0curve, and let D ⊂ X 0 (K 0 ) be a finite set. Let X = X 0 ⊗ K 0 K, let V be a strongly semistable vertex set of (X, D), let = (X, V ∪ D), and let ϕ : → be a tame covering of metrized complexes of curves, as in the statement of Theorem 7.4. Let ϕ : X → X be a lifting of ϕ to a tame covering of (X, V ∪ D). Whereas we take the data of the morphism → to be geometric, i.e. only defined over K, the covering X → X is in fact defined over a finite, separable extension of K 0 . This follows from the fact that if U = X \ D and U = X \ ϕ −1 (D), then ϕ : U → U is a tamely ramified cover of U over X relative to D (see Remarks 2.20(1) and 4.32(1)), along with the following lemma.
Lemma 7.12. Let K 0 be any field, let K be a separably closed field containing K 0 , let X 0 be a smooth, projective, geometrically connected K 0 -curve, and let D ⊂ X 0 (K 0 ) be a finite set. Let X = X 0 ⊗ K 0 K, let ϕ : X → X be a finite morphism with X smooth and (geometrically) connected, and suppose that ϕ is branched only over D, with all ramification degrees prime to the characteristic of K. Then, there exists a finite, separable extension K 1 of K and a morphism ϕ 1 : X 1 → X 0 ⊗ K 0 K 1 descending ϕ.
Proof. Let U 0 = X 0 \ D, and let U = X \ D and U = X \ ϕ −1 (D). First, suppose that K 0 is separably closed. By [40], Exposé XIII, Corollaire 2.12, the tamely ramified étale fundamental groups π t 1 (U 0 ) and π t 1 (U) are isomorphic (with respect to some choice of base point). Since ϕ : U → U is a tamely ramified cover of U over X relative to D, it is classified by a finite-index subgroup of π t 1 (U) = π t 1 (U 0 ), so there exists a tamely ramified cover ϕ 0 : U 0 → U 0 of U 0 over X 0 relative to D descending ϕ. Now, we drop the hypothesis that K 0 is separably closed. By the previous paragraph, we may assume that K is a separable closure of K 0 . By general principles, the projective morphism X → X descends to a subfield of K which is finitely generated (i.e. finite) over K 0 . Remark 7.13. With the notation in (7.11), suppose that K 0 is a complete valued field with value group 0 = val(K × 0 ) and algebraically closed residue field k; that is 'rational over K 0 ' in that it comes from a (split) semistable formal R 0 -model of X 0 in the sense of Section 5, and that has edge lengths contained in 0 . With some extra work, it is possible to carry out the gluing arguments of Theorem 7.7 directly over the field K 0 (in this context Lemma 6.10(1) still holds), which shows that the cover X → X is in fact defined over K 0 . In the case of a discrete valuation, this also follows from [47] or from [39] if D = ∅.

Liftings of tame harmonic morphisms
Theorem 7.7 implies the existing of liftings for (finite) tame harmonic morphisms of metrized complexes which are not necessarily tame coverings, the difference being generic étaleness. See Definition 2.21 for both definitions. Proof. For any finite vertex x of at which ϕ, seen as a morphism of augmented metric graphs, is ramified, let q 1 , . . . , q r ∈ C v be all the branch points of ϕ x which do not correspond to any edge of , and let p ij denote the preimages of q i under ϕ x . We modify and by attaching infinite edges e i to at x = ϕ(x ) for each q i , infinite edges e ij to at x for each p ij , and defining red x (e i ) = q i and red x (e ij ) = p ij . Since each ϕ x is tamely ramified, the harmonic morphism ϕ naturally extends to a tame covering ϕ : → between the resulting modifications. Enlarging D to D by choosing points in X(K) \ D with reduction q i , we can assume that = (X, V ∪ D). The result now follows from Remark 7.16. As mentioned above, in Proposition 7.15 we do not require that ϕ : → be generically étale. This corresponds to not requiring that D ⊂ X(K) contain the branch locus of the lift ψ : X → X: indeed, by Lemma 4.33, if D contains the branch locus then ϕ is a tame covering. In the situation of Proposition 7.15, the set of liftings of to a tame covering of (X, V ∪ D) can be infinite. For example, if = {p } and = {p} are both points and the morphism ϕ p : C p ∼ = P 1 → C p ∼ = P 1 is z → z 2 , with (X, V ) a minimal triangulation of P 1 and D = ∅, then there are infinitely many such lifts, corresponding to the different ways of lifting the critical points and critical values of ϕ p from k to K.

Application: component groups of Néron models
In contrast to the rest of the paper, we assume in this section that R is a complete discrete valuation ring with fraction field K and algebraically closed residue field k. In this case, we take the value group to be Z.
There is a natural notion of harmonic 1 forms on a metric graph of genus g (see [37]). The space 1 ( ) is a g-dimensional real vector space which can be canonically identified with H 1 ( , R), but we write elements of 1 ( ) as ω = e ω e de as in [7], where the sum is over all edges with respect to a fixed vertex set for . There is a canonical lattice 1 Z ( ) of integer harmonic 1-forms inside 1 ( ); these are the harmonic 1-forms for which every ω e is an integer. A harmonic morphism ϕ : → induces a natural pullback map on Recall that a Z-metric graph (i.e. a -metric graph for = Z) with no infinite vertices is a (compact and finite) metric graph whose edge lengths are all positive integers (or equivalently, having a vertex set with respect to which all edge lengths are 1). If X/K is a smooth, proper, geometrically connected analytic curve and X is a semistable R-model for X, the skeleton X of X is naturally a Z-metric graph. Moreover, as we have seen earlier in this paper, a finite morphism of semistable models induces in a natural way a harmonic morphism of Z-metric graphs.
Let be a Z-metric graph with no infinite vertices. We define the regularized Jacobian Jac reg ( ) of to be the group Jac(G), where G is the 'regular model' for (the unique graph induced by a vertex set with all edge lengths equal to 1). The group Jac(G) can be described explicitly as Jac(G) = 1 (G) # /H 1 (G, Z), where 1 (G) # denotes the linear functionals 1 (G, R) → R of the form α with α ∈ C 1 (G, Z) (see [7]). There is a canonical isomorphism between Jac(G) and Pic 0 (G), the group of divisors of degree 0 on G modulo the principal divisors (see [5]) as well as a canonical isomorphism where H 1 (G, Z) # = ω ∈ 1 (G, R) : γ ω ∈ Z ∀γ ∈ H 1 (G, Z) .
As an application of Theorem 8.4 and our results on lifting harmonic morphisms, one can construct many examples of harmonic morphisms of Z-metric graphs for which ϕ * is not surjective. For example, consider the following question posed by Ken Ribet in a 2007 email correspondence with the second author (Baker): Suppose f : X → X is a finite morphism of semistable curves over a complete discretely valued field K with g(X) ≥ 2. Assume that the special fiber of the minimal regular model of X consists of two projective lines intersecting transversely. Is the induced map f * : X → X on component groups of Néron models necessarily surjective?
We now show that the answer to Ribet's question is no. Example 8.6. Consider the 'banana graph' B( 1 , . . . , g+1 ) consisting of two vertices and g + 1 edges of length i . This is the reduction graph of a semistable curve whose reduction has two P 1 's crossing transversely at singular points of thickness 1 , . . . , g+1 . If we set G = B (1, 1, 1, 1) and G = B(1, 2, 2) and let , be the geometric realizations of G and G, respectively, then there is a degree 2 harmonic morphism of Z-metric graphs ϕ : → taking e 1 and e 2 to e 1 , e 3 to e 2 , and e 4 to e 3 . The homomorphism ϕ * is nonsurjective since |Jac reg ( )| = 4 and |Jac reg ( )| = 8. The map ϕ can be enriched to a harmonic morphismφ of metrized complexes of curves by attaching a P 1 to each vertex and letting each morphism P 1 → P 1 be z → z 2 , with 0, ±1, and ∞ being the marked points upstairs. By Remark 7.13, the morphismφ lifts to a morphism ψ : X → X of curves over K. Since all edges of G have length 1, the minimal proper regular model of X has a special fiber consisting of two P 1 's crossing transversely. By Theorem 8.4, the induced covariant map on component groups of Néron models is not surjective.
Endnotes a This isomorphism is not canonical. However, there is an isomorphism Aut A (A ) ∼ = μ δ which is canonical up to the action of Z/2Z on Aut A (A ) sending σ to σ −1 . Equivalently, the parameters t and t −1 give rise to different homomorphisms σ → σ (t ±1 )/t ±1 : Aut A (A ) → μ δ , but the choice of an orientation for (the skeleton of ) A induces a canonical isomorphism Aut A (A ) ∼ = μ δ . b This δ need not be prime to char(k). c We thank the referee for greatly simplifying our original proof of Lemma 6.19.