Étale covers with a prescribed exterior sheet

Proof. The spaces in question are locally path connected, and their connected components are open and closed. For the compact domain U this also follows from a finite affinoid cover and the finite component decompositions of affinoids. An open-and-closed restriction of a finite étale map is finite étale: on affinoids it is given by an idempotent summand of the finite algebra. Its rank is locally constant, hence constant and positive on the connected base D.

Choose a classical point of D. Its geometric fibre is a free transitive G-set. Every component meets that fibre, which proves transitivity on components. Two points of the fibre lying in the same component are related by an element preserving that component. Thus its fibre is a free transitive B-set, and its rank is |B|.□

Proof. Positive-genus paths. All the algebraic data descend to a finite extension of k. Choose a split semistable model of X with, for each i, two distinct smooth closed special-fibre points whose tubes lie inside the fixed V_i. To arrange this without altering V_i, insert a rational-radius type-two point strictly inside V_i as a vertex of a semistable refinement. Choose two smooth residue directions at this vertex away from the direction toward the exterior; their tubes lie in V_i. After a finite extension their centres are rational. The finite set of these type-two points can be inserted simultaneously, and the correspondence between vertex sets and models gives the required model [1, Proposition 3.13(3) and Theorem 4.11].

Apply the positive-genus resolution input at each of the two chosen closed points. The resulting positive-genus stable vertices map into their respective tubes, hence to two distinct type-two points x_{i1},x_{i2}\in V_i. Take a connected finite étale Galois cover Z_0\to X dominating the finitely many covers so obtained. Every point of Z_0^{\rm an} over every x_{ij} has positive residual genus. Indeed, some lift dominates the selected positive-genus point, and the associated inclusion of residue function fields prevents its genus from being zero by Lüroth’s theorem. Galois transitivity then gives the assertion for all lifts. These points are vertices of the stable skeleton \Gamma_0 of Z_0 [1, Theorem 4.22(2)].

For each i, choose a component above V_i. By Lemma 2.1, it contains lifts of both x_{i1} and x_{i2}. Join them by a path within that component. Its retracted image in \Gamma_0 is contained in the original path image: to reach a point of an attached open disk from either skeletal endpoint, the path must pass through that disk’s attachment point. The retracted path therefore still lies above V_i. Removing repetitions gives a simple edge path P_i of positive length, with distinct positive-genus endpoints, lying above V_i.

A residue torsor creates cycles. Fix a prime \ell\ne p. We seek a cyclic étale torsor of degree \ell on the special fibre which is connected on the normalized components at both ends of every P_i. Each endpoint will then have one vertex above it, while each edge has \ell lifts. This excess of edges forces a cycle; Figure 1 illustrates the one-edge case. The proof of [6, Proposition 2.6] also uses cyclic component covers, gluing, and deformation. Here we prescribe the endpoint characters simultaneously through the normalization sequence.

Let C be the geometric special fibre of the stable model of Z_0, and let \nu denote its normalization. For the constant étale sheaf the normalization sequence is

The last term is a skyscraper sheaf with vanishing H^1, and the finite normalization has no higher direct images of this sheaf. Consequently H^1(C,\mathbb F_\ell) surjects onto the product of the H^1 of its normalized components. Each smooth positive-genus component D admits a nonzero \mathbb F_\ell-character. Indeed, over the algebraically closed residue field, the Kummer sequence gives H^1(D,\mu_\ell)=\operatorname{Pic}(D)[\ell] =\operatorname{Jac}(D)[\ell]. This group is nonzero by [5, Theorem 8.2 and Remark 8.4], and a choice \mu_\ell\simeq\mathbb F_\ell identifies it with the character group in question. Prescribe such a character at each required endpoint component and lift this tuple to H^1(C,\mathbb F_\ell). The resulting torsor is connected: its character is nonzero, and \ell is prime.

After finite constant extension this torsor is defined on the special fibre over the residue field. Finite étale covers of a proper scheme over a complete Noetherian local ring are equivalent to those of its special fibre [3, Exposé IX, Théorème 1.10]. Full faithfulness also lifts the action and the torsor identity, giving a finite étale cover of the stable model. Its source is semistable and normal after every finite constant extension. Its generic fibre Z_1 is geometrically connected: otherwise, after some finite constant extension, a decomposition of that fibre would extend to open-and-closed components of the normal model. Each component meets the special fibre by properness, contradicting its geometric connectedness.

The source model is also stable. On normalized components its map is finite étale, so Riemann–Hurwitz forces any genus-zero source component to have degree one over a genus-zero component, retaining at least three node branches. A genus-one component lies over a genus-one component and retains at least one node branch. Thus no stabilization can contract the cycles that we now find in its skeleton.

If P_i has m\ge1 edges, its inverse graph has E=m\ell edges and V\le(m-1)\ell+2 vertices. Every node has \ell split lifts; each endpoint has just one vertex above it by the connectedness imposed on its normalization. Write c\ge1 for the number of connected components of this inverse graph. Then

It therefore contains an embedded circle, entirely above V_i.

Galois refinement preserves the cycles. Deck transformations of Z_0/X extend to its stable model. Take the product over that model of all conjugates of the cyclic cover. After a finite constant extension, choose a geometric connected component of its generic fibre and its open-and-closed closure in the normal product model. On geometric generic fibres this corresponds to the intersection of the conjugate normal index-\ell subgroups of \pi_1(Z_0), which is normal in \pi_1(X). The resulting cover Z\to X is therefore Galois. The chosen component is finite étale and surjective over the model of Z_1.

Above a circle with q edges and q vertices, a degree-e finite étale model cover has eq edges and at most eq vertices. Its nonempty inverse graph consequently satisfies b_1=E-V+c\ge1. The source model is stable by the preceding argument, so its skeleton contains an embedded circle above each V_i, as required.□

Proof. Choose an embedded circle C_i\subset\Gamma lying above V_i, and select one of its edges. We will construct a cyclic graph cover by gluing sheets only across a cut in this edge. To make its pullback split over all of U_Z, the cut must avoid not just U_Z but its image under the retraction.

Choose a point \eta_i strictly inside the selected edge at an irrational position for the normalized valuation. Vertex positions and edge lengths are rational, so such a point exists. It is of type three and satisfies

Indeed side disks attach only at type-two points, whose coordinates on this edge have valuation in \mathbb Q. Since \eta_i lies above V_i, its singleton retraction fibre misses U_Z. Thus \tau(U_Z) avoids \eta_i.

Orient the selected edge and cut it open at \eta_i, giving two new endpoints \eta_i^- and \eta_i^+ on its incoming and outgoing halves. Take p copies of the cut graph, labelled by a\in\mathbb F_p, and identify \eta_i^- in copy a with \eta_i^+ in copy a+1. This gives a cyclic graph cover of degree p. Away from \eta_i it is the disjoint union of the p labelled copies of \Gamma\setminus\{\eta_i\}. Traversing C_i in the selected orientation sends the sheet label a to a+1, so the cover of C_i is connected.

Pull this graph cover back by \tau. It splits over the entire U_Z because \tau(U_Z) misses the cut. On the component above V_i containing C_i, the loop C_i permutes all p sheets transitively. The restriction to that component, and hence the pulled-back cover itself, is connected.

It remains to realize this topological cover algebraically. A finite topological covering of a Berkovich curve obtains an analytic structure by pulling back the structure sheaf on its locally trivial sheets [2, Lemma 2.6]. The local product algebras glue by their sheet permutations, giving a coherent finite étale analytic algebra. Proper GAGA algebraizes this algebra, its unit and multiplication, with their identities preserved by full faithfulness [7, Appendix A, Theorem A.1(ii)]; étaleness is detected after analytification [9, Fact 5.1.5]. The deck action also algebraizes. Invariance under the algebraically closed extension \bar k\subset\Omega descends this cover and its action to \bar k [3, Exposé X, Corollary 1.8]. The resulting cyclic finite étale cover has the two restriction properties just established.□

Proof of Theorem 1.1. Take the covers of Lemma 3.1 for all i, their conjugates under \Gal(Z/X), and a connected component of their fibre product over Z. Call the resulting curve T. The intersection of their kernels in \pi_1(Z) is normal in \pi_1(X), so T\to X is connected, finite étale and Galois. Put

The group P is elementary abelian of exponent p, since it embeds in the product of the cyclic deck groups. All conjugate covers split over U_Z: each deck transformation of Z/X preserves U_Z. Their composite P-cover therefore splits over U_Z as well.

Let H\subset G be the stabilizer of a connected component of the inverse image of U in T^{\rm an}. Splitting over U_Z gives

Indeed P permutes freely the trivial sheets above each component of U_Z.

For a fixed i, choose a component W above V_i in Z on which the cyclic cover from Lemma 3.1 is connected. Let A be a component above W in T, and let B\subset G be its stabilizer. The stabilizer for T\to Z is B\cap P, so the same fibre calculation as in Lemma 2.1 gives \deg(A/W)=|B\cap P|. The map A\to W factors through the connected restriction of the chosen cyclic cover. The map to that restriction is surjective: it is finite étale, and its nonzero locally constant rank is positive everywhere on the connected target. Consequently p divides |B\cap P|. Lemma 2.1 gives transitivity of G on all components over V_i, and P is normal in G. Consequently

Set Y=T/H. The two quotients fit into the commutative square

The curve Y is smooth, proper and connected, and the induced map g:Y\to X is finite étale. We use one component calculation for both required assertions. Let D be U or one of the V_i, and let A be a component of its inverse image in T, with stabilizer B\subset G. The inverse image in T of the image of A in Y is the union of its H-translates. This union is open and closed in the inverse image of D, so that image is a connected component of g^{-1}(D). An H-translate of A is either A or disjoint from it; hence the image is A/(B\cap H). By Lemma 2.1, its degree over D is

For the chosen exterior component, B=H, so its image has degree one over U. A finite étale map of degree one is an isomorphism, which proves assertion (i).

Now take any component over V_i in Y, and choose a component above it in T with stabilizer B. Put N=B\cap P. By (4.2), N is a nontrivial normal p-subgroup of B. By (4.1), N\cap(B\cap H)=1. Hence B\cap H injects into B/N, and

is divisible by p. This proves assertion (ii) for every component above every fixed disk and completes the proof.□