Mustafin varieties, moduli spaces and tropical geometry

Mustafin varieties are flat degenerations of projective spaces, induced by a choice of an $n-$tuple of lattices in a vector space over a non-archimedean field. They were introduced by Mustafin in the 70s in order to generalise Mumford's groundbreaking work on the unformisation of curves to higher dimension. These varieties have a rich combinatorial structure as can be seen in pioneering work of Cartwright, H\"abich, Sturmfels and Werner. In this paper, we introduce a new approach to Mustafin varieties in terms of images of rational maps, which were studied by Li. Applying tropical intersection theory and tropical convex hull computations, we use this method to give a new combinatorial description of the irreducible components of the special fibers of Mustafin varieties. This enables connections to various topics. In particular, we see that any multiview variety appears as an irreducible component of the special fiber of some Mustafin variety. Furthermore, we use an interpretation of Mustafin varieties as a moduli functor introduced by Faltings to relate them to certain moduli functors, called linked Grassmannians. These objects are featured in limit linear series theory. The focal point of study regarding linked Grassmannians are so-called \textit{simple points}. As a direct consequence of the new combinatorial description of Mustafin varieties, we prove that the simple points of linked Grassmannians are dense in every fiber. Finally, we use the connection to linked Grassmannians, to relate the special fibers of Mustafin varieties to certain local models of unitary Shimura varieties.

Mustafin varieties are flat degenerations of projective spaces induced by choosing an n-tuple of lattices in the Bruhat-Tits building B d associated to PGL(V ) over a non-archimedean field K. These objects were introduced by Mustafin in [Mus78] in order to generalise Mumford's groundbreaking work on uniformisation of curves to higher dimensions [Mum72]. Since then they have been repeatedly studied under the name Deligne schemes (see e.g. [Fal01], [KT06], [CS10]). By studying degenerations of projective spaces, we give a framework for the study of degeneations of projective subvarieties. In his original work, Mustafin studied the case of so-called convex point configurations in B d as defined in Definition 2.14. An approach to study arbitrary point configurations was developed in [CHSW11], where the total space of this type of degenerations was named Mustafin variety for the first time. There it was proved that if the lattices in the point configuration have diagonal form with respect to a common basis (i.e. they lie in the same apartment), the corresponding Mustafin variety is essentially a toric degeneration given by mixed subdivisions of a scaled simplex. These mixed subdivision are beautiful combinatorial objects that are known to be equivalent to tropical polytopes and triangulations of products of simplices. For point configurations that do not obey this property some first structural results were proved. In this paper, we give a new combinatorial description of the special fibers of Mustafin varieties, which yields a complete classification of the irreducible components of special fibers of Mustafin varieties. Moreover, we uncover a connection between Mustafin varieties and so-called (pre)linked Grassmannians -objects that show up in the theory of limit linear series. Using this connection, we relate Mustafin varieties to the standard local model of Shimura varieties (as studied e.g. in [Gör01]).
Let R be a discrete valuation ring, K the quotient field and k the residue field. We fix a uniformiser π. As an example take K = C((π)) as the ring of formal Laurent series over C with discrete valuation v( n≥l a n π n ) = l for l ∈ Z and a n ∈ C with a l = 0. Then R = { n≥l a n π n : k ∈ Z ≥0 } and k = C. Moreover, let V be vector space of dimension d over K. We define P(V ) = ProjSym(V * ) as paramatrising lines through V . We call free R−modules L ⊂ V of rank d lattices and define P(L) = ProjSym(L * ), where L * = Hom R (L, R). Note, that we will only consider lattices up to homothety, i.e. L L if L = c · L for some c ∈ K × . Definition 1.1 (Mustafin varieties). Let Γ = {L 1 , . . . , L n } be a set of rank d lattices in V . Then P(L 1 ), . . . , P(L n ) are projective spaces over R whose generic fibers are canonically isomorphic to P(V ) P d−1 K . The open immersions P(V ) → P(L i ) give rise to a map P(V ) −→ P(L 1 ) × R · · · × R P(L n ). We denote the closure of the image endowed with the reduced scheme structure by M(Γ). We call M(Γ) the associated Mustafin variety. Its special fiber M(Γ) k is a scheme over k.
While the generic fiber of such a scheme is isomorphic to P d−1 , the special fiber has many interesting properties. The main tool in this paper is the study of closures images of rational maps of the form f : P(W ) P W W 1 × · · · × P W W n , where W is a vector space over k of dimension d and (W i ) i∈[n] is a tuple of sub-vector spaces W i ⊂ W , such that W i = 0 [Li17]. We denote the closure of the above map by X(W, W 1 , . . . , W n ).
We proceed as follows: Let Γ be a point configuration in the Bruhat-Tits building B d associated to PGL(V ). We consider the convex hull conv(Γ) (see Definition 2.14), which is a set of lattices. To each lattice class [L] ∈ conv(Γ), we associate a variety X Γ, [L] of the form X(k d , W 1 , . . . , W n ) for some W i depending on [ where V (Γ) is the set of polyhedral vertices of conv(Γ). The following is one of the main results of this paper.
Theorem 1.2. The irreducible components of Mustafin varieties are related to images of rational maps as follows: (1) If Γ is a an arbitrary point configuration, we have M(Γ) k = M(Γ).
(2) If Γ is a point configuration in one apartment, we have In each case, it is easy to see that the right hand side is contained in the special fiber of the Mustafin variety. For the other direction, we have to identify those lattice points that actually contribute an irreducible component. This is done by means of tropical intersection theory and multidegrees. Using this description we get a complete classification of the irreducible components of special fibers of Mustafin varieties as each variety of the form X(k d , W 1 , . . . , W n ) occurs as an irreducible component. Thus, we obtain the following application of Theorem 1.2: Theorem 1.3. The varieties X(k d ; W 1 , . . . , W d ) classify all irreducible components of special fibers of Mustafin varieties, i.e.
(1) Any irreducible component of the special fibers of a Mustafin varieties is a variety of the form X(k d ; W 1 , . . . , W d ), and (2) every variety X(k d ; W 1 , . . . , W n ) appears as an irreducible component of M(Γ) k for some Γ.
As mentioned before, we show that Mustafin varieties are closely related to so-called (pre)linked Grassmannians. Linked Grassmannians were introduced in [Oss06] in the context of limit linear series. Osserman introduced a new theory of limit linear series that in a certain sense compactifies the Eisenbud-Harris limit linear series theory. A generalisation of this notion was introduced in [Oss14], as so-called prelinked Grassmannians. (Pre)linked Grassmannians are degenerations of Grassmannians induced by a graph with additional data at the vertices at edges. If this graph is just a path, we call these objects linked Grassmannians and they were proved to be flat with reduced fibers in [HO08]. The focal objects in studying (pre)linked Grassmannians are so-called simple points. For a convex point configuration Γ we associate a linked Grassmannian as a scheme LG(1, Γ) over R. In [Fal01], Faltings introduced a moduli functor for Mustafin varieties. Using this result, we can interpret Mustafin varieties as the moduli space for the linked Grassmannian problem. Furthermore, this proves that the class of linked Grassmannians induced by the data Γ is flat with reduced fibers. Moreover, using a connection between Mustafin varieties and the simple points of a linked Grassmannian, the following theorem is a direct application of Theorem 1.2.
Theorem 1.4. The locus of simple points of LG(1, Γ) is dense in every fiber over R.
Finally, we use our moduli space interpretation of Mustafin varieties to give a connection between Mustafin varieties and so-called local models of Shimura varieties associated to an (EL)-datum as studied in [Gör01]. Shimura varieties can be thought of as modular curves in higher dimension and are of importance in Langlands program. This paper is structured as follows: In section 2 we give a quick review of the tools needed to prove our theorems. In particular, we will give a summary for the notion of tropical convexity and the relation between Bruhat-Tits Buildings and tropical convexity. We state that for a fixed point configuration Γ in one apartment, there is a bijection between the vertices in the convex hull of Γ and the lattice points in the tropical hull of certain points in the tropical torus associated to Γ. We summarize some of the structural results for Mustafin varieties and introduce the necessary basis for (pre)linked Grassmannians in 2.5. We finish the Preliminaries with a brief introduction to local models of Shimura varieties. Section 3 consists of the proof of Theorem 1.2 and Theorem 1.3. In subsection 3.1, we construct the varieties M(Γ) and M r (Γ). We prove Theorem 1.2 for the special case of |Γ| = 2 in subsection 3.2. Moreover, we prove Theorem 1.2 (1) in subsection 3.3, Theorem 1.2 (2) in subsection 3.4 and Theorem 1.3 in subsection 3.5. In section 4, we connect Mustafin varieties to prelinked Grassmannians by means of a moduli functor introduced by Faltings and prove Theorem 4.4. Moreover, we give an explicit description of the locus of simple points of the linked Grassmannians in terms of the special fibers of Mustafin varieties and prove Theorem 1.4, Finally in subsection 5 we interpret Mustafin varieties as a global realisation of the standard local model for Shimura varieties associtaed to an (EL)-datum.
Acknowledgements. We thank Hannah Markwig for her guidance and proof-reading throughout the preparation of this paper. Furthermore, we would like to thank Sarah Brodsky, Michael Joswig, Brian Osserman, Johannes Rau, Frank-Olaf Schreyer, Kristin Shaw, Bernd Sturmfels and Annette Werner for many interesting discussions, comments and suggestions concerning this topic. The first author acknowledges partial support by the DFG collaborative research center TRR 195, project A11 (INST 248/235-1).

Preliminaries
2.1. Tropical Geometry. In this subsection, we recall some basics of tropical geometry required for this paper. Our main combinatorial tool in this paper is the notion of tropical convexity. We restrict ourselves to basic notions and results and refer to [MS15] Chapter 5.2 for a more detailed introduction. Our proof of Theorem 1.2 involves the identification of certain lattice points in so-called tropical convex hulls. We achieve this by means of tropical intersection theory of tropical linear spaces (i.e. tropical varieties of degree 1). Tropical intersection theory is a well-developed theory, for more details see e.g. [AR10] or [MS15].
2.1.1. Tropical Convexity. In a sense tropical convexity is the notion of convexity over the tropical semiring (R, ⊕, ), where R = R ∪ {∞}, a ⊕ b = min(a, b) and a b = a + b. We make this more precise in the following definition: Definition 2.1. Let S be a subset of R n . We call S tropically convex, if for any choice x, y ∈ S and a, b ∈ R we get a x ⊕ b y ∈ S.
The tropical convex hull of a given subset V of R n is given as the intersection of all tropically convex sets in R n containing S. We denote the tropical convex hull of V by tconv(V ).
This definition implies that every tropical convex set S is closed under tropical scalar multiplication. Thus, if x ∈ S then so is x + λ1, where λ ∈ R and 1 = (1, . . . , 1). Therefore, we will usually identify S with its image in (n − 1)-dimension tropical torus R n R1 . We are interested in tropical convex hulls of a finite number of points. We begin by treating the case of two points.
Proposition 2.2 ( [DS04]). The tropical convex hull of two points x, y ∈ R n R1 is a concatenation of at most n − 1 ordinary line-segments. The direction of each line segment is a zero-one-vector.
The proof of this proposition is constructive and describes the points in the tropical convex hull explicitly. We will use this fact in section 3.2 to prove our statements in the case of a 2−point configuration. To give this explicit description for x = (x 1 , . . . , x n ) and y = (y 1 , . . . , y n ), we note that after relabelling and adding multiples of 1, we may assume 0 = y 1 − x 1 ≤ y 2 − x 2 ≤ · · · ≤ y n − x n . Then the tropical convex hull consists of the concatenation of the lines connecting the following points: (y n−1 − x n−1 ) x ⊕ y = (y 1 , y 2 , . . . , y n−1 , y n−1 − x n−1 + x n ) (y n − x n ) x ⊕ y = (y 1 , . . . , y n ) Some of these points might coincide, however they are always consecutive points on the line segment. Next, we introduce a useful description of tropical convex hulls in terms of bounded cells of a tropical hyperplane arrangement: Fix a set Γ = {v 1 , . . . , v n } of R n R1 , with v i = (v i1 , . . . , v in ). Consider the standard tropical hyperplane at v i in the max-plus algebra: Taking the common refinement, we obtain a polyhedral complex structure to R n R1 , i.e. a subdivision into convex polyhedra. The union of the bounded cells of this complex coincides with the tropical convex hull tconv(Γ) (see e.g. chapter 5.2 in [MS15]).
In the following example we compute the tropical convex hull of three points in the tropical torus.  Remark 2.4. The tropical convex hull of finitely many points is also called a tropical polytope. Tropical polytopes can be thought of as tropicalisations of polytopes over the field of real Puiseux series R{{t}} (see Proposition 2.1 in [DY07]). One can generalise this notion to arbitrary tropical polyhedra and polyhedra over R{{t}}, which in turn has applications in linear programming and complexity theory (see e.g. [ABGJ15]).
2.1.2. Tropical linear spaces. There are two notions of tropical linear spaces: An intrinsic purely combinatorial one given by matroid language and an extrinsic algebro-geometric one given by the image of a valuation map of a linear space over K. We will only talk about the extrinsic notion of so-called tropicalised linear spaces in this paper and refer to [MS15] for a more detailed discussion.
Definition 2.5. Let K be a field with discrete valuation v : K → Q ∪ {∞}. Then we obtain a map trop : K n → R n R1 given by the componentwise valuation map. For a linear space X ⊂ T n , we call the Euclidean topology closure trop(X) the tropicalisation of X. Moreover, trop(X) is a rational weighted balanced polyhedral complex, where all weights are 1. (see [MS15]) Example 2.6. Let L be the linear space in K 2 given by x + y = 1 = 0. The tropicalisation is the standard tropical line illustrated in Figure 2.  products as defined in [AR10]. However, it was proved in [Rau16] and [Kat12] that those two notions are equivalent. We start by defining stable intersection.
Example 2.8. We illustrate the difference between set-theoretic intersection and stable intersection in the example of two lines not in tropical general position. The two lines in Figure  3 intersect set-theoretically in the half-bounded line segment as illustrated in the upper right. However, the stable intersection only yields a single point as illustrated in the lower right.
In algebraic geometry, two general linear spaces of respective codimension m 1 and m 2 intersect in codimension m 1 + m 2 . A similar fact holds for tropical linear spaces. Theorem 2.10. [MS15] Let X 1 , X 2 be linear spaces contained in the n−dimensional torus T and let Σ 1 , Σ 2 be weighted balanced rational polyhedral complexes whose supports are trop(X 1 ) and trop(X 2 ) respectively. There exists a Zariski dense subset U ⊂ T , consisting of elements t = (t 1 , . . . , t n ) with val(t) = 0, such that Let H be the standard tropical hyperplane at a point v in R n R1 , then we denote the k−fold stable self-intersection (i.e. H ∩ st · · · ∩ st H k times ) by H k . Moreover, for a polyhedral complex P of dimension d, we call its subcomplex P consisting of all polyhedra in P of dimension smaller or equal to k, where k < d, the k−skeleton of P . The following fact was proved in [AR10].
Lemma 2.11. Let H be the standard tropical hyperplane in R n R1 rooted at a point v. Its k−fold stable self-intersection H k is given by its (d − k)-skeleton with all weights equal to 1.
Remark 2.12. Let v 1 , . . . , v n ∈ Z d Z1 , let m 1 , . . . , m n ∈ Z ≥0 and let H i be the standard tropical hyperplane at v i . Then the stable intersection coincides with set-theoretic intersection in the sense that Finally, since the standard tropical hyperplane is a tropicalised linear space, we obtain the following Proposition as a consequence of the previous discussion.
{pt} is exactly one point with multiplicity one. If v 1 , . . . , v n are in tropical general position, the set-theoric intersection H m 1 1 ∩ · · · ∩ H mn n = {pt} coincides with the intersection product.

Bruhat-Tits Buildings and Tropical Convexity.
In this section, we recall some of the relations between Bruhat-Tits buildings and tropical convexity. For a short summary of Bruhat-Tits buildings, we refer to Section 2 of [CHSW11], for more details on the relation between buildings and tropical convexity see e.g. [Wer11]. We denote the Bruhat-Tits building associated to PGL(V ) by B d and by (1) We write tconv(Γ) for the convex hull of the image of the point configuration under this map.
Definition 2.14. We call a point configuration The convex hull conv(Γ) of a point configuration Γ is the intersection of all convex sets containing Γ.
The following Lemma essentially goes back to [KT06], is a special case of Lemma 21 in [JSY07] and can be found in this version as Lemma 4.1 in [CHSW11].
Lemma 2.15. Let Γ be a point configuration in one apartment. The map in (1) induces a bijection between the lattices in conv(Γ) in the Bruhat-Tits building B d and the lattice points in tconv(Γ).
Remark 2.16. We call a point configuration Γ ⊂ B 0 d that is contained in the same apartment A in tropical general position, if the point configuration In [Li17], images of rational maps of the form were studied. In particular, the Hilbert function and the multidegrees were computed. For every non-empty I ⊂ [n] := {1, . . . , n} we define Remark 2.17. There are several equivalent notions of the multidegree of a variety X in a product of projective spaces P d−1 k n . One possibility is to consider the multigraded Hilbert polynomial h X : Let x u be a monomial of maximal degree in h X and c u be the coefficient. The multidegree function takes value cu u! at u, where u! = u 1 ! · · · u n !.
Another way to describe the multidegree is to consider the intersection of X with a system of u i general linear equations in the i−th factor of n i=1 P k d W i , where we choose the u i such that the intersection is finite. Then the value of the multidegree function at u = (u 1 , . . . , u n ) is the cardinality of this intersection product. For a more thorough introduction in terms of Chow classes, see e.g. [CLZ16]. We denote the set of multidegrees by multDeg(X) = {u ∈ Z n ≥0 : the multidegree function is non-zero at u}. Theorem 2.18. [Li17] Assume k is algebraically closed. Set p = max{h : M (h) = 0}. The dimension of X(W, W 1 , . . . , W n ) is p. Its multidegree function takes value one at the integer vectors in M (p) and 0 otherwise. The Hilbert function of X(W, W 1 , . . . , W n ) is where the u i are the variables and S,i is the smallest i−th components of all elements of S. Moreover, X(W, W 1 , . . . , W n ) is Cohen-Macaulay.
We end this section with an example.
2.4. Mustafin Varieties. Recall Definition 1.1 for a Mustafin variety M(Γ) associate to a point configuration Γ in B 0 d . In this subsection, we review the theory developed in [CHSW11], where many interesting structural results about M(Γ) and its special fiber were proved. We state results needed for our approach and refer to [CHSW11] for a more detailed discussion.
(1) For a finite set of lattices Γ, the Mustafin variety M(Γ) is an integral, normal, Cohen-Macaulay scheme which is flat and projective over R. Its generic fiber is isomorphic to P d−1 K and its special fiber is reduced, Cohen-Macaulay and connected. All irreducible components are rational varieties and their number is at most n+d We now choose coordinates in the same way as in [CHSW11]. Consider the diagonal map The image of ∆ is the subvariety of of P(V ) n cut out by the ideal generated by the 2×2 minors of a matrix X = (x ij ) i=1,...,d j=1,...,n of unknowns, where the jth column corresponds to coordinates in the jth factor. Start with an element g ∈ GL(V ), it is represented by an invertible n × n matrix over K. It induces a dual map g t : V * → V * and thus a morphism g : P(V ) → P(V ). For n elements g 1 , . . . , g n ∈ GL(V ), the image of is the subvariety of P(V ) n cut out by the multihomogeneous prime ideal where (g 1 , . . . , g n )(X) is the matrix whose jth column is given by Consider a reference lattice L = Re 1 + · · · + Re d . For any configuration Γ = {L 1 , . . . , L n } of lattices in B 0 d , we choose g i , such that g i L = L i for all i. The following diagram commutes: It follows immediately that the Mustafin Variety M(Γ) is isomorphic to the subscheme of P(L) n ∼ = (P d−1 R ) n cut out by the multihomogeneous ideal  (1) The configuration Γ is in general position.
(2) The special fiber M(Γ) k is of monomial type. Thus, in the case of tropical general position, the number of components of M(Γ) k is n+d−2 d−1 . 2.5. (Pre)linked Grassmannians. In this subection, we discuss the basic theory surrounding (pre)linked Grassmannians, developed in the Appendix of [Oss06] and the Appendix of [Oss14] in the context of the theory of limit linear series. We begin with the following situation: Let S be any base scheme and E 1 , . . . , E n vector bundles on S, each of rank d. We have morphisms f i : E i → E i+1 , g i : E i+1 → E i and a positive integer r < d. For an S−scheme T , we denote the pull-backs of the vector bundles by E i,T and the maps induced by the pull-backs by f i,T : E i,T → E i+1,T . The functor studied in [Oss06] is described as follows: This functor is representable by a projective scheme: over S, which is naturally a closed subscheme of a product G of Grassmannians schemes over S; G is smooth and projective over S of relative dimension nd(d − r). More precisely: Let G i be the Grassmannian of rank r sub-bundles of E i , then LG is a closed subscheme of G = G 1 × · · · × G n . Remark 2.25. If there is no confusion about the data, we denote LG(r, LG.
In order to study the dimension of LG, additional hypothesis were made in [Oss06].
Definition 2.26. We say that LG(r, Cohen-Macaulay and the following conditions on f i , g i are satisfied: (II) On the fibers of the E i at any point in the zero-locus of s, we have Kerf i = Img i and Kerg i = Imf i .
Equivalently, for any i and given integers r 1 and r 2 such that r 1 + r 2 < d, the closed subscheme of S obtained as the locus where f i has rank less than or equal to r 1 and g i has rank less than or equal to r 2 is empty. (III) At any point of S, we have Equivalently, for any integer r and any i, we have locally closed subschemes of S corresponding to the locus where f i has rank exactly r and f i+1 f i has rank less than or equal to r − 1 (similarly for g i ). Then we require all those subschemes to be empty. We will call a tuple (E i , {f i , g i } i ) satisfying these conditions an s−linked chain.
An important notion for our discussion are the exact points of a linked Grassmannian.
The following Proposition was proved in Lemma A.11 and Lemma A.12 in [Oss06].
Proposition 2.28. For linked Grassmannians, we have the following description of exact points: (i) The exact points form an open subscheme of LG and are naturally described as the complement of the closed subscheme on which rkf i | V i + rkg i | V i+1 < r for some i. (ii) In the case s = 0, we can describe exact points as those with rkf i | V i + rkg i | V i+1 = r for all i (which is even true for arbitrary scheme-valued points). Exact points have the following properties: (iii) The exact points are dense in LG and indeed dense in every fiber of LG → S. (iv) Every exact point x ∈ LG is a smooth point of LG over S. The non-exact points of the fibers may also be described nicely: (v) The non-exact points of a fiber are precisely the intersections of the components of that fiber.
Remark 2.29. In our case, linked Grassmannians will always be cosindered over S = Spec(R), where R is a DVR. Thus, we will only talk about exact points and non-exact points of the special fiber, since the generic fiber is a Grassmannian as seen in the proof of Lemma 2.24 in [Oss06].
The following theorem was proved in [HO08].
Theorem 2.30 ( [HO08]). Let S be integral and Cohen-Macaulay and let LG be any linked Grassmannian over S. Then LG is flat over S, reduced and Cohen-Macaulay, with reduced fibers.
In the Appendix of [Oss14], the notion of linked Grassmannians of length n was generalized to arbitrary underlying graphs. We make this more precise in the following definition and give a quick summary of the results we will need in Section 4.
Definition 2.31. Let G be a finite directed graph, connected by (directed) paths, d an integer and S a scheme. Suppose we are given the following data: E • , consisting of vector bundles E v of rank d over S for each v ∈ V (G) and morphisms f e : E v → E v for each edge e ∈ E(G), where e points from v to v . For a directed path P in G, we denote by f P the composition of the morphisms f e along th edges e in the path. Given an integer r < d suppose further that we also have the following condition satisfied: For any two paths P, P in G with the same head and tail, there are sections s, s ∈ Γ(S, O S ) such that s · f P = s · f P . Moreover, if P (resp. P ) is minimal (i.e. a shortest path with respect to the canonical graph metric), then s (resp. s) is invertible. Then define the prelinked Grassmannian LG(r, E • ) to be the scheme representing the functor associating to an S−scheme T the set of all collections (F v ) v∈V (G) of rank-r subbundles of the E v,T satisfying the property that for all edges e ∈ E(G), we have f e (F v ) ⊂ F v , where e points from v to v . It was proved in [Oss14] that this functor is representable. Moreover, LG(r, E • ) is a projective scheme over S and compatible with base change.
(1) Note that we allow the path consisting of one vertex and consider f P = id in this case. The condition implies that for any cycle P , we obtain f P = s · id for some scalar s, which corresponds to Definition 2.26 (I).
(2) When G is the graph consisting of vertices v 1 , . . . , v n and directed edges e i,i+1 pointing from v i to v i+1 and e i+1,i pointing from v i+1 to v i we are in the situation of Definition 2.23.
The notion of exact points generalises easily to prelinked Grassmannians, but they are harder to study. To overcome this obstacle, the notion of simple points was introduced in [Oss14]. When the graph G together with the data E • , f • is an s−linked chain, the notion of simple points coincides with the notion of exact points (see [Oss14]).
Definition 2.33. Let k be a field over S and (F v ) v a k − valued point of a prelinked Grassmannian LG(r, E • ). We say that (F v ) v is simple if there exist v 1 , . . . , v r ∈ V (G) and s i ∈ F v i for i = 1, . . . , r, such that for every v ∈ V (G), there exist paths P v i with each P v i going from v i to v and such that f P v 1 (s 1 ), . . . , f P vr (s r ) form a basis for F v .
We end this section by giving the two main statements for simple points: Proposition 2.34 ( [Oss14]). The following statements hold for simple points of a prelinked Grassmannian: (1) The simple points form an open subset of LG(r, E • ).
(2) On the locus of simple points LG(r, E • ) is smooth over S of relative dimension r(d−r).
2.6. Local models of Shimura varieties. In this subsection, we recall some of the basic theory around certain local models of Shimura varieties studied in [Gör01]. Shimura varieties can be viewed as higher dimensional analogous of modular curves. For a detailed introduction see for example [PRS13]. For the rest of this subsection, we assume that k is a perfect field.
In [Gör01] a local model for unitary Shimura varieties was studied. The standard local model is defined to be the R−scheme representing the following functor: Fix a number 0 < r < d and let e 1 , . . . , e d be the canonical basis of K d . We fix lattices L 0 , . . . , L d−1 as follows: L 0 = Re 1 + · · · + Re d and L i = Rπe 1 + · · · + Rπe i + Re i+1 + · · · + Re d . We obtain the following chain complex by the natural inclusion maps For each R−scheme S, the S−valued points of M loc are the isomorphism classes of the following commutative diagram: where L i,S = L i ⊗ S and the F are sub-vector bundles of rank r of L i,S .
Theorem 2.35 ( [Gör01]). The local model M loc is flat over R and its special fiber is reduced.

Special fibers of Mustafin Varieties
The goal of this section is to prove Theorem 1.2 and Theorem 1.3. We begin by constructing the varieties M(Γ) and M r (Γ).  L = π m 1 Re i 1 + · · · + π m d Re i d and L i = π n i 1 Re i 1 + · · · + π n i d Re i d . (This is always possible, see e.g. the discussion prior to Proposition 6.111 in [AB08].) There is a canonical map g i , such that g i [L] = [L i ], which up to homothety (i.e. multiplication by a scalar) is given by the matrix d by G i . We define an invertible matrix G i by This induces a rational map over the special fiber:  include a slight abuse of notation as X Γ, [L] is contained in the special fiber of P(L) n and X Γ, [L ] is contained in the special fiber of P(L ) n . In order to take the union , we observe that the isomorphism P(L ) n → P(L) n induces an isomorphism of special fibers and thus maps X Γ, [L ] into the special fiber of P(L) n , where we can take the union. The key observation for the proof of Theorem 1.2 is that the maps f Γ,[L] factorise as follows: (2) Thus, each variety Im( f L ) is a variety of the form X(k d , W 1 , . . . , W n ), where W i = ker( g −1 i ). Note that n i=1 W i might not be trivial. However, the components of M(Γ) k are equidimensional. The vertices with n i=1 W i = 0 contribute varieties of lower dimension and thus are contained in an irreducible component by Lemma 3.4. This irreducible component is contributed by a vertex satisfying n i=1 W i = 0 . Before we prove Theorem 1.2, we illustrate our construction in the following example: Example 3.3. By Lemma 2.15, there is a natural correspondence between lattice points in Z d Z1 and lattices in V over R. In Example 2.2 in [CHSW11], the special fiber of the Mustafin variety corresponding to the vertices in Example 2.3 was computed to be the union of the following irreducible components P 2 × pt × pt, pt × P 2 × pt, pt × pt × P 2 , P 1 × P 1 × pt, P 1 × pt × P 1 , pt × P 1 × P 1 Our construction yields the same variety as seen in the following computations. For v = v 3 , we obtain the map  Modding out π, we immediately see that where [L] is the lattice class corresponding to v Analogously, we obtain pt × P 2 × pt and P 2 × pt × pt for v = v 2 and v = v 1 respectively.
The following Lemma implies the easier containment for the equalities of Theorem 1.2. (v) and let g ∈ I(Γ, [L]) k , we want to prove that g(w) = 0. We choose a polynomial g ∈ I(Γ, [L]), which specialises to g and define W = f Γ, [L] (v) ∈ M(Γ) ⊂ P(L) n . We see immediately that g (W ) = 0. Moreover, since we chose the component-wise maps G −1 i , such that G −1 i ∈ Mat(R, d×d) and such that there are elements in G −1 i of valuation 0, we see that the constant term of g (W ) is given by g(W ). However, since g (w ) = 0, the constant term vanishes as well and thus g(w) = 0. We obtain f Γ, [L] (v) ∈ M(Γ) k and thus f Γ,[L] ⊂ M(Γ) k as desired. By definition Im( f Γ,L ) is reduced and irreducible and by [CHSW11] the same is true for the irreducible components of M(Γ) k . Therefore, the second statement follows.
3.2. A basic case: Two lattices. In this section we use results from [EO13] to prove Theorem 1.2 (2) if Γ consists of two lattices. That is, we show M r (Γ) = M(Γ) k , demonstrating some of the basic ideas for the proof of Theorem 1.2 (2) in subsection 3.4. By Lemma 3.4, the variety M r (Γ) is contained in M(Γ) k . Since both schemes are reduced, we can deduce equality if their bivariate Hilbert polynomials coincide. Each pair of two lattices is contained in a common apartment. Moreover, their convex hull forms a path between the two lattices we started with. Using the maps over the special fiber as in Construction 3.1, we obtain a complex as follows: where each B i = k d (and each B i corresponds to a polyhedral vertex). We use the notation of the B i s in order to keep track of the position. This is the same situation as in [EO13]: EO13]). Let f i : P(B i ) P(B 1 ) × P(B n ) be the induced map obtained by composing the maps above along the shortest path to the extremal vertices. We define n i=1 Im(f i ) to be the associated Esteves-Osserman variety. Remark 3.6. A similar discussion concerning the connection between more general Esteves-Osserman varieties and linked Grassmannian can be found in [HL]. However, since this work is still in preparation, we decided to include the arguments needed for our case.
In [EO13], it is proved that M(Γ) has the same bivariate Hilbert polynomial as P d−1 if the maps in the complex fulfil the following exactness condition: Lemma 3.7. Let Γ = {L 1 , L 2 } and apply Construction 3.1 to obtain a complex as follows: where B i = k d for all i ∈ {1, . . . , n}. Then this complex fulfils the conditions (1)-(4) above.
Proof. The two vertices lie in a common apartment. We see immediately that over R the maps A i → A i+1 are given by diagonal matrices A = diag(a 1 , . . . , a d ), where a j = 1 for j ∈ J i and a j = π for j ∈ J c i , where J i is some subset of [d]. Moreover, over R the maps A i+1 → A i are given by diagonal matrices where b j = π for j ∈ J i and b j = 1 for j ∈ J c i . Thus, the first two conditions follow immediately after modding out π. The third and fourth condition follow from the fact that J i ⊂ J i+1 which is a consequence of the structure of the tropical convex hull (see Lemma 2.2).
Thus, we obtain Theorem 1.2 restricted to the case where n = 2.
3.3. Proof of Theorem 1.2 (1). This subsection is devoted to proving Theorem 1.2 (1). The strategy of the proof is to show that for convex point configurations Γ and recover the general case by using Lemma 2.4 of [CHSW11].
Let Γ be a convex point configuration. We need to following Corollary, which was proved in [CHSW11], which is essentially a consequence of Proposition 2.20 (5). The key idea in proving Theorem 1.2 (1) for arbitrary point configurations is to observe that Construction 3.1 commutes with projections, i.e. for Γ = conv(Γ) the following diagramm commutes: We have proved that To see that To see the other direction, we compare the multidegrees. Since M(Γ) k is a flat degeneration of the diagonal of (P d−1 ) n , we know (see e.g. [CS10]) that (5) multDeg(M(Γ) k ) = (m 1 , . . . , m n ) ∈ Z n ≥0 : Moreover, the multidegree function takes value 1 at each multidegree. It is easy to see that If we prove that the same holds true for the multidegree function of M r (Γ), we can deduce equality. Thus, we have to prove two statements: • The special fiber of the Mustafin variety and the variety we constructed have the same multidegrees: multDeg( M r (Γ)) = multDeg(M(Γ) k ), where we know the right hand side of the equation. • The multidegree function takes value one at each (m 1 , . . . , m n ) ∈ multDeg( M r (Γ)). The basic idea is as follows: For every tuple (m 1 , . . . , m n ) as above, we find a vertex in tconv(Γ) whose associated variety contributes multidegree (m 1 , . . . , m n ). We use basic tropical intersection theory. By our method, it is immediate that there is only one vertex that can contribute a variety of this multidegree.
We begin by treating the case of lattices being in tropical general position. The case of arbitrary point configurations will follow using stable intersection.
3.4.1. Point configurations in tropical general position. For a point configuration Γ in tropical general position, the number of polyhedral vertices in the respective tropical convex hull is n+d−2 d−1 as proved in [DS04]. This coincides with the cardinality of multDeg(M(Γ) k ). Therefore, the natural candidates to for the correct multidegrees are those vertices.
We begin by describing the lattices corresponding to the lattice points of the polyhedral complex associated to tconv(Γ) and start by defining a map where C(v) i is the largest k, such that v lies in the codimension k-skeleton of the hyperplane rooted at at the lattice L i ∈ Γ. It is not obvious why this map is well defined. This is where tropical intersection theory comes into play.
As mentioned before the number of vertices of tconv(Γ) coincides with the number of tuples (m 1 , . . . , m n ) ∈ Z ≥0 . If we construct one pre-image for each such tuple under the map C in Equation (6), we are done. Let (m 1 , . . . , m n ) be such a tuple. We identify a vertex in tconv(Γ) as follows: Let Γ = {L 1 , . . . , L n } and v 1 , . . . , v n the corresponding vertices in the tropical torus. We intersect the standard max-hyperplanes H i at v i H m 1 1 ∩ st · · · ∩ st H mn n = H m 1 1 ∩ · · · ∩ H mn n = {v} as in Proposition 2.13. This intersection point v is vertex of tconv(Γ) and thus C is welldefined. In fact we have proved that the map C is bijective. Now, we study the maps f Γ,[L] = ( g −1 1 , . . . , g −1 n ) as in Construction 3.1, in particular the associated numerical data d I = dim i∈I ker g −1 i .
Lemma 3.9. Let v ∈ tconv(v 1 , . . . , v n ) such that v is contained in the codimension m i skeleton at v i , but not in the codimension m i + 1 skeleton, in the tropical torus. Let L be a lattice in the class corresponding to v, then where g −1 1 is the map to the i−th factor in Construction 3.1.
Since v is contained in the codimension m i skeleton at v i , but not in the codimension m i +1 skeleton, the minimum of v i0 −v 1 , . . . , v id −v d is attained m i + 1 times. As described in Construction 3.1, the map g −1 1 over the special fiber is given by where a j = 1, if the minimum is attained at v ij − v 1 and a j = 0 else. Thus, the dimension of the kernel of the map is given by d − (m i + 1).
where g −1 1 is the map to the i−th factor in Construction 3.1. Proof. Let v be the intersection point of the max-hyperplanes at v 1 , . . . , v n as before. Moreover, let C i be the codimension m i cone of the hyperplane at v i in which v is contained. Let H I be the intersection of the C i for i ∈ I. We can describe each C i as follows: for some J i ⊂ {1, . . . , d} such that |J i | = m i + 1. Since v 1 , . . . , v n are in general position, (v i ) i∈I are in general position as well. Moreover, with the same arguments as before, the codimension m i skeleta at v i for i ∈ I intersect in codimension i∈I m i . Thus, we have i∈I J i = i∈I m i + 1. Now the kernel of the map f Γ,[L] i is generated by the basis vectors e j , such that j / ∈ J i . Thus, the intersection of the kernels is generated by those basis vectors e j such that j / ∈ J i for all i, that is Thus the Lemma follows.
Example 3.11. Let v 1 , v 2 and v 3 as in Example 2.3. We want to find the vertex which contributes the multidegree (1, 0, 1). In order to do this, we intersect the codimension 1 skeleton at v 1 , the codimension 0 skeleton at v 2 and the codimmension 1 skeleton at v 3 as illustrated in Figure 5. We obtain the vertex (0, −1, −4), which by Example 3.3 contributes the variety P 1 × pt × P 1 , which has multidegree (1, 0, 1).
We are now ready to prove Theorem 1.2 for the case that Γ is in tropical general position.
Proof of Theorem 1.2 (2) for Γ in tropical general position. Fix a multidegree (m 1 , . . . , m n ) ∈ multDeg(M(Γ) k ) (see Equation (5)  To see that the multidegree function takes value 1 at each element in multDeg( M r (Γ)) we use Theorem 2.18: For every variety X = X(V, V 1 , . . . , V n ) the multidegree function takes value 1 at each element of multDeg(X). Thus we need to prove that for each multidegree, there is exactly one variety contributing it. However, this is true by the arguments we needed for the set-theoric equality of the multidegrees.
3.4.2. Point configurations in arbitrary position. We now use the theory of stable intersection to deduce Theorem 1.2 (2) for arbitrary point configurations.
Proof of Theorem 1.2 (2) for Γ in arbitrary position. Let Γ be a point configuration in arbitrary position. We consider a slight perturbation of the vertices in Γ to obtain a point configuration Γ in general position. Taking the limit Γ → Γ, we see that every vertex v of the polyhedral complex tconv(Γ) is the limit of several vertices w 1 , . . . , w l in tconv( Γ) and vice versa. Let [L], [L 1 ], . . . , [L l ] be the lattice classses corresponding to v, w 1 , . . . , w l . We claim that then we have multDeg( M r (Γ)) = multDeg( M r ( Γ)) = multDeg(M( Γ) k )) = multDeg(∆(P(V ))) = multDeg(M(Γ) k )) and together with the same argument about the multidegree function as in the tropical general position case, the theorem follows. Note that multDeg( M r (Γ)) = where each [L]∈conv(Γ) multDeg( f Γ, [L] ) is a set of multidegrees as in Equation (7) and the union is disjoint.
To prove multDeg( f Γ,[L] ) = {C(w 1 ), . . . , C(w n )}, we observe that for all i and j and there exists j i , such that This follows from the following consideration: Let Γ = {L 1 , . . . , L n } and let v i be the vertex corresponding to L i . For each i and j, there exists j i , such that there is a homeomorphism between tconv(v, v i ) and tconv(w j i , v i ). For j = j i , the natural map from tconv(w j i , v i ) to tconv(v, v i ) is surjective but shrinks edges. This translates to the relation between the kernels described above.
We Thus we obtain set-theoric equality of the multidegrees. As mentioned before, the multidegree functions coincide follows by the same arguments as in the tropical general position case and the fact that the union in Equation (8) is disjoint.
Example 3.12. Let v 1 = (0, 0, 0) and v 2 = (0, 1, 1). Let Γ = {L 1 , L 2 } be the point configuration with L i being the lattice corresponding to v i . We compute where x 1i , x 2i , x 3i are the coordinates in the i−th factor. We see that v 1 contributes the variety corresponding to x 22 , x 32 of multidegree {(2, 0)} and that v 2 contributes the variety corresponding to x 11 , x 21 x 32 − x 31 x 22 of multidegree {(1, 1), (0, 2)}. We want to see these multidegrees in the combinatorics of the tropical convex hull. In order to do this, we pertubate the vertex v 2 as illustrated in Figure 6, to obtain the point configuration Γ corresponding to v 1 = (0, 0, 0), v 2 = (0, 1, 2). We see that v 1 is the limit of v 1 and v 2 is the limit of v 2 and (0, 1, 1) by reversing the pertubation. We note that v 1 contributes a variety to M(Γ ) k of multidegree {(2, 0)}, (0, 1, 1) contributes a variety of multidegree {(1, 1)} and v 2 contributes a variety of multidegree {(0, 2)}. Then we see as in the proof that the multidegree of a vertex v in conv(Γ) is given by the union of the multidegrees of the vertices whose limit is v by reversing the pertubation. Another question raised in [CHSW11] is how many irreducible components the special fiber of a Mustafin variety has. In the one apartment case, this count is inherent in the combinatorial data inherent in the tropical convex hull, which was observed in Theorem 4.4 in [CHSW11] (and also follows from Theorem 1.2 (2)). The next remark comments on the number of irreducible components for arbitrary point configurations.
Remark 3.13. We note that Theorem 1.2 yields a linear algebra algorithm for counting the number of irreducible components of M(Γ) k for arbitrary point configuration. Namely, computing the convex hull conv(Γ) and computing the number of lattice classes [L] ∈ conv(Γ), such that dim(Im f Γ,[L] = d − 1, which only depends on the numerical data given by the d I . This number coincides with the number of irreducible components. In the one apartment case, we can express the number of irreducible components as combinatorial data inherent in the tropical convex hull. We believe that a similar description is possible for arbitrary point configurations using the tropical description of convex hulls in a Bruhat-Tits building given in [JSY07]. Computing the tropical convex hull in the Bergmann fan in the associated matroid, we can perform similar intersection products in the respective tropical linear space. More precisely, the tropical linear space will be of dimension d − 1. Thus, for a fixed multidegree (m 1 , . . . , m n ), intersecting the linear space with the codimension m i skeleta at the i−th vertices, we will obtain exactly one polyhedral vertex in the associated tropical convex hull. However, proving that the variety associated to the polyhedral vertex contributes in fact an irreducible component with multi-degree (m 1 , . . . , m n ) is not as simple as in the one apartment case. More precisely, the first step in computing the dimension of the intersections of the corresponding kernels requires knowing the basis vectors of each lattice, which in general is not as evident from the combinatorial structure as in the one apartment case.

Mustafin varieties and linked Grassmannians
Let Γ ⊂ B 0 d be a convex point configuration in one apartment. We associate a prelinked Grassmannian LG(r, Γ) of rank r subbundles as follows: (1) The base scheme is R. Remark 4.1. We note the following two subtleties of the above situation: (1) In Definition 2.31, we required a vector bundle at each vertex, whereas in the situation above, we have homothety classes at each vertex. However, the construction does not depend on the representatives as long as they are adjacent in the building corresponding to GL(V ). For the rest of the paper, let LG(r, Γ) be the prelinked Grassmannian of rank r associated to this data. Moreover, let Simp(Γ) be the locus of simple points in the special fiber LG(1, Γ) k . 4.1. Linked Grassmannians and Faltings' functor. In [Fal01], a moduli functor description for M(Γ) was introduced. We begin by relating this functor to the linked Grassmannian problem. We note that Faltings' notion of projective space is the dual construction to the one used here, i.e. in [Fal01] P(V ) parametrises hyperplanes instead of lines. This yields the notion of quotient bundles in loc. cit. instead of sub-bundles, which we use here: Fal01]). Let Γ be a convex point configuration. Then M(Γ) represents the following functor: To every R−scheme S we associated the set of all tuples of line bundles (l(L)) L∈Γ , such that each inclusion L i ⊂ L j (for L i , L j ⊂ Γ) maps l(L i ) to l(L j ).
Since the maps along the tropical convex hull in the linked Grassmannian we associated to Γ at the beginning of this section coincide with the inclusion maps, we obtain the following result as a Corollary, interpreting Mustafin varieties as a moduli space in limit linear series theory:  Most results on (pre)linked Grassmannians required the study of the simple points as elaborated in Section 2. Using Theorem 1.2, we can relate the simple points to Mustafin varieties. We note that it is clear that the simple points are dense in the generic fiber, since the maps between vertices over the generic fiber are isomorphisms. In the next subsection, we prove that the simple points are dense in the special fiber as well, whenever k is algebraically closed.

Mustafin varieties and simple points.
4.2.1. A motivating example: Two lattices. A first example for Theorem 4.4 was given in Warning A.16 in [Oss06]. We pick S = Spec(k), n = d = 2, r = 1. (Note that this corresponds to picking any two adjacent lattices in dimension 2 as each two adjacent lattices will yield the same maps.) Thus the vector space associated to v 1 and v 2 is k 2 with maps f (v 1 ,v 2 ) = 1 0 0 0 and f (v 2 ,v 1 ) = 0 0 0 1 . Thus the linked Grassmannian consists of points (V 1 = X 0 X 1 , V 2 = Y 0 Y 1 ), such that f 1 (V 1 ) ⊂ V 2 and vice versa. The condition of being linked is given by X 0 Y 1 = 0. In fact LG(1, Γ) k is scheme-theoretically cut out by this equation in P 1 × P 1 , which translates to a pair of P 1 's attached at X 0 = Y 1 = 0 (which is the only not exact point). Given Γ as mentioned above, we can compute the special fiber of the Mustafin variety, which by Theorem 1.2 is given by Im(f 1 ) ∪ Im(f 2 ) for f (v 1 ,v 2 ) : P 1 P 1 × P 1 given by 1 0 0 1 × 1 0 0 0 and f (v 2 ,v 1 ) : P 1 P 1 × P 1 given by 0 0 0 1 × 1 0 0 1 .
Thus, by Theorem 1.2, M(Γ) k is a pair of P 1 s attached at X 0 = Y 1 = 0 as well and we can conclude LG(1, Γ) k = M(Γ) k in this case.

Special fibers of Mustafin Varieties and prelinked Grassmannians.
The key step in proving Theorem 1.4 is the following proposition.
Proposition 4.6. Let Γ be a convex point configuration. Then the closure of the locus of simple points in LG(1, Γ) k is set-theoretically given by the special fiber of the Mustafin varieties: Proof. The statement is about the special fiber, thus we base change to Spec(k). Moreover, we can choose coordinates on the linked Grassmannian as for Mustafin varieties by performing a linear coordinate change LG(1, Γ) ⊂ P(L 1 ) × · · · × P(L n ) g 1 ×···×gn − −−−−− → P(L) × · · · × P(L) for a reference lattice L and linear maps g i , such that g i L = L i . We observe that the maps between adjacent lattices over k coincide with the respective maps in Construction 3.1. First, we have to check that G = G(Γ) together with the associated data at the vertices and edges actually satisfies the conditions in Definition 2.31. The only thing to check is the condition on the paths. Let g P be the map obtained by the composition of maps corresponding to edges along the path P . We make the following claim from which the required conditions follow immidiately: Lemma 4.7. Let P be a path and P a minimal path between [L] and [L ]. Then we have g P = g P or g P ≡ 0.
Proof. In order to prove this claim, we consider the maps before base changing. As maps over R, we observe that there exists n ∈ N, such that t n g P = g P .
If n = 0, we obtain g P = g P over the special fiber and g P ≡ 0 if n > 0. Therefore, G = G(Γ) satisfies the conditions in Definition 2.31.
In order to link the simple points to the special fiber, we need the following lemma.
Lemma 4.8. The tropical convex hull between two lattice classes [L] and [L ] is a minimal path.
In the rank 1 case, simple points translate to the following situation: Let (F v ) v∈V (G) be a simple point. Then there exists a v ∈ V (G), such that taking minimal paths P v ,v from v to v ∈ V (G) for each v, we obtain F v = f P v ,v (F v ) and we say (F v ) v∈V (G) is rooted at v . Thus we can classify the set Simp(Γ) v of all simple points rooted at v as the image of the following rational map

Mustafin varieties and local models of Shimura varieties
We interpret the standard local model M loc described in subsection 2.6 as a Mustafin variety by the following theorem: where M loc is the local model for r = 1.
Remark 5.2. By a similar argument M loc is shown to coincide with a different class of linked Grassmannian for rank r subbundles in [HL] .
Proof. We prove that M loc = LG(1, Γ) and follow with Theorem 4.4 that M loc = M(Γ). Fix an R−scheme S and let (F 0 , . . . , F d−1 ) be a tuple of rank 1 sub-bundles of (L 1,S , . . . , L d−1,S ) parametrised by LG(1, Γ). Since the maps L 0 → · · · → L d−1 appear in underlying graph of the linked Grassmannian associated to Γ, we see that the tuple (F 0 , . . . , F d−1 ) is parameterised by M loc as well. For the other direction, let (F 0 , . . . , F d−1 ) be a tuple as above parametrised by M loc . In order to see that this tuple is parametrised by LG(1, Γ), we need to prove that for i, j ∈ {0, . . . , d − 1}, the map L i → L j maps F i to F j . However, the map L i → L j is given by the composition L i → L i+1 · · · → L j−1 → L j . Since the tuple (F i ) i is parametrised by M loc the chain of maps F i → F i+1 → · · · F j−1 → F j is well defined, F i is mapped to F j as well.
Since Mustafin varieties are flat with reduced fibers by Proposition 2.20, we can use Theorem 5.1 to obtain a new proof of Theorem 2.35 for r = 1.