Mustafin varieties, moduli spaces and tropical geometry

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Marvin Anas Hahn ·Binglin Li

Mustafin varieties, moduli spaces and tropical geometry

Received: 18 April 2019 / Accepted: 3 August 2020 / Published online: 24 August 2020 Abstract. Mustafin varieties are flat degenerations of projective spaces, induced by a set of lattices in a vector space over a non-archimedean field. They were introduced by Mustafin (Math USSR-Sbornik 34(2):187, 1978) 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 et al. (Selecta Math 17(4):757–793, 2011). In this paper, we introduce a new approach to Mustafin varieties in terms of images of rational maps, which were studied in Li (IMRN, 2017). 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. Finally, we outline a first application of our results in limit linear series theory.

Contents

1. Introduction . . . . 2. Preliminaries . . . . 2.1. Tropical geometry . . . . 2.2. Bruhat-Tits buildings and tropical convexity . . . . 2.3. Images of rational maps. . . . 2.4. Mustafin varieties . . . . 3. Special fibers of Mustafin varieties . . . . 3.1. Constructing the varietiesM() andM^r() . . . . 3.2. Proof of Theorem 1.2 (1) . . . . 3.3. Proof of Theorem 1.2 (2) . . . . 3.4. Classification of the irreducible components of special fibers of Mustafin vari-

eties . . . . 4. A first application: prelinked Grassmannians . . . . References. . . . M. A. Hahn (

B

): Department of Mathematics, University of Tübingen, 72076 Tübingen, Germany. e-mail: marvin-anas.hahn@uni-tuebingen.de

B. Lin: Department of Statistics, University of Georgia, Athens, GA 30602, USA.

e-mail: binglinligeometry@uga.edu

Mathematics Subject Classification: Primary 14T05·14D06·14D20; Secondary 20E42· 20G25·52B99·14G35

https://doi.org/10.1007/s00229-020-01237-8

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1. Introduction

Mustafin varieties are flat degenerations of projective spaces induced by choosing a set of lattices in aK-vector spaceV. These objects were introduced by Mustafin in [22] in order to generalise Mumford’s groundbreaking work on uniformisation of curves to higher dimensions [21]. Since then they have been repeatedly studied under the nameDeligne schemes(see e.g. [7,10,16]). By studying degenerations of projective spaces, we give a framework for the study of degenerations of projective subvarieties. In his original work, Mustafin studied the case of so-calledconvex subsets ofBdas defined in Definition2.12. An approach to study arbitrary subsets ofBdwas developed in [5], where the total space of this type of degenerations was namedMustafin varietyfor the first time. There it was proved that if the lattices in the subset ofBdhave 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 a mixed subdivision of a scaled simplex. These mixed subdivisions are beautiful combinatorial objects that are known to be equivalent to tropical polytopes and triangulations of products of simplices. For subsets of Bdthat 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.

LetRbe a discrete valuation ring,Kthe quotient field andkthe residue field.

We fix a uniformiserπ. As an example takeK = C((π))as the ring of formal Laurent series overCwith discrete valuation v(

n≥lanπⁿ) = l forl ∈ Zand an ∈ Cwithal = 0. ThenR = {

n≥lanπⁿ :l ∈ Z≥0}andk =C. Moreover, letV be vector space of dimensiond overK. We defineP(V)=ProjSym(V^∗)as paramatrising lines throughV. We call freeR-modulesL ⊂V of rankd lattices and defineP(L) =ProjSym(L^∗), whereL^∗ =HomR(L,R). Note, that we will mostly consider lattices up to homothety, i.e.L LifL=c·Lfor somec∈K^×. We denote the homothety class ofLby[L].

Definition 1.1.(Mustafin varieties) Let = {[L1], . . . ,[Ln]}be a set of lattice classes inV. ThenP(L1), . . . ,P(Ln)are projective spaces overRwhose generic fibers are canonically isomorphic toP(V)P^d_K⁻¹. The open immersionsP(V) → P(Li)give rise to a map

P(V)−→P(L1)×R· · · ×RP(Ln).

We denote the closure of the image endowed with the reduced scheme structure by M(). We callM()theassociated Mustafin variety. Its special fiberM()kis a scheme overk.

While the generic fiber of such a scheme is isomorphic toP^d⁻¹, the special fiber has many interesting properties.

The main tool in this paper is the study of closures of images of rational maps of the form

f :P(W)P WW1

× · · · ×P WWn

,

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where W is a vector space over k of dimension d and (Wi)i∈[n] (with [n] = {1, . . . ,n}) is a tuple of sub-vector spacesWi ⊂W, such that

Wi = 0under- taken in [17]. We denote the closure of the above map byX(W,W1, . . . ,Wn).

We make the following assumption for the rest of the paper.

General Assumption. The residue field k is algebraically closed.

We proceed as follows: We denote byB⁰_dthe set of lattice classes up to homothety inV. Let ⊂ B⁰_d be a finite subset. To each lattice class [L] ∈ B⁰_d, we associate a varietyX_,[L]of the formX(k^d,W1, . . . ,Wn)for someWi depending on[L]and(see Construction 3.1). We consider the convex hull conv()(see Definition2.12), which is a set of lattices. Then we define a variety in

P^d_k⁻¹n

as follows

M() =

[L]∈conv()

X_,[L].

Moreover, when iscontained in a single apartment of B⁰_d, i.e. the lattices in are simulatenously diagonalisable, we identify a distinguished subsetV()of conv()and define

M^r()=

[L]∈V()

X_,[L].

The following is the main result of this paper.

Theorem 1.2.The irreducible components of the special fiber of Mustafin varieties are related to images of rational maps as follows:

(1)Ifis a an arbitrary finite set of lattice classes, we have M()k =M().

(2)Ifis a finite set of lattice classes in one apartment, we have M()k =M^r().

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 classes [L]that actually contribute an irreducible component. This is done by means of tropical intersection theory and multidegrees. Employing the techniques involved in the proof of Theorem1.2, we obtain the following theorem.

Theorem 1.3.The varieties X(k^d;W1, . . . ,Wd)withdim(X(k^d;W1, . . . ,Wd))= d−1classify all irreducible components of special fibers of Mustafin varieties, i.e.

(1)Any irreducible component of the special fiber of a Mustafin variety is a variety of the form X(k^d;W1, . . . ,Wd)and

(2)every variety X(k^d;W1, . . . ,Wn) with dim(X(k^d;W1, . . . ,Wd)) = d −1 appears as an irreducible component ofM()kfor some.

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Finally, we outline a first application of Theorem1.2in limit linear series theory.

This paper is structured as follows: In Sect.2we give a quick review of the tools needed to prove our theorems. In particular, we focus on the relation between the notions of tropical convexity and convexity in Bruhat–Tits buildings. We summarize some of the structural results for Mustafin varieties. Section3consists of the proof of Theorem1.2and Theorem1.3. In Sect.3.1, we construct the varietiesM() andM^r(). We prove Theorem1.2(1) in Sect.3.2, Theorem1.2(2) in Sect.3.3 and Theorem1.3in Sect.3.4. In Sect.4, we give a first application of our results in the theory of limit linear series.

2. 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 [20] Chapter 5.2 for a more detailed introduction. Our proof of Theorem1.2involves the identification of certain lattice points in so-calledtropical 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.

[2] or [20].

2.1.1. Tropical convexity In a sense tropical convexity is the notion of convexity over the tropical semiring(R,⊕,), whereR=R∪ {∞},a⊕b=min(a,b)and ab=a+b. We make this more precise in the following definition:

Definition 2.1.Let S be a subset of Rⁿ. We call S tropically convex, if for any choicex,y∈ Sanda,b∈Rwe getax⊕by∈S.

The tropical convex hull of a given subsetV ofRⁿis given as the intersection of all tropically convex sets inRⁿcontainingS. We denote the tropical convex hull of V by tconv(V).

This definition implies that every tropical convex setSis closed under tropical scalar multiplication. Thus, ifx ∈ S then so is x+λ1, where λ ∈ Rand1 = (1, . . . ,1). Therefore, we will usually identifySwith its image in(n−1)-dimension tropical torusRⁿR1.

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.[8] The tropical convex hull of two points x,y ∈ RⁿR1 is a concatenation of at most n−1ordinary 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. To give this explicit description forx=(x1, . . . ,xn)

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andy=(y1, . . . ,yn), we note that after relabelling and adding multiples of1, we may assume 0= y1−x1 ≤ y2−x2≤ · · · ≤ yn−xn. Then the tropical convex hull consists of the concatenation of the lines connecting the following points:

x =(y1−x1)x⊕y=(y1,y1−x1+x2,y1−x1+x3, . . . ,y1−x1+xn) (y2−x2)x⊕y=(y1,y2,y2−x2+x3, . . . ,y2−x2+xn)

· · · ·

(yn−1−xn−1)x⊕y=(y1,y2, . . . ,yn−1,yn−1−xn−1+xn) (yn−xn)x⊕y=(y1, . . . ,yn)

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= {v1, . . . , vn}ofRⁿR1, withvi = (vi1, . . . , vi n). Consider the standard tropical hyperplane at vi in the max-plus algebra:

H_v_i = {w∈RⁿR1:themaximumofw1−vi1, . . . , wn−vi n

is attained at least twice}.

Taking the common refinement, we obtain a polyhedral complex structure on RⁿR1, i.e. a subdivision into convex polyhedra, which we denote by Tconv().

The support of the bounded cells of Tconv()is equal to tconv()(see e.g. chapter 5.2 in [20]). Moreover, we denote byV()the set of vertices of the polyhedral complex Tconv().

In the following example we compute the tropical convex hull of three points in the tropical torus.

Example 2.3.We pick 3 points

v1=(0,−1,−2), v2=(0,−2,−4), v3=(0,−3,−6).

Viewed as points in the tropical torus, we can identify these points withv1 = (−1,−2),v1 =(−2,−4)andv3 =(−3,−6). The tropical convex hull is illustrated in Fig.1.

Remark 2.4.The tropical convex hull of finitely many points is also called atropical polytope. Tropical polytopes can be thought of as tropicalisations of polytopes over the field of real Puiseux seriesR{{t}}(see Proposition 2.1 in [9]). One can generalise this notion to arbitrary tropical polyhedra and polyhedra overR{{t}}, which in turn has applications in linear programming and complexity theory (see e.g. [1]).

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Fig. 1. The tropical convex hull of v1 = (0,−1,−2), v2 = (0,−2,−4) and v3 = (0,−3,−6)in the tropical torus

2.1.2. Stable intersection Tropical intersection theory is a well-developed field.

In this paper, we will only intersect standard tropical hyperplanes, which is why we restrict ourselves to this case. A more general discussion can be found in [2,20].

Definition 2.5.We fixn pointsv1, . . . , vn ∈ Z^dZ1 Hi be the standard tropical hyperplane atvi. We denote theset-theoretic intersection of H1, . . . ,Hnby

H1∩ · · · ∩Hn.

Moreover, we define thestable intersection of H1, . . . ,Hnby H1∩st· · · ∩st Hn= lim

1,...,n→0(H1+1·w1)∩ · · · ∩(Hn+n·wn), where+denotes the Minkowski sum andwi ∈Z^dZ1are generic vertices.

Remark 2.6.We note, that there are several definitions of stable intersection in tropical geometry, which all turn out to be equivalent. For various viewpoints, we refer to [2,14,15,25,26].

Example 2.7.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 Fig.2intersect 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.

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Fig. 2.The difference between set-theoretic and stable intersection

In algebraic geometry, two general linear spaces of respective codimensionm1

andm2intersect in codimensionm1+m2. A similar fact holds for tropical linear spaces. We first introduce a notion of points intropical general position.

Definition 2.8.A square matrixM ∈R^r^×^r istropically singular, if the minimum in

det(M)=

σ∈Sr

m1σ(1) · · · mrσ(r)

is attained at least twice. A subset {m1, . . . ,mn} ⊂ R^d is in tropical general position, if every maximal minor of the matrix((mi j)i j)is non-singular.

Let H be the standard tropical hyperplane at a point v ∈ RⁿR1, then we denote thek-fold stable self-intersection (i.e.H∩st· · · ∩ st H

ktimes

) byH^k. Moreover, for a polyhedral complexPof dimensiond, we call its subcomplexPconsisting of all polyhedra inPof dimension smaller or equal tok, wherek<d,the k-skeleton of P. The following fact was proved in [3].

Lemma 2.9.Let H be the standard tropical hyperplane inRⁿR1atv. Its k-fold stable self-intersection H^kis given by its(d−k)-skeleton.

Remark 2.10.Let v1, . . . , vn ∈ R^dR1 be in tropically general position, let m1, . . . ,mn ∈ R≥0 and let Hi be the standard tropical hyperplane at vi. Then the stable intersection coincides with set-theoretic intersection in the sense that

H₁^m¹∩st· · · ∩st H_n^mⁿ =H₁^m¹∩ · · · ∩H_n^mⁿ. We end this section, with the following proposition.

Proposition 2.11.Letv1, . . . , vn ∈ R^dR1and let m1, . . . ,mn ∈ R≥0, such that n

i=1mi =d−1. Furthermore, let Hibe standard tropical hyperplane atvi, then H₁^m¹∩st· · · ∩st H_n^mⁿ = {pt}

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is exactly one point. Letv1, . . . , vnbe in tropically general position, then the set- theoric intersection

H₁^m¹ ∩ · · · ∩H_n^mⁿ = {pt} coincides with the intersection product.

Proof. This follows immediately from theorem 5.7, theorem 5.8 in [2] and

Lemma2.9.

2.2. 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 [5], for more details on the relation between buildings and tropical convexity see e.g. [27]. We denote the Bruhat-Tits building associated to PGL(V)byBd. Furthermore, we denote byB⁰_dthe set of lattice classes inV. We note thatB⁰_dmay be viewed as the set of integral points ofBd. We call two lattice classes[L],[M] inB⁰_d adjacent if there exist representatives L ∈ [L]andM ∈ [M], such that πM⊂L⊂M.

We pick a basise1, . . . ,ed of V. The associated apartment Ais the set of lattice classes inB⁰_d which are diagonal with respect toe1, . . . ,ed. We observe that the following map is a bijection:

f :A−→Z^dZ1

{π^m¹Re1+ · · · +π^m^dRed} −→(−m1, . . . ,−md)+Z1. (1) For a subset ⊂ A, we denote Tconv() := Tconv(f())and tconv() :=

tconv(f()). Moreover, denote the set of lattices corresponding to the vertices of the polyhedral complex Tconv()byV().

Definition 2.12.We call a subset⊂B⁰_dconvex if for[L],[L] ∈, any vertex of the form[π^aL∩π^bL]is also in.

The convex hull conv()of a subset⊂B⁰_dis the intersection of all convex sets containing.

The following Lemma essentially goes back to [16], is a special case of Lemma 21 in [13] and can be found in this version as Lemma 4.1 in [5].

Lemma 2.13.Let⊂B⁰_dbe contained in one apartment A. The map in Eq.(1) induces a bijection between the lattices in conv() and the integral points in tconv().

Remark 2.14.We call a subset⊂B⁰_dthat is contained in the same apartmentA in tropical general position, if the subset

{f(a):a∈} ⊂Z^dZ1⊂R^dR1 is in tropical general position.

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2.3. Images of rational maps

LetW be ad-dimensional vector space overkand let(Wi)ⁿ_i₌₁be a tuple of sub- vectorspaces ofW with

n i=1

Wi = 0. In [17], images of rational maps of the form

P(W)P(WW1)× · · · ×P(WWn)

were studied. In particular, the Hilbert function and the multidegrees were computed. For every non-emptyI ⊂ [n] := {1, . . . ,n}we define

dI =dim

i∈I

Wi. For everyh∈Z_≥0we defineM(h)to be the set

M(h)= {(m1, . . . ,mn)∈Zⁿ_≥0: n i=1

mi

=handd−

i∈I

mi >dI for every non-emptyI ⊂ [n]}.

Remark 2.15.There are several equivalent notions of the multidegree of a variety X in a product of projective spaces

P^d_k⁻¹n

. One possibility is to consider the multigraded Hilbert polynomialhX: Letx^ube a monomial of maximal degree in hXandcube the coefficient. The multidegree function takes value^c_u^u_! atu, where u! =u1! · · ·un!.

Another way to describe the multidegree is to consider the intersection of Xwith a system of ui general linear equations in thei-th factor of _n

i=1P k^dWi

, where we choose theui such that the intersection is finite. Then the value of the multidegree function atu=(u1, . . . ,un)is the degree of this intersection product.

For a more thorough introduction in terms of Chow classes, see e.g. [6]. We denote the set of multidegrees by

multDeg(X)= {u ∈Zⁿ_≥₀:the multidegree function is non-zero atu}.

Theorem 2.16.[17] Assume k is algebraically closed. Set p=max{h:M(h)= 0}. The dimension of X(W,W1, . . . ,Wn)is p. Its multidegree function takes value one at the integer vectors in M(p) and 0 otherwise. The Hilbert function of

X(W,W1, . . . ,Wn)is

S⊂M(p)

(−1)^|^S^|−¹ n j=1

ui+S,i

S,i

,

where the uiare the variables andS,iis the smallest i -th components of all elements of S. Moreover, X(W,W1, . . . ,Wn)is Cohen-Macaulay.

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We end this section with an example.

Example 2.17.LetV =C³ande1,e2,e3the standard basis. Moreover, letV1 = (0),V2=(e2,e3),V3=(e2,e3)then

X =X(V,V1,V2,V3)∼=P²×pt×pt.

In the notation of Theorem 2.16, we see p = 2 = dim(X)with multidegree multDeg(X)=M(p)=M(2)= {(2,0,0)}. The multidegree function takes value 1 at(2,0,0)and 0 else.

LetVande1,e2,e3as above and letV1=(e1),V2=(e1),V3=(e3). Furthermore, let(xi j)i,j=1,2,3be the coordinates on

P²3

, wherex1j,x2j,x3jare the coordinates on the j-th factor. ThenX(V,V1,V2,V3)is the subvariety of

P²3

cut out be the ideal

I =(x11,x12,x33,x21x32−x31x22).

In the notation of Theorem 2.16, we see p = 2 = dim(X)with multidegree multDeg(X)= M(p)= M(2)= {(1,0,1), (0,1,1)}. The multidegree function takes value 1 at(1,0,1)and(0,1,1). The multidegree function takes value 0 else.

Moreover, we see that under the projection to the second and third factor of P²3

, we have thatX(V,V1,V2,V3)is mapped isomorphically toP¹×P¹.

2.4. Mustafin varieties

Recall Definition1.1for a Mustafin varietyM()associate to a subset⊂B⁰_d. In this subsection, we review the theory developed in [5], where many interesting structural results aboutM()and its special fiber were proved. We state results needed for our approach and refer to [5] for a more detailed discussion.

Proposition 2.18.([5,22])

(1)For a finite set of lattice classes ⊂ B⁰_d, the Mustafin varietyM()is an integral, normal, Cohen-Macaulay scheme which is flat and projective over R.

Its generic fiber is isomorphic toP^d_K⁻¹and its special fiber is reduced, Cohen- Macaulay and connected. All irreducible components are rational varieties and their number is at most_n₊_d₋₂

d−1

, where n= ||.

(2)Ifis a convex set inB⁰_d, then the Mustafin variety is regular and its special fiber consists of n smooth irreducible components that intersect transversely. In this case the reduction complex ofM()is induced by the simplicial subcomplex ofBdinduced by.

(3)Ifis contained in one apartment, the components of the special fiber corre- spond to maximal cells of the subdivision of the simplex|| · d−1, which is dual to the polyhedral complexTconv(). Ifis in tropical general position, the irreducible components are products of projective linear spaces.

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(4)An irreducible component mapping birationally to the special fiber ofP(Li)is called a primary component. The other components are called secondary com- ponents. For each i=1, . . . ,n there exists such a unique primary component.

The primary components are pairwise distinct. A projective variety X arises as a primary component for some subset⊂Bdif and only if X is the blow-up ofP^d_k⁻¹at a collection of n−1linear subspaces.

(5)Let C be a secondary component ofM()k. There exists an element v in conv(), such that

M(∪ {v})k−→M()k

restricts to a birational morphismC˜ →C,whereC is the primary component˜ ofM(∪ {v})kcorresponding tov.

Remark 2.19.We note that by e.g. [18, Proposition 4.4.16] the special fiberM()k

is equidimensional of dimensiond−1. We note that the pairwise distinctness of primary components follows from [5, Theorem 5.3].

3. Special fibers of Mustafin varieties

The goal of this section is to prove Theorem1.2and Theorem1.3.

3.1. Constructing the varietiesM() andM^r()

Let = {L1, . . . ,Ln} ⊂ B⁰_d be a finite set of lattice classes in V. The goal of this subsection is to construct the varietiesM() andM^r(). Our construction is motivated by the following way of choosing global coordinates onM()introduced in [5].

Consider the diagonal map

:P(V)−→P(V)ⁿ=P(V)×K· · · ×K P(V).

The image of is the subvariety ofP(V)ⁿ cut out by the ideal generated by the 2×2 minors of a matrix X =(xi j)i=1,...,d

j=1,...,n

of unknowns, where the jth column corresponds to coordinates in the jth factor.

Start with an elementg∈GL(V), it is represented by an invertiblen×nmatrix over K. It induces a dual mapg^t :V^∗→ V^∗and thus a morphismg :P(V)→ P(V). Fornelementsg1, . . . ,gn∈GL(V), the image of

P(V)−→P(V)^{n g}−−−−−−→¹^×···×^gⁿ P(V)ⁿ

is the subvariety ofP(V)ⁿcut out by the multihomogeneous prime ideal I2((g₁⁻¹, . . . ,g_n⁻¹)(X))⊂K[X],

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where(g⁻₁¹, . . . ,g_n⁻¹)(X)is the matrix whose jth column is given by

g⁻_j¹

⎛

⎜⎝ x1j

...

xd j

⎞

⎟⎠.

Consider a reference lattice L = Re1 + · · · + Red. For any set = {[L1], . . . ,[Ln]}of lattice classes inB⁰_d, we choosegi, such thatgiLi =Lfor all i. The following diagram commutes:

P(V) P(V)ⁿ

RP(Li) P(L)ⁿ

(g1×···×gn)◦

(g₁×···×g_n)

It follows immediately that the Mustafin Variety M() is isomorphic to the subscheme of P(L)ⁿ ∼= (P^d_R⁻¹)ⁿ cut out by the multihomogeneous ideal I2

(g⁻₁¹, . . . ,g_n⁻¹)(X)

∩R[X]inR[X].

Motivated by this choice of coordinates onM(), we proceed with the following construction.

Construction 3.1.Let = {[L1], . . . ,[Ln]} ⊂ B⁰_d be a finite set of lattices.

Moreover, let L = Re1+ · · · +Red be a lattice inV fore1, . . . ,ed ∈ V. As in the choice of coordinates onM()above, we letg1, . . . ,gn ∈GL(V), such that giLi = L. We denoteg =(g1, . . . ,gn). In the following, we construct a variety X_[L],goverkassociated to this data.

(a) To begin with, we construct for open immersiongi :P(V)→P(L)a continuation to a rational map

g_i :P(L)P(L) with

g_i_|P(_V₎ ≡gi. (2)

In order to construct this continuation, we note that sinceLis a free rankdlattice, we have thate1, . . . ,edis a basis ofV. LetGi be the matrix representation of gi with respect to the basise1, . . . ,ed. We define

si =min_μ,ν=1,...,d(val((Gi)_μν))

andG_i =π⁻^sⁱGi. ThenG_iis defined overR, butπ⁻¹G_iis not. We first note thatG_iandGiinduce the same morphism

P(V)−→^gⁱ P(L)

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since they lie in the same equivalence class of PGL(V). Moreover,G_iinduces a linear mapg_i:L →L, which yields the desired rational map

P(L)^gⁱ P(L) (3) satisfying Eq. (2). We note thatg_iis independent of the choice of basise1, . . . ,ed

and the representative of[L].

(b) Secondly, we observe that by the above considerations, we have constructed a continuation ofP(V)−−−−−−−→⁽^g¹^,...,^gⁿ^)◦ P(L)ⁿto the rational map

F_[L],g:P(L)⁽^g¹^,...,^gⁿ^)◦ P(L)ⁿ.

We denote by(g_i)k:P(L)k P(L)kthe rational map over the special fibers induced by Eq. (3). Then, the map over the special fibers induced byF_[L],gis given by

f_[L],g:P(L)k

((g₁)k,...,(g_n)k)◦

P(L)ⁿ_k,

i.e. we have

F_[L],g

|P(L)k

≡ f_[L],g. We define the variety

X_[],g:=Im(f_[L],g).

Remark 3.2.We note that in the notation of Construction3.1, the basise1, . . . ,ed

of L induces a basise₁, . . . ,e_donk^d ≡Lmodπ. With respect to this basis, the map(g_i)k:P(L)k →P(L)kis given by the matrixG_imodπ.

In Construction3.1, our construction ofX_,gdepends on choices of elements gi ∈GL(V), such thatgiLi =L. In the following lemma, we show that while the construction depends on these choices, the variety itself does not.

Lemma 3.3.Let L and given as in Construction 3.1. Moreover, let g = (g1, . . . ,gn) ∈ GL(V)ⁿ and h = (h1, . . . ,hn) ∈ GL(V)ⁿ, such that giLi = hiLi =L. Then, we have

X_[L],g∼=X_[L],h.

Therefore, we will also denote the variety constructed in Construction3.1by X_[L],. Proof. We first observe that sincegiLi = hiLi = L, we have for li = hi ◦ g_i⁻¹thatliL = L. Thus, we obtain isomorphismsP(L)−→^lⁱ P(L), which induce isomorphismsP(L)k (li)k

−−→P(L)k over the special fibers. In particular, we obtain an isomorphism

P(L)ⁿ −−−−−→⁽^l¹^,...,^lⁿ⁾ P(L)ⁿ

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and again an isomorphism over the special fiber P(L)ⁿk

((l₁)k,...,(l_n)k)

P(L)ⁿk.

In the notation of Construction3.1, we consider the following commutative diagram

P(L) P(L)ⁿ

P(L)ⁿ

F_[L],g

F_[_L_],_h

(l₁,...,l_n)

which induces the commutative diagram P(L)k P(L)ⁿ_k

P(L)ⁿ_k

f_[L],g

f_[_L_],_h

((l1)k,...,(ln)k) (4)

As the vertical arrow in Eq. (4) is an isomorphism, the lemma follows.

We are now ready to defineM() andM^r().

Definition 3.4.We fix a reference latticeL, let= {[L1], . . . ,[Ln]} ⊂B⁰_dand gi ∈GL(V)such thatgiLi =L. Moreover, for any[L] ∈B⁰_dwith representative L, we choosehL ∈G L(V)withhLL=L.

For any[L] ∈B⁰_d, we consider the linear isomorphisms h˜_L :P(L)ⁿ−−−−−−−→⁽^h^L^,...,^h^L⁾ P(L)ⁿ. Then, forg_[_L_]=(h⁻_L¹g1, . . . ,h⁻_L¹gn)we define

M() =

[L]∈conv()

h˜L(X_[L],g

[L]).

Moreover, whenis contained in a single apartment, we define M^r()=

[L]∈V()

h˜L(X_[L],g

[L]).

Before relatingM() andM^r()toM()k, we make the following important remarks.

Remark 3.5.(1) By a similar argument as in the proof of Lemma3.3, we see that M() andM^r() do not depend on the reference lattice L or the chosen elementsgi,gj,[L]∈GL(V).

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(2) In the situation of Construction3.1, let the subset= {[L1], . . . ,[Ln]} ⊂B⁰_d be contained in one apartment Aassociated to the basise1, . . . ,ed. Let L be lattice with[L] ∈A, then we may choose the following representations

L=π^m¹Re1+ · · · +π^m^dRed and Li =πⁿⁱ¹Re1+ · · · +πⁿⁱ^dRed.

Therefore, we may choose the matricesGi in Construction3.1as

⎛

⎜⎜

⎝ π^m¹⁻ⁿⁱ¹

...

π^m^d⁻ⁿⁱ^d

⎞

⎟⎟

⎠

Then, we obtainG_i =π⁻^sⁱGi, wheresi =minj=1,...,d(mj−nⁱ_j). In particular, by Remark3.2the map f_[L],gis represented by(H1, . . . ,Hn), where

Hi =G_imodπ=

⎛

⎜⎝ a1

...

ad

⎞

⎟⎠,

where

aj =1, ifmj−nⁱ_j = min

l=1,...,d(ml−nⁱ_l) andaj =0 else. (see Example3.6).

(3) The key observation for the proof of Theorem1.2is that the mapsf_[L],gfactorise as follows:

P^d_k⁻¹ P

k^dKer((g₁)k)

× · · · ×P

k^dKer((gn)k)

P^d_k⁻¹n f_[_L_],_g

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Thus, each varietyX_[L],is a variety of the form X(k^d,W1, . . . ,Wn), where Wi = ker((g_i)k). Note that n

i=1Wi might not be trivial. However, by Remark 2.19 the components of M()k are equidimensional. The vertices withn

i=1Wi = 0contribute varieties of lower dimension and thus are contained in an irreducible component by Lemma3.7. This irreducible component is contributed by a vertex satisfyingn

i=1Wi = 0.

Before we prove Theorem1.2, we illustrate our construction in the following example:

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Example 3.6.By Lemma2.13, there is a natural correspondence between lattice points inZ^dZ1and lattice classes inV overR. In Example 2.2 in [5], the special fiber of the Mustafin variety corresponding to the vertices in Example2.3 was computed to be the union of the following irreducible components

P²× pt×pt, pt×P²×pt, pt×pt×P², P¹

×P¹×pt, P¹×pt×P¹, pt×P¹×P¹

Our construction yields the same variety as seen in the following computations.

Forv=v3, we obtain the map

⎛

⎝1 0 0 0π² 0 0 0 π⁴

⎞

⎠×

⎛

⎝1 0 0 0π 0 0 0 π²

⎞

⎠×id

Modding outπ, we immediately see that

X_[L], =pt×pt×P²,

where [L]is the lattice class corresponding tov3. Analogously, we obtain pt× P²×ptandP²×pt×ptforv=v2andv=v1respectively.

Forv=(0,−1,−4), we obtain the map

⎛

⎝1 0 0 0 1 0 0 0π²

⎞

⎠×

⎛

⎝π 0 0 0 1 0 0 0π

⎞

⎠×

⎛

⎝π²0 0 0 1 0 0 0 1

⎞

⎠.

Modding outπ, we immediately see that

X_[L], =P¹×pt×P¹,

where once again [L] is the lattice class corresponding to v. Analogously, we obtainP¹×P¹×pt andpt×P¹×P¹forv=(0,−1,−3)andv=(0,−2,−4) respectively.

The following Lemma implies the easier containment for the equalities of The- orem1.2.

Lemma 3.7.In the notation of Construction3.1, for any lattice class[L] ∈ B⁰_d, the following relation holds:

X_[L],g⊂M()k.

Moreover, the variety X_[L],gis an irreducible component ofM()kif and only if dim(X[L],g)=dim(M()k).

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Proof. To prove the first assertion, we observe that

X_[L],g=F_[L],g

P(V)

∩P(L)ⁿ_k,

while by the discussion at the beginning of this subsection, we have M()k=F_[L],g(P(V))∩P(L)ⁿ_k =F_[L],g(P(V))∩P(L)ⁿ_k.

SinceF_[L],gis a continuous map, we have thatF_[L],g

P(V)

⊂F_[L],g(P(V)). The second assertion follows immediately by Remark2.19.

To end this subsection, we give a purely tropical description of the map f_[L],g, wheneveris a finite set in one apartmentAand[L] ∈ A. In fact, we associate such a map and thus a variety to any finite subset of the tropical torusRⁿR1. Construction 3.8.Let= {[v1], . . . ,[vn]} ⊂RⁿR1be a finite subset of the tropical torus, wherevi =(v_i¹, . . . , v_iⁿ). For anyv∈RⁿR1, wherev=(v¹, . . . , vⁿ), we define rational maps

f_[v], :P^d_k⁻¹(P^d_k⁻¹)ⁿ where the mapg_jto the jth factor is given by

⎛

⎜⎝ a1

...

ad

⎞

⎟⎠,

where am = 1, ifv^m_j −v^m =minl(v^l_j −v^l)andam =0 otherwise. We further define X_[v],=Im(f_[v],)and define

M^r()=

[v]∈V()

X_v,.

For= {[L1], . . . ,[Ln]} ⊂B⁰_dbe finite set of lattice classes in one apartmentA and let= {[v1], . . . ,[vn]}be the corresponding subset of the tropical torus, then by the discussion in Remark3.5(2), we have

M()=M()and M^r()=M^r().

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3.2. Proof of Theorem1.2(1)

This subsection is devoted to proving Theorem1.2(1). The strategy of the proof is to show that

M()k=M()

for convex subset of lattice classes and recover the general case by using [5, Lemma 2.4].

We fix a convex subset= {[L1], . . . ,[Ln]} ⊂B⁰_d. By Proposition2.18(2), the special fiber consists ofnirreducible components. Thus, in order to prove

M()k=M()

we need to show thatM() hasnpairwise distinct irreducible components.

Let[Li] ∈and chooseg1, . . . ,gn∈GL(V)withgjLj =Li. In particular, we may choosegi =id. In the notation of Construction3.1, we consider for each

j∈ [n]the following commutative diagram P(Li)k P(Li)ⁿ_k

P(Li)k f_[L],g

id p_j

where pj denotes the projection to the j-th factor. There is a dense open subset Uj ⊂P(Li)k, such thatpj ◦ f_[Li],g|U ≡(g_j)k.

In particular, we obtain pj◦ f_[L_i],g|U ≡ id. In other words, for j =i the vertical arrow is a birational map betweenX_[Li],gandP(Li). Therefore, we have that dim(X_[Li],g)=d−1. Thus, by Lemma3.7it follows thatX_[Li],gis an irreducible component ofM()k, which is in fact thei-th primary component. Since by Propo- sition2.18(4), the primary components are pairwise distinct, this completes the proof of Theorem1.2(1) for convex sets.

Now, letbe any subset ofB⁰_d. Then by lemma 2.4 of [5], we can compare the special fibers ofM()andM(conv()): LetCbe an irreducible component ofM()k, then there exists a unique irreducible componentC˜ ofM(conv())k

which projects ontoCunder

p :

L∈conv()

P(L)k →

L∈

P(L)k.

The key idea in proving Theorem 1.2 (1) for arbitrary subsets ⊂ B⁰_d is to observe that Construction 3.1 commutes with projections. More precisely, let L be any lattice, = conv() = {[L1], . . . ,[Ln],[Ln+1], . . . ,[L_ν]}and we denote g1, . . . ,g_ν with giLi = L. Moreover, we set g = (g1, . . . ,g_ν) and

˜

g=(g1, . . . ,gn)the following diagramm commutes:

P(L)k

L∈P(L)k

L∈P(L)k f_[L],g

f_[_L_],˜_g

p