FAITHFUL TROPICALIZATION OF HYPERTORIC VARIETIES
by
MAX B. KUTLER
A DISSERTATION
Presented to the Department of Mathematics
and the Graduate School of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
June 2017
DISSERTATION APPROVAL PAGE
Student: Max B. Kutler
Title: Faithful Tropicalization of Hypertoric Varieties
This dissertation has been accepted and approved in partial fulfillment of the
requirements for the Doctor of Philosophy degree in the Department of Mathematics
by:
Nicholas Proudfoot Chair
Alexander Polishchuk Core Member
Vadim Vologodsky Core Member
Benjamin Young Core Member
Hank Childs Institutional Representative
and
Scott L. Pratt Dean of the Graduate School
Original approval signatures are on file with the University of Oregon Graduate
School.
Degree awarded June 2017
ii
c© 2017 Max B. Kutler
iii
DISSERTATION ABSTRACT
Max B. Kutler
Doctor of Philosophy
Department of Mathematics
June 2017
Title: Faithful Tropicalization of Hypertoric Varieties
The hypertoric variety MA defined by an arrangement A of affine hyperplanes
admits a natural tropicalization, induced by its embedding in a Lawrence toric variety.
In this thesis, we explicitly describe the polyhedral structure of this tropicalization
and calculate the fibers of the tropicalization map. Using a recent result of Gubler,
Rabinoff, and Werner, we prove that there is a continuous section of the tropicalization
map.
iv
CURRICULUM VITAE
NAME OF AUTHOR: Max B. Kutler
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene, OR
Harvey Mudd College, Claremont, CA
DEGREES AWARDED:
Doctor of Philosophy, Mathematics, 2017, University of Oregon
Master of Science, Mathematics, 2014, University of Oregon
Bachelor of Science, Mathematics, 2011, Harvey Mudd College
AREAS OF SPECIAL INTEREST:
Tropical and Non-Archimedean Geometry
Toric Varieties
Hypertoric Varieties
Matroids
PROFESSIONAL EXPERIENCE:
Graduate Teaching Fellow, University of Oregon, 2011-2017
PUBLICATIONS:
M. B. Kutler and C. R. Vinroot. On q-analogs of recursions for the number of
involutions and prime order elements in symmetric groups. J. Integer Seq.,
13(3) Article 10.3.6:12 pages, 2010.
v
ACKNOWLEDGEMENTS
First and foremost, I thank my advisor, Nick Proudfoot, for his guidance
and many helpful suggestions. The work presented here benefitted greatly from
conversations with Angie Cueto and Dhruv Ranganathan.
I am forever indebted to my teachers. I thank Tom Arend, Jeff Hammonds,
Nancy Judy, Jack Reynolds, Jeremy Shibley, and Kathleen Walsh for challenging
me in at Rex Putnam High School, and I thank Art Benjamin, Andy Bernoff,
Jon Jacobsen, Dagan Karp, Rachel Levy, Mike Orrison, Francis Su, and Ursula
Whitcher for warmly welcoming me into the mathematical community at Harvey
Mudd College. I am particularly grateful to Joe Roberts, for showing me the beauty
of mathematics for the first time; to Mike Orrison, for his mentorship; and to Dagan
Karp, for introducing me to tropical geometry and the wonderful community of
tropical geometers.
I thank Austin Anderson, Thomas Avila, Taylor Brown, Charlie Hankin, Nick
Howell, Esther Hwang, Ann Johnston, Craig Levin, Bronson Lim, Rob Muth, Michael
Park, Dhruv Ranganathan, Travis VanKrause, Adam Welly, and David Yabu, for their
support and friendship. I thank Jaimie Esaki for her love and companionship.
Finally, I thank my family, especially my parents Stephanie and Keith, for loving
and believing in me always.
vi
For Pa
vii
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
II. TORIC VARIETIES AND HYPERTORIC VARIETIES . . . . . . . . 6
2.1. Toric varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2. Hyperplane arrangements and matroids . . . . . . . . . . . . . 8
2.3. The Lawrence toric variety of an arrangement . . . . . . . . . 13
2.4. The hypertoric variety of a an arrangement . . . . . . . . . . . 19
III. ANALYTIFICATION AND TROPICALIZATION . . . . . . . . . . . 23
3.1. Affinoid algebras and analytic spaces . . . . . . . . . . . . . . 23
3.2. Tropicalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
IV. THE TROPICALIZATION OF A HYPERTORIC VARIETY . . . . . 30
4.1. Description of the tropicalization . . . . . . . . . . . . . . . . . 30
4.2. Fibers of tropicalization . . . . . . . . . . . . . . . . . . . . . . 35
viii
Chapter Page
V. FAITHFUL TROPICALIZATION . . . . . . . . . . . . . . . . . . . . 38
5.1. The theorem of Gubler-Rabinoff-Werner . . . . . . . . . . . . 38
5.2. Faithful tropicalization of hypertoric varieties . . . . . . . . . . 40
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
ix
CHAPTER I
INTRODUCTION
In this dissertation, we study the tropicalization of the hypertoric variety MA
defined by an arrangement A of affine hyperplanes. Hypertoric varieties were first
studied by Bielawski and Dancer [BD00]. They are “hyperka¨hler analogues” of toric
varieties, and examples of conical symplectic resolutions. The relationship between
the variety MA and the arrangement A is analogous to that between a semiprojective
toric variety and its polyhedron. See, e.g., [Pro08] for an overview of this relationship.
The hypertoric variety MA is not, in general, a toric variety. However, it is naturally
defined as a closed subvariety of a toric variety, the Lawrence toric variety BA. The
Lawrence embedding allows us to define a tropicalization of MA.
Given a closed embedding of a variety X in a toric variety, there is a
corresponding tropicalization Trop(X), which is the continuous image of the
Berkovich space Xan under the tropicalization map. The tropicalization may be
endowed with the structure of a finite polyhedral complex. A single variety X may
yield many distinct tropicalizations, each given by a different choice of embedding
into a toric variety. When we speak of the tropicalization of X, it is always with
respect to a chosen embedding.
By a result of Foster, Gross, and Payne [FGP14, Pay09], if X has at least one
embedding into a toric variety, then the inverse system of all such embeddings induces
an inverse system of tropicalizations, and the limit of this system in the category
of topological spaces is Xan. This raises the question of how well a particular
tropicalization approximates the geometry of the analytic space. To this end, a
tropicalization is said to be faithful if the tropicalization map Xan → Trop(X)
1
admits a continuous section, realizing Trop(X) as (homeomorphic to) a closed subset
of Xan.
If X is embedded in a torus, then Trop(X) is the support of a finite polyhedral
complex, which is balanced when the polyhedra are weighted by tropical multiplicity.
Gubler, Rabinoff, and Werner have proved that such a tropicalization is faithful if all
tropical multiplicities are equal to one [GRW16]. Moreover, in this case, the section
of tropicalization is uniquely defined. This generalizes work of Baker, Payne, and
Rabinoff, who obtained the first results on faithful tropicalization in the case where
X is a curve [BPR16].
In the more general situation where X is embedded in a toric variety, Trop(X) is
the union of the tropicalizations Trop(X∩O) as O ranges over all torus orbits. In this
case, tropical multiplicity one is no longer sufficient to imply faithfulness: it is possible
that the continuous sections defined on each of the strata Trop(X ∩O) do not glue to
a continuous section on the entire tropicalization [GRW15, Example 8.11]. However,
Gubler, Rabinoff, and Werner [GRW15, Theorem 8.14] have recently proved that if X
is embedded in a toric variety with dense torus T , then the resulting tropicalization
is faithful, with uniquely defined continuous section, if certain conditions on the
embedding and the polyhedral structure of the resulting tropicalization are satisfied.
We state a simplified version of this result as Theorem 5.1. While the first results on
faithful tropicalizations [CHW14, DP16], required careful study of Berkovich spaces
and their skeleta, that analysis is now absorbed into the proof of this theorem,
so that faithfulness may be checked by exclusively working “downstairs,” with the
tropicalization.
In this thesis, we apply this this theorem of Gubler, Rabinoff, and Werner
to prove that an arbitrary hypertoric variety MA is faithfully tropicalized by its
2
embedding in the Lawrence toric variety MA. We thus obtain many new examples,
in every even dimension, of varieties which are faithfully tropicalized by a “natural”
embedding into a toric variety. These examples include the cotangent bundle of
projective space and the cotangent bundle of a product of projective spaces, as well
as many singular varieties. To our knowledge, this is the first application of Gubler,
Rabinoff, and Werner’s theorem to a class of tropicalizations for which faithfulness
was previously unknown.
Furthermore, we shall see that, in all but the most trivial case, the hypertoric
variety MA in its Lawrence embedding does not meet all torus orbits in the expected
dimension (Corollary 2.12). This is in contrast to several other known examples
of “nice” tropicalizations, including the moduli space M0,n of stable rational curves
[GM10, Tev07], some alternate compactifications of M0,n [CHMR16], and the space of
logarithmic stable maps to a projective toric variety [Ran15]. By [GRW15, Corollary
8.15], a variety which meets all torus orbits in the expected dimension, or not at all,
is faithfully tropicalized if it has multiplicity one everywhere. Since this result is not
available to us, we must find a polyhedral structure on Trop(MA) to work with. Our
first main result describes such a polyhedral structure in terms of the combinatorics
of the defining arrangement A.
Theorem 4.1. The tropicalization Trop(MA) of the hypertoric variety is the union
of cones C
(F,R)
F indexed by a flat F of M, a face R of the localization AF , and a flag
of flats F in the restriction MF . These cones satisfy
dimC
(F,R)
F = d− codimR+ `(F).
3
This gives Trop(MA) the combinatorial structure of a finite polyhedral complex, under
the closure relation
C
(F ′,R′)
F ′ ⊆ C(F,R)F (1.1)
if and only if the following conditions hold:
– F ⊆ F ′;
– R′ ⊆ R;
– F ′ is a flat in F , and truncF ′(F) is a refinement of F ′.
Moreover, this gives each stratum Trop(MA)∩ N˜R(σF,R) the structure of a polyhedral
fan, which is balanced when all cones are given weight one.
Equipped with Theorem 4.1, we can describe the interplay between the fan of
the toric variety BA and the fan Trop(MA ∩ T˜ ), where T˜ is the dense torus of BA.
The cones of each of these two fans are described in terms of the combinatorics of
the arrangement A. By examining these combinatorics, we see that the conditions of
Theorem 5.1 are satisfied, proving the tropicalization is faithful.
Theorem 5.4. There is a unique continuous section of the tropicalization map
ManA → Trop(MA).
The rest of the dissertation is outlined as follows. In Chapter II, we recall basic
facts about toric geometry and hyperplane arrangements, and we define the Lawrence
toric variety and hypertoric variety associated to an arrangement. We also prove some
technical lemmas. Chapter III serves as a brief overview of non-Archimedean analytic
spaces and tropicalization. We describe the tropicalization of a linear space (Example
3.1), which we will later use to define the polyhedral structure on Trop(MA). In
4
Chapter IV, we prove Theorem 4.1. We calculate the fibers of tropicalization, and
show that they are affinoid subdomains of ManA containing a unique Shilov boundary
point. Finally, in Chapter V we state Theorem 5.1, due to Gubler, Rabinoff, and
Werner, and use it to prove Theorem 5.4.
5
CHAPTER II
TORIC VARIETIES AND HYPERTORIC VARIETIES
In this chapter, we briefly review the theories of toric varieties and hyperplane
arrangements, and we set notation and definitions we will use in the sequel. We define
the Lawrence toric variety and the hypertoric variety associated to an arrangement.
We note that the Lawrence toric variety is defined here in terms of its fan, whereas
typically in the literature it is defined as a GIT quotient. The equivalence of our
approach with the standard definition follows from Lemmas 2.6, 2.7, and 2.8. For
further background reading on these topics, the interested reader is referred to [Ful93]
and [CLS11] on toric varieties; [Oxl11] on matroids; [BD00] and [HS02] on Lawrence
toric varieties and hypertoric varieties; and [PW07] and [Pro08] on the relationship
between these varieties and the associated hyperplane arrangements.
Throughout the remainder of this dissertation, we fix a lattice M ∼= Zd and an
algebraically closed field K, complete with respect to a non-Archimedean valuation
ν : K → R ∪ {∞}, which may be trivial. The dual lattice to M is N = Hom(M,Z),
and we set MR = M ⊗Z R and NR = N ⊗Z R = Hom(M,R). Let T = SpecK[M ] be
the split K-torus with character lattice M and cocharacter lattice N .
2.1. Toric varieties
Let Σ be a (pointed) rational polyhedral fan in NR. Each cone σ ∈ Σ defines an
affine toric variety Yσ = SpecK[σ
∨ ∩M ] with dense torus T . For τ ≺ σ in Σ, Yτ
is naturally an open subvariety of Yσ. Gluing along these identifications, we obtain
the T -toric variety YΣ =
⋃
σ∈Σ Yσ defined by Σ.
6
The action of T partitions YΣ into torus orbits. These orbits are in bijection with
cones in Σ, with σ ∈ Σ corresponding to the orbit O(σ) = SpecK[σ⊥∩M ]. The orbit
O(σ) is a torus of dimension equal to codimσ, with character lattice M(σ) = σ⊥∩M
and cocharacter lattice N(σ) = N/(〈σ〉 ∩N), where 〈σ〉 = Rσ is the linear span of σ
in NR. We have the set-theoretic decomposition YΣ =
⊔
σ∈Σ O(σ).
We set MR(σ) = M(σ) ⊗Z R and NR(σ) = N(σ) ⊗Z R. If τ ≺ σ, then we have
the projection NR(τ)→ NR(σ), which by abuse of notation we denote piσ. For a cone
τ ∈ Σ, the orbit closure O(τ) is a toric variety with dense torus O(τ). Its fan is the
set of cones star(τ) = {piτ (σ) | σ  τ} in NR(τ).
A homomorphism of tori φ : T → T ′ is uniquely determined by the corresponding
homomorphism φ∗ : M ′ → M of character lattices, or equivalently by the dual
homomorphism φ∗ : N → N ′ of cocharacter lattices. Note that φ is injective if and
only if φ∗ is surjective (if and only if φ∗ is injective). Dually, φ is surjective if and
only if φ∗ is injective (if and only if φ∗ is surjective). We say that an injective or
surjective morphism of tori is split if the corresponding map of character lattices (or,
equivalently, of cocharacter lattices) is split.
If Σ and Σ′ are fans in NR and N ′R, respectively, then a homomorphism φ : T → T ′
extends to an equivariant morphism of toric varieties YΣ → YΣ′ if and only if for each
cone σ ∈ Σ there exists σ′ ∈ Σ′ such that φ∗(σ) ⊆ σ′.
Following [Gro15], we define a linear subvariety L of the torus T to be a
subvariety in some choice of torus coordinates. That is, there exists an isomorphism
M ∼= Zd, inducing K[M ] ∼= K[Zd] = K[x±11 , . . . , x±1d ], such that the ideal of L is
generated by linear forms in the xi.
Lemma 2.1. If φ : T → T ′ is a split surjection of tori, and L ⊆ T ′ is a linear
subvariety, then φ−1(L) is a linear subvariety of T with codimT φ−1(L) = codimT ′ L.
7
Proof. Let {x1, . . . , xd} be an integral basis of M ′. Then {φ∗(x1), . . . , φ∗(xd)} is
linearly independent in M because φ∗ is injective, and each φ∗(xi) is primitive because
xi is primitive and φ
∗ is split. Therefore, {φ∗(x1), . . . , φ∗(xd)} may be extended to an
integral basis of M .
If the ideal of L is generated by linear forms in the variables xi, then the ideal of
φ−1(L) is generated by linear forms in the variables φ∗(xi). This shows that φ−1(L) is
a linear subvariety of T . Moreover, by injectivity of φ∗, the ideal of φ−1(L) is generated
by the same number of independent linear forms as is the ideal of L, proving that
codimT φ
−1(L) = codimT ′ L.
2.2. Hyperplane arrangements and matroids
Given a finite set E, a tuple a = (ae) ∈ NE of nonzero primitive elements, and
r = (re) ∈ ZE, we define the corresponding arrangement A = A(a, r) to be the
multiset of affine integral hyperplanes
He = {m ∈MR | 〈m, ae〉+ re = 0} (e ∈ E)
in MR. If a generates the lattice N , then a is a primitive spanning configuration.
Each hyperplane He is cooriented by the integral normal vector ae, with “positive”
and “negative” closed halfspaces,
H+e = {u ∈MR | 〈u, ae〉+ re ≤ 0}
and
H−e = {u ∈MR | 〈u, ae〉+ re ≤ 0},
8
respectively.
The arrangement A is simple if the intersection of any k hyperplanes is either
empty or has codimension k, and A is unimodular if every collection of d linearly
independent normal vectors {ae1 , . . . , aed} is an integral basis of N . An arrangement
which is both simple and unimodular is smooth.
If r = 0, so that each hyperplane He is a linear subspace of MR, then we call the
arrangement central. Given A = A(a, r), we let A0 = A(a, 0) be the centralization
of A. We denote by (He)0 the translation of He to the origin.
For each relation
∑
e∈E ceae in N satisfied by the configuration a, we have the
corresponding linear form
∑
e∈E
cexe ∈ K[xe | e ∈ E].
We let L = L(a) be the d-dimensional linear subspace of AE defined by the vanishing
of these linear forms. The dependencies among points in the configuration a are
encoded in the underlying matroid M = M(a) on E. A matroid on E is a
combinatorial structure, defined by declaring a collection of subsets of E to be
independent (the subsets which are not independent are called dependent). The
collection of independent subsets must satisfy certain axioms inspired by the linear
algebraic notion of linear independence.
The rank function of M defines the rank rkS of a subset S ⊆ E to be the
dimension of the subspace of NR spanned by {ae | e ∈ S}. Equivalently, the rank of
S is equal to the codimension of the intersection
⋂
e∈S(He)0 of all central hyperplanes
indexed by S. Observe that the rank of M, defined to be rkE, is equal to d if and
only if a is a primitive spanning configuration.
9
A subset F ⊆ E is a flat of M if it is maximal for its rank; that is, if S ⊇ F ,
then either S = F or rkS > rkF . A flag of flats in M is a chain
F = (∅ = F0 ( F1 ( · · · ( Fk−1 ( Fk = E)
where each Fi is a flat. The length of such a flag, denoted `(F), is the number k
of nonempty flats in F . Since the rank must increase at each step, a maximal flag
of flats will have length equal to the rank of M. By inserting flats, any flag may be
refined into a maximal flag.
It is clear from the definition that the collection of all flats is uniquely determined
by the rank function. It is a basic result of matroid theory that the reverse is true,
and that the rank function and the lattice of flats each individually determines the
collection of all independent sets. Thus, a matroid may be “cryptomorphically”
defined in terms of either its rank function or its flats, with each of these structures
being subject to appropriate axioms. See [Oxl11] for these axioms and other
equivalent characterizations of matroids.
Given a flat F of M, we define the restriction of the central arrangement A0
to F , denoted AF0 , to be the arrangement of hyperplanes {(He)0∩HF | e /∈ F} in the
vector space HF =
⋂
e∈F (He)0. We let L
F ⊆ AErF denote the corresponding linear
subspace, obtained from L by setting xe = 0 for all e ∈ F . The underlying matroid
of AF0 , denoted MF , is the matroid on E r F obtained from M by deleting F . The
flats ofMF are precisely the sets F ′rF for F ′ ⊇ F a flat ofM; we therefore identify
flats of MF with flats of M which contain F , and flags of flats in MF with flags in
M which begin at F .
As a dual construction, we define the localization of the arrangement A at any
subset S ⊆ E to be the arrangement of hyperplanes {He | e ∈ S} in the vector space
10
MR. The centralization of AS is (A0)S, and we writeMS for its underlying matroid.
The ground set of MS is S, and its flats are precisely those flats of M which are
contained in S.
Remark 2.2. There are notable differences in our definitions of restriction and
localization.
(1) While it is possible to define the restriction to a non-flat S, the resulting matroid
will contain loops. In this document, we shall only need to restrict to flats, and
so we will limit our attention to that case. On the other hand, in order to
combinatorially describe the fan of the Lawrence toric variety in Section 2.3, it
will be necessary to localize A at every subset S ⊆ E.
(2) There is no canonical way to lift the construction of the restriction AF0 to
the non-central arrangement A, which is why we defined restriction on the
centralization. By contrast, the localization of A0 at S uniquely determines the
localization of A at S.
Remark 2.3. We have defined localization so that AS is an arrangement in MR.
As a result, the normal vectors of the hyperplanes in AS do not necessarily span N ,
even if the original arrangement A was defined by a primitive spanning configuration.
Alternatively, one may wish to define AS to be an arrangement in MR/HS; the normal
vectors of this arrangement will then be spanning if the normal vectors of A are
spanning (see, e.g., [PW07, §2]). However, it will be convenient for our purposes to
have all localizations living in the same vector space MR.
11
An arrangement A assigns a sign vector sgnA(m) ∈ {+, 0,−}E to each m ∈MR,
via
sgnA(m)e =

+ if m ∈ H+e rHe,
0 if m ∈ He,
− if m ∈ H−e rHe.
A nonempty fiber of sgnA : MR → {+, 0,−}E is called a face of the arrangement
A. A vertex of A is a face consisting of a single point. Each face R defines sets
E+(R) = {e ∈ E | R ⊆ H+e } and E−(R) = {e ∈ E | R ⊆ H−e }. Note that
E = E+(R) ∪ E−(R). We set E0(R) = E+(R) ∩ E−(R) = {e ∈ E | R ⊆ He}.
Notice that the closure of a face is the intersection of all halfspaces which contain
it:
R =
( ⋂
e∈E+(R)
H+e
)
∩
( ⋂
e∈E−(R)
H−e
)
. (2.1)
It follows that the codimension of R in MR is the codimension of the intersection of
all hyperplanes containing it:
codimR = codim
⋂
e∈E0(R)
He = codim
⋂
e∈E0(R)
(He)0 = rkME0(R). (2.2)
The above discussion of faces applies to localizations of A as well. If S ⊆ E is
any subset, then a face R of AS determines sets S+(R), S0(R), and S−(R), with
S = S+(R) ∪ S−(R) and S0(R) = S+(R) ∩ S−(R). Furthermore, R and codimR
are computed as in (2.1) and (2.2), respectively, with E replaced by S.
Lemma 2.4. Let S ⊆ E. If R is a face of A and R′ is a face of the localization AS,
then R ⊆ R′ if and only if E+(R) ⊇ S+(R′) and E−(R) ⊇ S−(R′).
12
Proof. Suppose R ⊆ R′. Since the halfspaces H+e and H−e are closed, any halfspace
which contains R′ also contains R′, hence contains R. That is, E+(R) ⊇ S+(R′) and
E−(R) ⊇ S−(R′)
Conversely, suppose E+(R) ⊇ S+(R′) and E−(R) ⊇ S−(R′). Then for each
e ∈ S+(R′), we also have e ∈ E+(R) and therefore R ⊆ H+e . Similarly, R ⊆ H−e
for each e ∈ S−(R′). By the formula (2.1) applied to the face R′, we conclude that
R ⊆ R′.
2.3. The Lawrence toric variety of an arrangement
We now describe how an arrangement A = A(a, r) in MR, where a is a primitive
spanning configuration, gives rise to a toric variety BA, called the Lawrence toric
variety. While we continue to fix the torus T = SpecK[M ], the variety BA is not a
T -toric variety; rather, it is a T˜ -toric variety, where T˜ is a (split) extension of GEm by
T .
The configuration a defines a homorphism ZE → N , where the generator δe is
mapped to ae. This map is surjective because a is spanning, and its kernel Λ is a
lattice because a is primitive.
Let ∆: ZE → ZE ⊕ ZE denote the antidiagonal embedding: If we denote by δ+e
the generators of the first copy of ZE and by δ−e the generators of the second copy
of ZE, then ∆(δe) = δ+e − δ−e . The composition of ∆ with the ι : Λ → ZE gives an
inclusion of Λ into ZE⊕ZE. Let N˜ denote the quotient, so that we have the following
commutative diagram with exact rows.
0 Λ ZE N 0
0 Λ ZE ⊕ ZE N˜ 0
id
ι
∆
a
(2.3)
13
The images ρ+e and ρ
−
e in N˜ of the generators δ
+
e and δ
−
e , respectively, provide a
natural spanning set of N˜ . Note that N˜ is a lattice of rank |E|+ d, and we have the
short exact sequence
0 N N˜ ZE 0 (2.4)
where N → N˜ is the vertical map from (2.3) and N˜ → ZE is defined by ρ±e 7→ δe. In
particular, for each relation
∑
e∈E ceae = 0 in N , we have
∑
e∈E
ce(ρ
+
e − ρ−e ) = 0
in N˜ , and these are all relations among the generators ρ±e .
For each pair (S,R), where S ⊆ E and R is a face of the localization AS, define
σS,R to be the cone in N˜R with rays generated by the integral vectors
{ρ+e | e ∈ S+(R)} ∪ {ρ−f | f ∈ S−(R)}.
Let ΣA = {σS,R | S ⊆ E and R is a face of AS} be the collection of all such cones.
Lemmas 2.6, 2.7, and 2.8 establish that ΣA is a fan in N˜R, called the Lawrence fan,
whose maximal cones are precisely the (|E| + d)-dimensional cones σE,ξ indexed by
vertices ξ of A. Let M˜ be the dual lattice to N˜ , and let T˜ = SpecK[M˜ ] be the
Lawrence torus. Then the T˜ -toric variety BA = YΣA is called the Lawrence toric
variety associated to A.
Remark 2.5. In the literature, BA is usually defined as the GIT quotient of the
cotangent bundle T ∗AE ∼= AE×AE by G = SpecK[Λ∨] with respect to the character
α = ι∗(r) ∈ Λ∨. Hausel and Sturmfels [HS02, Proposition 4.3] identify the fan of this
14
GIT quotient as the collection of cones σE,ξ together with all of their faces. Hence,
this fan is ΣA.
However, the maximal cones in [HS02] are indexed not by vertices of A, but by
bases of the dual matroid ofM. Given such a basis, the intersection of all hyperplanes
He indexed by the dual basis of M yields a vertex of A. Every vertex arises in this
way, but unless A is simple, one vertex may arise from multiple bases. (In the extreme
example where A is central, there may be many bases, but each produces the single
vertex of A.) Therefore, it is more natural to index the maximal cones by vertices of
the arrangement, as we do here.
To our knowledge, no description of the non-maximal cones of ΣA appears in the
literature.
Lemma 2.6. The cone σS,R has dimension |S|+ codimR.
Proof. It suffices to prove this in the case S = E.
The dimension of σE,R is equal to the dimension of its real span 〈σE,R〉 ⊆ N˜R.
Define
V1 = R〈ρ+i , ρ−j | i ∈ E+(R)r E0(R), j ∈ E−(R)r E0(R)〉
and
V2 = R〈ρ+i , ρ−j | i, j ∈ E0(R)〉.
Then we clearly have 〈σE,R〉 = V1 + V2.
Since every relation among the elements ρ±e is of the form
∑
e∈E
ce(ρ
+
e − ρ−e ) = 0,
15
any nontrivial linear dependence among the generators of σE,R must occur among the
vectors {ρ+i , ρ−j | i, j ∈ E0(R)} generating V2. It follows that dimV1 = |E r E0(R)|
and 〈σE,R〉 = V1 ⊕ V2.
Choose any basis B ⊆ E0(R) of the matroid ME0(R). We claim that
{ρ+i , ρ−j | i ∈ B, j ∈ E0(R)} (2.5)
is a basis for V2. Indeed, any nontrivial linear dependence among these generators
must occur among the subset {ρ+i , ρ−j | i, j ∈ B}, but any such dependence must be
trivial because B is independent. Thus, the set in (2.5) is linearly independent. On
the other hand, for any i ∈ E0(R)r B, there is a unique expression ai =
∑
b∈B cbab,
where cb ∈ Z. This implies
ρ+i = ρ
−
i +
∑
b∈B
cb(ρ
+
b − ρ−b ),
and therefore the set in (2.5) spans V2. By (2.2), we have codimR = rkME0(R).
Therefore,
dimV2 = |B|+ |E0(R)| = rkME0(R) + |E0(R)| = codimR+ |E0(R)|,
and hence
dimσE,R = dimV1 + dimV2 = codimR+ |E|.
Lemma 2.7. Every face of σS,R is of the form σS′,R′, and we have the face relations
σS′,R′ ≺ σS,R if and only if S ⊇ S ′ and R ⊆ R′.
16
Proof. Again, we may assume that S = E and R is a face of A.
Suppose τ ≺ σE,R is a face. Then τ = u⊥ ∩ σE,R for some u ∈ σ∨E,R ⊆ M˜R. We
also know that τ is generated by a subset of the rays of σE,R. That is, τ is the cone
generated by
{ρ+i | i ∈ I} ∪ {ρ−j | j ∈ J}
for some subsets I ⊆ E+(R) and J ⊆ E−(R). Set S ′ = I ∪ J .
Note that the dual ∆∗ of the antidiagonal map of (2.3) maps M˜R onto MR. Fix
any point p ∈ R. Set
m = ∆∗(u) + p ∈MR,
where  > 0 will be fixed shortly. Observe that
〈m, ae〉+ re = 〈u,∆(ae)〉+ 〈p, ae〉+ re = (〈u, ρ+e 〉 − 〈u, ρ−e 〉) + (〈p, ae〉+ re)
for any e ∈ E. Since the sign of 〈p, ae〉 + re for each e is determined by R, and E is
a finite set, it is possible to choose  > 0 sufficiently small so that
sgnAS′ (m)e =

+ if e ∈ I r (I ∩ J),
0 if e ∈ I ∩ J,
− if e ∈ I r (I ∩ J).
That is, m lies in a face R′ of AS′ such that S ′+(R′) = I and S ′−(R′) = J . This
shows that τ = σS′,R′ . Since I ⊆ E+(R) and J ⊆ E−(R), Lemma 2.4 implies that
R ⊆ R′.
17
Conversely, let S ′ ⊆ E and let R′ be a face of AS′ with R ⊆ R′. We shall
show that σS′,R′ is a face of σE,R. By Lemma 2.4, we have E+(R) ⊇ S ′+(R′) and
E−(R) ⊇ S ′−(R′).
Fix some p ∈ R and m ∈ R′, and for each k ∈ E r S ′, choose positive real
numbers ck and dk such that ck − dk = 〈m− p, ak〉. Define a functional on ZE ⊕ ZE
by
u =
∑
i∈S′−(R′)
〈m− p, ai〉x+i −
∑
j∈S′+(R′)rS′0(R′)
〈m− p, aj〉x−j +
∑
k∈ErS′
(ckx
+
k − dkx−k ),
where {x±e } is the dual basis to {δ±e }. A priori, u ∈ R〈x±e 〉. However, we have defined
u so that
〈∆∗(u), δe〉 = 〈u,∆(δe)〉 = 〈u, δ+e − δ−e 〉
is equal to 〈m − p, ae〉 for all e ∈ E. Since {ae, e ∈ E} spans NR, it follows that
∆∗(u) = m− p ∈MR, and therefore u ∈ M˜R. By construction, we have u ∈ σ∨E,R and
u⊥ ∩ σE,R = σS′,R′ , proving that σS′,R′ ≺ σE,R.
Lemma 2.8. If ξ and ζ are two vertices of A, then σE,ξ ∩ σE,ζ is a cone of the form
σS,R. Moreover, every cone σS,R is the face of σE,ξ for some vertex ξ.
Proof. We say that a hyperplane He separates the vertices ξ and ζ if ξ and ζ lie in
opposite halfspaces of He and neither lies on He. Let S ⊆ E be the set of all e such
that He does not separate ξ and ζ. Then there is a unique face R of AS such that
R contains every point in the interior of the line segment connecting ξ and ζ, and
σE,ξ ∩ σE,ζ = σS,R.
Conversely, if σS,R is any cone in ΣA, then R is a union of faces of A. Since a
is spanning, each face of A contains at least one vertex in its closure, so R has this
18
property as well. Let ξ be any vertex of A contained in R. Therefore σS,R ≺ σE,ξ by
Lemma 2.7.
2.4. The hypertoric variety of a an arrangement
Consider the surjection N˜ → ZE in (2.4). Tensoring with R, we obtain a linear
map Φ∗ : N˜R → RE with Φ∗(σS,R) = RS≥0. We thus obtain a surjective map of toric
varieties Φ: BA → AE. We define the hypertoric variety of the arrangement
A, denoted MA, to be the preimage of the linear space L under the map Φ. It is
irreducible of dimension 2d.
Remark 2.9. Since ZE⊕ZE surjects onto N˜ (cf. (2.3)), the map Φ lifts to a surjection
T ∗(AE) ∼= AE ×AE → AE (this is the moment map for the hamiltonian action of GEm
on AE). If we work over K = C, then the complex points of AE/L can be identified
with the dual Lie algebra of the torus G = SpecK[Λ∨], and the composition of Φ with
the projection AE → AE/L is then the moment map µ for the hamiltonian action
of G on T ∗AE. This endows MA = µ−1(0) α G with a canonical Poisson structure.
Arbo and Proudfoot have proved that this Poisson structure makes MA a symplectic
variety in the sense of Beauville [AP16, Proposition 4.14]. The torus T acts on MA
via its antidiagonal embedding in the Lawrence torus T˜ . This action is hamiltonian
with moment map Φ|MA .
By [BD00, Theorems 3.2 & 3.3], the hypertoric variety MA has at worst orbifold
singularities if and only if the arrangement A is simple, and is smooth if and only if
A is smooth. The variety MA0 is affine, and if A is simple then MA → MA0 is an
orbifold resolution of singularities. By [HP04, Lemma 2.2], MA is independent of the
coorientations of the hyperplanes in A.
19
Example 2.10. If A is the arrangement of coordinate hyperplanes in MR, then the
associated hypertoric variety is MA ∼= T ∗Ad ∼= A2d. The polytope of Pd is a d-
simplex in MR cut out by d+ 1 affine hyperplanes. The hypertoric variety associated
to the arrangement consisting of these hyperplanes is isomorphic to T ∗Pd. The
same procedure realizes the cotangent bundle of a product of projective spaces as
a hypertoric variety. In general, if Y is a projective toric variety with polyhedron P ,
then the arrangement of hyperplanes cutting out P defines a hypertoric variety which
contains T ∗Y as a dense open subset [BD00, Theorem 7.1].
Since Φ∗(σS,R) = RS≥0, the restriction of Φ to O(σS,R) is a surjection onto the
torus orbit GErSm in AE. This surjection is split because T˜ → GEm is split. Moreover,
Φ−1(GErSm ) =
⊔
R
O(σS,R),
where the (set-theoretic) disjoint union is taken over all faces R of AS.
Proposition 2.11. Let F ⊆ E be a subset and let R be a face of AF . The intersection
MA∩O(σF,R) is nonempty if and only if F is a flat ofM, in which case it is a linear
subvariety of O(σF,R) of dimension 2d−rkF −codimR. In particular, MA∩O(σF,R)
is irreducible when it is nonempty.
Proof. Because MA ∩ O(σF,R) is the preimage of L ∩ GErFm under the surjection
Φ|O(σF,R) : O(σF,R)→ GErFm , we have MA ∩O(σF,R) 6= ∅ if and only if L∩GErFm 6= ∅,
and this occurs if and only if F is a flat of M.
Suppose then that F is a flat of M. Since L ∩ GErFm is a linear subvariety
of GErFm and Φ|O(σF,R) is a split surjection, we may apply Lemma 2.1 to conclude
that MA ∩ O(σF,R) is a linear subvariety of O(σF,R) of codimension equal to the
codimension of L ∩GErFm in GErFm .
20
Since dim(L ∩GErFm ) = dimLF = rkMF = d− rkF , we have
codimO(σF,R)(MA ∩O(σF,R)) = |E r F | − d+ rkF.
By Lemma 2.6,
dim(MA ∩O(σF,R)) = dimO(σF,R)− (|E r F | − d+ rkF )
= (d− codimR+ |E r F |)− (|E r F | − d+ rkF )
= 2d− rkF − codimR.
In general, if X is a subvariety of a toric variety, then the expected dimension of
the intersection of X with a torus orbit O(σ) is dimX − dimσ. If the dimension of
the intersection is equal to the expected dimension, then we say that X intersects the
torus orbit properly. Proposition 2.11 implies that a hypertoric variety does not, in
general, intersect all Lawrence torus orbits properly.
Corollary 2.12. The hypertoric variety MA does not meet each torus orbit of BA
properly unless MA ∼= A2d.
Proof. Let F be a flat ofM, so that MA ∩O(σF,R) is nonempty. By Lemma 2.6, the
expected dimension of this intersection is
dimMA − dimσF,R = 2d− |F | − codimR.
By Proposition 2.11, dim(MA ∩ O(σF,R)) agrees with the expected dimension if and
only if |F | = rkF . This will hold for every flat ofM if and only if |E| = d andM is
21
the uniform matroid of rank d. In this case, the primitive spanning configuration a
will be an integral basis for N , and therefore MA ∼= A2d.
22
CHAPTER III
ANALYTIFICATION AND TROPICALIZATION
This chapter contains a brief overview of Berkovich’s theory of non-Archimedean
analytic spaces, and the definition of the (Kajiwara-Payne) tropicalization of
subvarieties of a toric variety. We include Example 3.1, describing the tropicalization
of a linear space, which will be of use to us in describing the tropicalization of the
hypertoric variety MA in Chapter IV. For further reading, see [Ber90] for foundations
of Berkovich spaces, [CHW14, Section 5] for a practical discussion of affinoid algebras,
[MS15] for a general treatment of tropicalizations, and [Gub13] for a comprehensive
treatment of tropicalization from the perspective of non-Archimedean geometry.
Write T = R ∪ {∞}, which we shall consider as a monoid under addition and
as a topological space homeomorphic to (0, 1]. Recall that our ground field K is
equipped with a non-Archimedean valuation ν : K → T. Let | · | = exp(−ν( · )) be
the associated norm on K.
3.1. Affinoid algebras and analytic spaces
For r = (r1, . . . , rn) ∈ Rn>0, we have the weighted Gauss norm || · ||r on the
polynomial ring K[x1, . . . , xn], defined by∣∣∣∣∣
∣∣∣∣∣ ∑
u∈Nn
cux
u
∣∣∣∣∣
∣∣∣∣∣
r
= max
u∈N
|cu|ru
23
(where cu = 0 for all but finitely many u, and r
u = ru11 · · · runn ). The completion of
K[x1, . . . , xn] with respect to || · ||r is the generalized Tate algebra
K〈r−11 x1, . . . , r−1n xn〉 =
{ ∑
u∈Nn
cux
u
∣∣∣ |cu|ru → 0 as |u| → ∞}
which can be thought of as the ring of convergent power series on the polydisc of
radius r in An. It is a Banach algebra, equipped with the norm || · ||r.
A K-affinoid algebra is a Banach algebra (A , || · ||), where A is isomorphic
to a quotient K〈r−11 x1, . . . , r−1n xn〉/I and || · || is equivalent to the quotient norm.
The Berkovich spectrum M (A ) of a K-affinoid algebra A is the set of bounded
multiplicative seminorms γ on A , equipped with the coarsest topology such that
eva : M (A )→ R≥0, γ 7→ γ(a) is continuous for every a ∈ A .
Similar to the construction of the generalized Tate algebra, we may complete the
Laurent polynomial ring K[x±11 , . . . , x
±1
n ] with respect to || · ||r to obtain
K〈r−11 x1, r1x−11 . . . , r−1n xn, rnx−1n 〉 =
{∑
u∈Zn
cux
u
∣∣∣ |cu|ru → 0 as |u| → ∞},
which is also a K-affinoid algebra.
Given a K-affinoid algebra A , with norm || · ||, and s ∈ R>0, define
A 〈s−1x, sy〉 =
{∑
i,j≥0
cijx
iyj
∣∣∣ cij ∈ A , ||cij||si−j → 0 as i+ j →∞}.
For any element f ∈ A , we can then define the affinoid algebra
A 〈s−1f, sf−1〉 = A 〈s−1x, sy〉/(x− f, fy − 1).
24
Iterating this construction, given s1, . . . , sn ∈ R>0 and f1, . . . , fn ∈ A , we
may define the affinoid algebra A 〈s−11 f1, s1f−11 , . . . , s−1n fn, snf−1n 〉. Its Berkovich
spectrum M (A 〈s−11 f1, s1f−11 , . . . , s−1n fn, snf−1n 〉) includes into M (A ) as the set of
all seminorms γ such that γ(fi) = si for i = 1, . . . , n.
A Berkovich K-analytic space is, roughly speaking, a topological space
equipped with a sheaf of analytic functions which is locally isomorphic to the
Berkovich spectrum of a K-affinoid algebra, where the ring of analytic functions
on M (A ) is A . For details, see [Ber90, Chapters 2 & 3].
Given a K-variety X, there is an analytification functor which associates to X a
K-analytic space Xan. As a topological space, Xan can be described without reference
to Berkovich spectra or affinoid algebras. If X = SpecA is affine, then Xan is the set
of ring valuations A→ T extending the valuation ν on K (or, equivalently, as the set
of multiplicative seminorms on A extending | · |). We give Xan the coarsest topology
such for every a ∈ A, the evaluation map eva : Xan → T, val 7→ val(a) is continuous.
For general X, we may take a cover of X by affine open subschemes {Ui}, and the
analytifications Uani glue to form X
an. (Equivalently, Xan can be expressed as a set
of equivalence classes of L-valued points, as L varies over all valued field extensions
of K [Gub13, Remark 2.2].)
The functor X → Xan possesses many nice properties [Ber90, Sections 3.4 & 3.5].
For instance, Xan is compact if and only if X is proper, and the topological dimension
of Xan is equal to the algebraic dimension of X. If ϕ : X → Y is a morphism of K-
varieties, then many properties (e.g. smooth, e´tale, flat, finite) of ϕ are inherited by
ϕan. Of particular use to us is that ϕan is a closed (resp. open) immersion if and only
if ϕ is.
25
3.2. Tropicalization
Let Σ be a fan in NR, as in Section 2.1. For a cone σ ∈ Σ, we define NσR to
be the set of monoid homomorphisms Hom(σ∨ ∩M,T). We give NσR the topology of
pointwise convergence. If τ ≺ σ, then N τR is naturally identified with the open subset
of N
σ
R consisting of maps which are finite on τ
⊥ ∩ σ∨ ∩M (in particular, NR = N{0}R
is an open subset of each N
σ
R). Gluing along these identifications, we obtain N
Σ
R, a
partial compactification of NR. This mirrors the construction of the toric variety YΣ:
we have a decomposition N
Σ
R =
⋃
σ∈Σ N
σ
R analogous to the decomposition of YΣ into
affine toric varieties, and a (set-theoretic) decomposition N
Σ
R =
⊔
σ∈Σ N(σ) analogous
to the decomposition of YΣ into torus orbits. These two constructions are related by
the process of tropicalization.
The tropicalization map on the torus T = SpecK[M ] is the continuous
surjection
trop: T an → NR
which takes a valuation val : K[M ] → T to its restriction val |M : M → R. More
generally, for a cone σ ∈ Σ, we have a tropicalization map
trop: Y anσ → NσR = Hom(σ∨ ∩M,T)
which similarly maps a valuation val : K[σ∨ ∩M ] → T to its restriction to σ∨ ∩M .
These maps glue to give a tropicalization map trop: Y anΣ → N
Σ
R. This map is a
continuous and proper surjection, which has the property that its restriction to each
torus orbit is the usual tropicalization O(σ)an → NR(σ) for a torus.
Given a closed subvariety X ⊆ YΣ, the analytification Xan is a closed subspace
of Y anΣ . The tropicalization Trop(X) ⊆ N
Σ
R of X defined by its embedding in YΣ
26
is the image of Xan under trop. If X is a subvariety of the torus T , then Trop(X)
may be given the structure of a finite polyhedral complex, which is a (not necessarily
pointed) fan if X is defined over a subfield of K having trivial valuation. Moreover,
this polyhedral complex is of pure dimension dimX and is equipped with a positive
integer-valued weight function, the tropical multiplicity, with respect to which the
complex is balanced [MS15, Theorem 3.3.5].
We refer the interested reader to [OP13, Section 2] for a discussion of tropical
multiplicities. The basic idea is as follows. Each w ∈ NR defines a scheme T w over
the valuation ring of K with generic fiber T . We think of this as a degeneration of
T . The closure of X ⊆ T in T may or may not intersect the special fiber. This
intersection is a scheme over the residue field, called the initial degeneration inwX
of X at w. Part of the so-called Fundamental Theorem of Tropical Geometry [MS15,
Theorem 3.2.5] states that w ∈ Trop(X) if and only if inwX is nonempty. In this
case, the tropical multiplicity of Trop(X) at w is the multiplicity of inwX, i.e., its
number of irreducible components, counted with multiplicity.
In the general setting where X is a subvariety of a toric variety, Trop(X)
may be computed orbit-by-orbit: Trop(X) ∩ NR(σ) = Trop(X ∩ O(σ)). The
multiplicity of Trop(X) at w ∈ Trop(X) ∩NR(σ) is equal to the tropical multiplicity
of Trop(X ∩O(σ)) at w. Thus, Trop(X) is a partial compactification of the balanced
finite polyhedral complex Trop(X∩T ) by lower-dimensional balanced finite polyhedral
complexes. If X = X ∩ T (in particular, if X is irreducible and X ∩ T is nonempty),
then Trop(X) is the closure of Trop(X ∩ T ) in NσR [MS15, Corollary 6.2.16].
Tropicalization is functorial with respect to toric morphisms. Let f : YΣ → YΣ′
be such a map. For σ ∈ Σ, there exists σ′ ∈ Σ′ such that f∗(σ) ⊆ σ′. For such a σ′,
the restriction of f ∗ gives a map M(σ′) → M(σ), inducing NσR → Nσ
′
R . These maps
27
glue to give a map N
Σ
R → NΣ
′
R , denoted Trop(f). See [Pay09] for details. By [MS15,
Corollary 6.2.15], if X ⊆ YΣ, then Trop(f)(Trop(X)) = Trop(f(X)).
Example 3.1. Of particular importance to us will be the tropicalization of a linear
space. As in Section 2.2, let A be an arrangement, with associated linear space
L ⊆ AE = SpecK[xe | e ∈ E] and underlying matroid M.
The torus orbits of AE are indexed by subsets S ⊆ E, where S corresponds to the
torus GErSm ⊆ AE defined by xe = 0 if and only if e ∈ S. The intersection L ∩GErSm
of L with one of these orbits is nonempty if and only if S is a flat of M, in which
case it is the intersection of the restriction LS ⊆ AErS with the torus GErSm .
Given a flat F of M, we define δF =
∑
e∈F δe ∈ RE ⊆ TE, where δe ∈ RE is the
basis vector corresponding to e ∈ E. For a flag of flats
F = (∅ = F0 ⊂ F1 ⊂ · · · ⊂ Fk−1 ⊂ Fk = E),
we have the cone
βF = R≥0〈δF1 , · · · , δFk−1 ,±δE〉 ⊆ RE,
where dim βF = `(F) = k. We have βF ≺ βF ′ if and only if F ′ is a refinement of F .
The collection of cones βF defines a fan in RE, called the Bergman fan ofM (with
the fine fan structure of [AK06]), with support equal to Trop(L∩Gm). As discussed
in [DP16, Section 2], every initial degeneration of L is also a linear space. Therefore,
the Bergman fan ofM is a pure polyhedral fan of dimension rkM, which is balanced
when each cone is assigned weight one.
Note that every cone βF in the Bergman fan contains the diagonal copy of R, the
span of δE. Many authors take the quotient by this lineality space, which is equal to
the tropicalization of the projectivization of L. We shall not adopt this convention.
28
If F is a flat, then Trop(L ∩ GErFm ) is the support of the Bergman fan of the
restriction MF . Its cones, denoted β(F )F , are in correspondence with flags F of flats
in MF (such a flag is identified with a flag of flats in M beginning at F ). The full
tropicalization Trop(L) ⊆ TE, together with the fan structures on each of its strata,
is the extended Bergman fan of A. It is equal to the closure of
Trop(L ∩GEm) = Trop(L) ∩ RE
in TE.
29
CHAPTER IV
THE TROPICALIZATION OF A HYPERTORIC VARIETY
In this chapter, we describe the structure of the tropicalization of a hypertoric
variety induced by its canonical embedding in the Lawrence toric variety. The main
result is Theorem 4.1, which describes a polyhedral structure on the tropicalization.
We also calculate the fibers of the tropicalization map. The main tool to obtain
these results is the moment map Φ, which by functoriality of tropicalization gives
a map from the tropicalization of the hypertoric variety to the Bergman fan of the
underlying matroid of the arrangement.
Fix an arrangement A = A(a, r) in MR, where a is a primitive spanning
configuration. As in Section 2.2, let L ⊆ AE be the associated linear space and
M the underlying matroid.
4.1. Description of the tropicalization
Let Φ: BA → AE be the moment map from Section 2.4. By definition of MA, Φ
maps MA onto L, and by functoriality of tropicalization, Trop(Φ) gives a surjection
Trop(MA)→ Trop(L). Given a flat F ofM, the stratum Trop(L∩GErFm ) of Trop(L)
is the Bergman fan of the restrictionMF . As noted in example 3.1, this fan has cones
β
(F )
F indexed by flags F of flats inMF . Given such a flag F , let C(F,R)F be the preimage
of β
(F )
F under the surjection Trop(MA ∩ O(σF,R)) → Trop(L ∩ GErFm ). Since every
Bergman fan is balanced when every maximal cone is given weight one, each stratum
Trop(MA ∩ O(σF,R)) = Trop(MA) ∩ N˜R(σF,R) inherits this structure of a balanced
polyhedral fan with cones C
(F,R)
F and all weights equal to one. Our main theorem
describes how these fans are pieced together.
30
Theorem 4.1. The tropicalization Trop(MA) of the hypertoric variety is the union
of cones C
(F,R)
F indexed by a flat F of M, a face R of the localization AF , and a flag
of flats F in the restriction MF . These cones satisfy
dimC
(F,R)
F = d− codimR+ `(F).
This gives Trop(MA) the combinatorial structure of a finite polyhedral complex, under
the closure relation
C
(F ′,R′)
F ′ ⊆ C(F,R)F (4.1)
if and only if the following conditions hold:
– F ⊆ F ′;
– R′ ⊆ R;
– F ′ is a flat in F , and truncF ′(F) is a refinement of F ′.
Moreover, this gives each stratum Trop(MA)∩ N˜R(σF,R) the structure of a polyhedral
fan, which is balanced when all cones are given weight one.
Given a flat F and a face R of AF , there are two fans which live in
Trop(O(σF,R)) = N˜R(σF,R): the fan Trop(MA) ∩ N˜R(σF,R) and the fan of the orbit
closure O(σF,R). The former fan has cones C
(F,R)
F indexed by flags of flats in MF ,
while the latter consists of the projections of the cones σS,R′ with σS,R′  σF,R (by
Lemma 2.7, this is equivalent to S ⊇ F and R′ ⊆ R). The following lemma relates
these two fans.
31
Lemma 4.2. Let F be a flat of M and R a face of AF . Given a set S ⊆ E which
contains F and a face R′ of AS contained in R, the intersection
C
(F,R)
F ∩ relint(piσF,R(σS,R′)),
for F a flag of flats inMF , is nonempty if and only if S is a flat ofM which appears
in the flag F . In this case,
C
(F,R)
F ∩ N˜R(σS,R′) = C(S,R
′)
truncS(F).
Proof. First, because Φ(σF,R) = RF≥0, we have that
Trop(Φ)|N˜R(σF,R) : N˜R(σF,R) = Trop(O(σF,R))→ RErF = RE/RF
is given by [v] 7→ [Φ(v)]. (Here we are identifying RE/RF , the vector space spanned by
the cocharacter lattice of O(RF≥0) = GErFm ⊆ AE with RErF primarily for notational
convenience.) In other words, the square
N˜R RE
N˜R(σF,R) RErF
Trop(Φ)
piσF,R
Trop(Φ)
(4.2)
commutes.
Now, suppose v ∈ C(F,R)F ∩ relint(piσF,R(σS,R′)) ⊆ N˜R(σF,R). Every vector in
σS,R′ can be written as a linear combination, with non-negative coefficients, of the
generators ρ+e , ρ
−
f for e ∈ S+(R′), f ∈ S−(R′). Then v is the image of such a
vector under piσF,R , which kills all generators of σS,R′ indexed by elements of F (since
32
F±(R) ⊆ S±(R′) by Lemma 2.4). In order for v to be in the relative interior of
piσF,R(σS,R′), therefore, it must be that coefficient of ρ
+
e (resp. ρ
−
f ) is positive for
e ∈ S+(R′)rF+(R) (resp. f ∈ S−(R′)rF−(R)). Since the square (4.2) commutes,
it follows that Trop(Φ)(v) ∈ RErF will lie in RSrF>0 ∩ β(F )F . By the definition of β(F )F
(cf. Example 3.1), this intersection is nonempty if and only if S is a flat in the flag
F .
Conversely, suppose S is a flat in F . For e ∈ S r F , define
ve =

1
2
(ρ+e + ρ
−
e ) if e ∈ S0(R′),
ρ+e if e ∈ S+(R)r S0(R′),
ρ−e if e ∈ S−(R)r S0(R′),
and let v =
∑
e∈SrF ve. Then, by design, we have v ∈ relint(piσF,R(σS,R′)) and
Trop(Φ)(v) = δSrF ∈ βF . , it follows that v ∈ Trop(Φ)−1(βF) = C(F,R)F , and therefore
C
(F,R)
F ∩ relint(piσF,R(σS,R′)) 6= ∅.
For the final part of the lemma, we assume that S is a flat in the flag F . We
have
C
(F,R)
F ∩ N˜R(σS,R′)) = piσS,R′ (C(F,R)F )
by [OR13, Lemma 3.9], so we need only prove that this projection coincides with
C
(S,R′)
truncS(F). The square
N˜R(σF,R) RErF
N˜R(σS,R′) RErS
Trop(Φ)
piσS,R′
Trop(Φ)
commutes for the same reason that that (4.2) commutes. Thus, we see that Trop(Φ)
maps piσS,R′ (C
(F,R)
F ) onto β
(S)
truncS(F), which shows that piσS,R′ (C
(F,R)
F ) ⊆ C(S,R
′)
truncS(F).
33
On the other hand, if w ∈ C(S,R′)truncS(F) and v is a preimage of w under piσS,R′ , then
Trop(Φ)(v) need not lie in β
(F )
F , so that v need not be in C
(F,R)
F . However, we can
choose η ∈ RSrF so that Trop(Φ)(v) + η ∈ β(F )F . Since η ∈ RSrF , there exists a
preimage v′ ∈ N˜R(σF,R) of η under Trop(Φ), such that v′ is expressed as a sum of
generators ρ+e and ρ
−
f with e, f ∈ S r F . Then piσS,R′ (v′) = 0, and therefore v + v′ ∈
C
(F,R)
F projects to w. This shows the reverse inclusion C
(S,R′)
truncS(F) ⊆ piσS,R′ (C
(F,R)
F ).
We are now ready to prove the main theorem.
Proof of Theorem 4.1. Suppose that C
(F ′,R′)
F ′ ⊆ C(F,R)F . Then necessarily the
intersection C
(F,R)
F ∩ N˜R(σF ′,R′) is nonempty. By Lemma 4.2, this implies F ′ ⊇ F is
a flat in F and R′ ⊆ R. In this case,
C
(F,R)
F ∩ N˜R(σF ′,R′) = C(F
′,R′)
truncF F
will contain C
(F ′,R′)
F ′ as a face if and only if truncF ′(F) is a refinement of F ′.
Let C
(F,R)
F be a cone in Trop(MA) ∩ N˜R(σF,R). Since
Trop(Φ): NR(σF ,R)→ RErF
is a linear surjection of relative dimension d − codimR, and C(F,R)F is defined to be
the preimage of β
(F )
F , it follows that
dimC
(F,R)
F = d− codimR+ dim β(F )F = d− codimR+ `(F).
34
Remark 4.3. By Theorem 4.1, a cone C
(F,R)
F is inclusion-maximal in the stratum
Trop(MA) ∩ N˜R(σF,R) if and only if the flag F is maximal. A maximal flag in MF
has length rkMF = d− rkF , so that
dimC
(F,R)
F = d− codimR+ `(F) = 2d− rkF − codimR.
Thus dimC
(F,R)
F agrees with dim(MA ∩ O(σF,R)) = dim(Trop(MA) ∩ N˜R(σF,R)) by
Proposition 2.11. This shows that the inclusion-maximal cones in each stratum of
Trop(MA) are precisely the dimension-maximal cones, which should be expected
because tropicalizations are always pure-dimensional.
4.2. Fibers of tropicalization
In this section, we calculate the fiber of the tropicalization mapMA → Trop(MA)
over a point θ ∈ Trop(MA), following the approach of [CHW14]. Suppose that
θ ∈ Trop(MA) ∩ N˜R(σF,R) for some flat F of M and face R of the localization
AF . We shall write trop−1(θ) ⊆ O(σF,R)an to denote the fiber over θ of the map
trop: O(σF,R)an → N˜R(σF,R), and trop−1MA(θ) = trop−1(θ) ∩ManA for the fiber over θ
of ManA → Trop(MA). Let η = (ηe) ∈ RErS be the image of θ under Trop(Φ).
We defined the moment map Φ: BA → AE as an extension to BA of a split
surjection of tori T˜ → GEm. This surjection is given by the map N˜ → ZE from (2.4)
or equivalently by an injection of characters Φ∗ : ZE → M˜ . The image under Φ∗ of
a generator xe ∈ ZE is the primitive diagonal element x+e + x−e ∈ M˜ ⊆ ZE ⊕ ZE.
Inspired by the standard notation for the homogeneous coordinate ring of BA, we
denote by zewe the monomial in K[M˜ ] corresponding to x
+
e + x
−
e .
35
The diagonal element x+e + x
−
e is in M˜(σF,R) if and only if e ∈ E r F , and the
restriction of Φ to O(σF,R) is given by restricting Φ∗ to obtain ZErF → M˜(σF,R).
We may extend the set {x+e + x−e | e ∈ E r F} to an integral basis of M˜(σF,R). By
Lemma 2.6 we must add d− codimR = dimR primitive elements ui. We write yi for
the monomial corresponding to ui, so that
K[O(σF,R)] = K[M˜(σF,R)] ∼= K[(zewe)±1, y±1i | e ∈ E r F, i = 1, . . . , dimR].
An element of N˜R(σF,R) = Hom(M˜(σF,R),R) is uniquely determined by its values on
the integral basis {x+e +x−e , ui | e ∈ ErF, i = 1, . . . , dimR} of M˜(σF,R). In particular,
a valuation val ∈ O(σF,R)an lies in trop−1(θ) if and only if val(zewe) = 〈x+e +x−e , θ〉 = ηe
for all e ∈ E r F and val(yi) = 〈ui, θ〉 for all i = 1, . . . , dimR.
Let B be a basis of MF which has maximal η-weight; that is, ∑e∈B ηe is
maximized at B, and consider the subring
A = K[(zewe)
±1, (yi)±1 | e ∈ B, i = 1, . . . , dimR]
of K[O(σF,R)]. Since B is a basis ofMF , for each f ∈ E r (F ∪B), there is a unique
element
pf =
∑
e∈B
ce(zewe) ∈ A
such that pf − zfwf ∈ K[O(σF,R)] lies in the ideal of MA ∩ O(σF,R). Since these
relations generate the ideal, this shows that A is isomorphic to the coordinate ring
K[MA ∩O(σF,R)]. Furthermore, because the basis B is η-maximal, any valuation val
on A with val(zewe) = ηe for every e ∈ B must necessarily satisfy val(pf ) = ηf for all
f ∈ E r (F ∪ B) as well.
36
We can therefore identify trop−1MA(θ) with the set of valuations on A such that
val(zewe) = ηe for all e ∈ B and val(yi) = 〈ui, θ〉 for i = 1, . . . , dimR. This is a
satisfying description of the fiber; however, it is useful to identify this fiber with the
Berkovich spectrum of a particular K-affinoid algebra.
Set re = exp(−ηe) and si = exp(−〈ui, θ〉). Define the affinoid algebras
A = K〈r−1e (zewe), re(zewe)−1, s−1i yi, siy−1i | e ∈ B, i = 1, . . . , dimR〉.
and
B = A 〈r−1f pf , rfp−1f | f ∈ E r (F ∪ B)〉.
Then by construction, each seminorm γ ∈M (B) restricts to a seminorm on A with
γ(zewe) = re and γ(yi) = si for all e ∈ E r F and i = 1, . . . , dimR. (Equivalently,
− log γ(−) is a valuation on A with − log γ(zewe) = ηe and − log γ(yi) = 〈ui, θ〉.)
In fact, this is a bijective correspondence, and any such seminorm on A extends
uniquely to B. We will not prove this statement, but we refer the reader to the proof
of [CHW14, Proposition 5.6], which can be adapted to prove the following.
Proposition 4.4. The fiber trop−1MA(θ) is M (B) ⊆ManA .
It turns out that M (B) has a unique Shilov boundary point: a seminorm
γ ∈M (B) such that evb : M (B)→ R≥0 is maximized at γ for every b ∈ B. Again,
we shall not prove this, but we refer to [CHW14, Theorem 5.8 & Corollary 5.9] for
an outline.
The significance of this result is that the section Trop(MA)→ManA which we will
construct in Chapter V will map each θ ∈ Trop(MA) to the unique Shilov boundary
point of trop−1MA(θ).
37
CHAPTER V
FAITHFUL TROPICALIZATION
In this chapter, we prove that each hypertoric variety is faithfully tropicalized by
its Lawrence embedding. We do so by using Theorem 4.1 to show that the conditions
of a theorem of Gubler, Rabinoff, and Werner are satisfied.
5.1. The theorem of Gubler-Rabinoff-Werner
Let X be a suvariety of a torus. Gubler, Rabinoff, and Werner have shown that
there exists a unique continuous section to the tropicalization map Xan → Trop(X)
if Trop(X) has tropical multiplicity one at every point [GRW16, Theorem 10.6].
In the general case, where X is a subvariety of a toric variety YΣ which is not
necessarily a torus, we can apply the above result on each torus orbit: If Trop(X) has
multiplicity one at every point, then there is a unique section of tropicalization which
is continuous on Trop(X ∩ O(σ)) = Trop(X) ∩NR(σ) for each σ ∈ Σ. However, this
section may fail to be continuous on all of Trop(X). An example of an irreducible
hypersurface in A3 for which this section is not continuous is given in [GRW15,
Example 4.9].
In [GRW15], Gubler, Rabinoff, and Werner provide the following sufficient
criteria for continuity of this section.
Theorem 5.1 ([GRW15, Proposition 8.8 & Theorem 8.14]). Let Σ be a pointed
rational fan in NR, and let X ⊆ YΣ be a subvariety. Suppose that
(1) X ∩ T is dense in X;
38
(2) for all σ ∈ Σ, the subscheme X ∩ O(σ) is either empty or equidimensional of
dimension dσ;
(3) Trop(X) has tropical multiplicity one at every point;
(4) Trop(X) ∩ NR can be covered by finitely many d0-dimensional polyhedra with
the following property: If the recession cone of P meets the relative interior of
σ, then piσ(P ) = P ∩NR(σ) has dimension dσ.
Then there is a unique continuous section of the tropicalization map Xan → Trop(X).
Remark 5.2. It follows from [GRW15, Proposition 8.3] that for each point in the
tropicalization of multiplicity one, the fiber of the tropicalization map over that point
contains a unique Shilov boundary point. The section produced by Theorem 5.1 maps
each point of Trop(X) to the unique Shilov boundary point in its fiber.
Although the proof of Theorem 5.1 requires careful study of the analytification
Xan, the criteria (1)–(4) can be checked purely by inspecting Trop(X) (and X
itself). We remark that while Theorem 5.1 is a powerful tool, it does not trivialize
the problem of finding a faithful tropicalization of X. Indeed, it does not give
any indication as to how to find a faithful tropicalization of X, nor does it imply
that one must even exist. Rather, Theorem 5.1 transforms a difficult problem in
non-Archimedean geometry—that of verifying that a particular tropicalization is
faithful—into a difficult combinatorics problem. For example, as outlined in [GRW15,
Example 8.16], Theorem 5.1 can be used to prove that the Plu¨cker embedding yields
a faithful tropicalization of Gr(2, n); however, many ingredients of the original proof
in [CHW14] remain necessary to establish conditions (2)–(4).
Remark 5.3. Faithful tropicalization is easy to verify in one situation. If X is
irreducible and intersects each torus orbit properly or not at all, then by [GRW15,
39
Theorem 8.15] the resulting tropicalization is faithful if Trop(X) has multiplicity one
everywhere. By Corollary 2.12, a hypertoric variety MA fails to possess this nice
property outside of the trivial case MA = A2d. (The Grassmannian Gr(2, n) also
does not intersect torus orbits properly.)
5.2. Faithful tropicalization of hypertoric varieties
Let MA be any hypertoric variety. As in section 4.1, let Trop(MA) denote the
tropicalization induced by the Lawrence embedding MA ⊆ BA. We now use Theorem
5.1 to prove that this is a faithful tropicalization.
Theorem 5.4. There is a unique continuous section of the tropicalization map
ManA → Trop(MA).
Proof. We shall show that the four conditions of Theorem 5.1 hold. The intersection
MA ∩ T˜ is nonempty, and therefore dense in MA because MA is irreducible. By
Proposition 2.11, the intersection of MA with any torus orbit in BA is either empty
or it is a linear space. In particular, each of these intersections (when nonempty) is
equidimensional and the tropical multiplicity of Trop(MA) is one at every point.
It remains to show that (4) holds. We equip Trop(MA) with the polyhedral
structure described in Theorem 4.1. Then Trop(MA) ∩ N˜R is covered by the 2d-
dimensional cones C
(∅,M˜R)
F , where F is a maximal flag of flats inM. For convenience,
we will write CF instead of C
(∅,M˜R)
F . By Lemma 4.2, the cone CF meets the relative
interior of a Lawrence cone σF,R if and only if F is a flat of M which appears in the
flag F . In this case, Lemma 4.2 gives
piσF,R(CF) = CF ∩ N˜R(σF,R) = C(F,R)truncF (F).
40
Now, truncF (F) is a maximal flag of flats in MF , and therefore C(F,R)truncF (F) is an
inclusion-maximal cone of the fan Trop(MA)∩ N˜R(σF,R). It follows (cf. Remark 4.3)
that C
(F,R)
truncF (F) has dimension equal to dim(MA ∩O(σF,R)). We may therefore apply
Theorem 5.1 to conclude that there is a unique continuous section of tropicalization
defined on all of Trop(MA).
Remark 5.5. We conclude by noting that there is a more general notion of hypertoric
variety than we have discussed in this dissertation. Arbo and Proudfoot [AP16] have
recently shown how to construct a hypertoric variety from a zonotopal tiling T . Such
a hypertoric variety is also embedded in a (generalized) Lawrence toric variety, and
agrees with the variety constructed in Section 2.4 in the case where T is a regular
tiling and hence normal to some affine arrangement. We suspect that Theorem 5.4
remains true in this more general setting.
41
REFERENCES CITED
[AK06] Federico Ardila and Caroline J. Klivans. The Bergman complex of a matroid
and phylogenetic trees. J. Combin. Theory Ser. B, 96(1):38–49, 2006.
[AP16] Matthew Arbo and Nicholas Proudfoot. Hypertoric varieties and zonotopal
tilings. Int. Math. Res. Not., 2016(23):7268–7301, 2016.
[BD00] Roger Bielawski and Andrew S. Dancer. The geometry and topology of toric
hyperka¨hler manifolds. Comm. Anal. Geom., 8(4):727–760, 2000.
[Ber90] Vladimir G. Berkovich. Spectral theory and analytic geometry over non-
Archimedean fields, volume 33 of Mathematical Surveys and Monographs.
American Mathematical Society, Providence, RI, 1990.
[BPR16] Matthew Baker, Sam Payne, and Joseph Rabinoff. Nonarchimedean
geometry, tropicalization, and metrics on curves. Algebr. Geom., 3(1):63–105,
2016.
[CHMR16] Renzo Cavalieri, Simon Hampe, Hannah Markwig, and Dhruv
Ranganathan. Moduli spaces of rational weighted stable curves and tropical
geometry. Forum Math. Sigma, 4 e9:35 pages, 2016.
[CHW14] Maria Angelica Cueto, Mathias Ha¨bich, and Annette Werner. Faithful
tropicalization of the Grassmannian of planes. Math. Ann., 360(1-2):391–437,
2014.
[CLS11] David A. Cox, John B. Little, and Henry K. Schenck. Toric varieties, volume
124 of Graduate Studies in Mathematics. American Mathematical Society,
Providence, RI, 2011.
[DP16] Jan Draisma and Elisa Postinghel. Faithful tropicalisation and torus actions.
Manuscripta Math., 149(3-4):315–338, 2016.
[FGP14] Tyler Foster, Philipp Gross, and Sam Payne. Limits of tropicalizations. Israel
J. Math., 201(2):835–846, 2014.
[Ful93] William Fulton. Introduction to toric varieties, volume 131 of Annals of
Mathematics Studies. Princeton University Press, Princeton, NJ, 1993. The
William H. Roever Lectures in Geometry.
[GM10] Angela Gibney and Diane Maclagan. Equations for Chow and Hilbert
quotients. Algebra Number Theory, 4(7):855–885, 2010.
42
[Gro15] Andreas Gross. Intersection theory on linear subvarieties of toric varieties.
Collect. Math., 66(2):175–190, 2015.
[GRW15] Walter Gubler, Joseph Rabinoff, and Annette Werner. Tropical skeletons.
Preprint, arXiv:1508.01179, 2015.
[GRW16] Walter Gubler, Joseph Rabinoff, and Annette Werner. Skeletons and
tropicalizations. Adv. Math., 294:150–215, 2016.
[Gub13] Walter Gubler. A guide to tropicalizations. In Algebraic and combinatorial
aspects of tropical geometry, volume 589 of Contemp. Math., pages 125–189.
Amer. Math. Soc., Providence, RI, 2013.
[HP04] Megumi Harada and Nicholas Proudfoot. Properties of the residual circle
action on a hypertoric variety. Pacific J. Math., 214(2):263–284, 2004.
[HS02] Tama´s Hausel and Bernd Sturmfels. Toric hyperka¨hler varieties. Doc. Math.,
7:495–534, 2002.
[MS15] Diane Maclagan and Bernd Sturmfels. Introduction to tropical geometry,
volume 161 of Graduate Studies in Mathematics. American Mathematical
Society, Providence, RI, 2015.
[OP13] Brian Osserman and Sam Payne. Lifting tropical intersections. Doc. Math.,
18:121–175, 2013.
[OR13] Brian Osserman and Joseph Rabinoff. Lifting nonproper tropical intersections.
In Tropical and non-Archimedean geometry, volume 605 of Contemp. Math.,
pages 15–44. Amer. Math. Soc., Providence, RI, 2013.
[Oxl11] James Oxley. Matroid theory, volume 21 of Oxford Graduate Texts in
Mathematics. Oxford University Press, Oxford, second edition, 2011.
[Pay09] Sam Payne. Analytification is the limit of all tropicalizations. Math. Res.
Lett., 16(3):543–556, 2009.
[Pro08] Nicholas Proudfoot. A survey of hypertoric geometry and topology. In Toric
topology, volume 460 of Contemp. Math., pages 323–338. Amer. Math. Soc.,
Providence, RI, 2008.
[PW07] Nicholas Proudfoot and Benjamin Webster. Intersection cohomology of
hypertoric varieties. J. Algebraic Geom., 16(1):39–63, 2007.
[Ran15] Dhruv Ranganathan. Skeletons of stable maps i: Rational curves in toric
varieties. Preprint, arXiv:1506.03754 (to appear in J. London Math. Soc.), 2015.
[Tev07] Jenia Tevelev. Compactifications of subvarieties of tori. Amer. J. Math.,
129(4):1087–1104, 2007.
43