ZONOTOPES AND HYPERTORIC VARIETIES
by
MATTHEW ARBO
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
September 2015
DISSERTATION APPROVAL PAGE
Student: Matthew Arbo
Title: Zonotopes and 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
Arkady Berenstein Core Member
Benjamin Elias Core Member
Benjamin Young Core Member
Jiannbin Shiao 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 September 2015
ii
c© 2015 Matthew Arbo
iii
DISSERTATION ABSTRACT
Matthew Arbo
Doctor of Philosophy
Department of Mathematics
September 2015
Title: Zonotopes and Hypertoric Varieties
Hypertoric varieties are a class of conical symplectic resolutions which are very
computable. In the current literature, they are only defined constructively, using
hyperplane arrangements. We provide an abstract definition of a hypertoric variety and
a new construction using zonotopal tilings and relate the zonotopal construction to the
hyperplane construction.
iv
CURRICULUM VITAE
NAME OF AUTHOR: Matthew Arbo
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene
Southeastern Louisiana University, Hammond
University of New Orleans, New Orleans, Louisiana
Tulane University, New Orleans, Louisiana
DEGREES AWARDED:
Doctor of Philosophy, 2015, University of Oregon
Master of Science, 2012, University of Oregon
Bachelor of Science, 2007, Southeastern Louisiana University
v
ACKNOWLEDGEMENTS
First and foremost, I would like to thank my advisor, Nick Proudfoot, for his
guidance and support, as well as the rest of the professors in the Mathematics Department
who have taught me. I would also like to thank my fellow graduate students who have
spent the last seven years with me, and the office staff for all their help.
I would also like to thank all of the professors who taught me before graduate
school, especially Alan Cannon, Kent Neuerberg, and Randall Wills, and the faculty
mentors at the IMMERSE and Willamette Valley REU-RET programs.
Last, but certainly not least, I would like to thank my parents for being there
whenever things got rough, and my family for all of their encouragement and support.
vi
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
II. COMBINATORICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1. Sign Vectors and Oriented Matroids . . . . . . . . . . . . . . . . 7
2.2. Zonotopes and Linear Hyperplane Arrangements . . . . . . . . . 9
2.3. Zonotopal Tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
III. VARIETIES FROM ZONOTOPES . . . . . . . . . . . . . . . . . . . . . 16
3.1. Varieties from Arrangements and Signed Arrangements . . . . . 17
3.2. Zonotopes and the Extended Core . . . . . . . . . . . . . . . . . 22
3.3. The Extended Core . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4. Smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
IV. VARIETIES FROM TILINGS . . . . . . . . . . . . . . . . . . . . . . . . 27
4.1. Refinements and Partial Resolutions . . . . . . . . . . . . . . . . 28
vii
Chapter Page
V. THE MAIN THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.1. The Extended Core . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.2. Line Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.3. Alternate Weights . . . . . . . . . . . . . . . . . . . . . . . . . . 32
VI. RECOVERING THE TILING . . . . . . . . . . . . . . . . . . . . . . . . 34
VII. A WORKED EXAMPLE . . . . . . . . . . . . . . . . . . . . . . . . . . 36
7.1. Constructing T ∗CP 2 . . . . . . . . . . . . . . . . . . . . . . . . 36
7.2. Recovering the Tiling . . . . . . . . . . . . . . . . . . . . . . . . 37
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
viii
LIST OF FIGURES
Figure Page
2.1. A zonotope with vertices labeled, and a relabeling of the same zonotope.
To avoid confusion, no axes are drawn, and vectors are based at the
origin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2. Two different tilings of the same zonotope. . . . . . . . . . . . . . . . . . 13
2.3. The bold line segments are an equivalence class. Together with the gray
two-dimensional zonotopes, they form a zone of the tiling. The tiling
which includes the dashed edges and the three parallelotopes which
contain them is called the non-Pappus tiling, and is well-known to be
non-regular. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
7.1. The fans for the toric varieties V−−−, V+−−, and V++−. . . . . . . . . . . 37
ix
LIST OF TABLES
Table Page
1.1. A comparison of results on toric and hypertoric varieties . . . . . . . . . 6
x
CHAPTER I
INTRODUCTION
Hypertoric varieties were introduced by Bielawski and Dancer under the name
“toric hyperka¨hler manifolds” in 2000 [BD00]. The definition they give is constructive:
given a suitable rational hyperplane arrangement in the dual of the Lie algebra of a given
torus, one can construct a manifold, and manifolds which arise this way are called toric
hyperka¨hler manifolds. Hypertoric varieties arising from this construction have been
studied in many papers, including [HS02, HH05, PW07, Pro08, Kon00]. In this paper,
we provide an abstract definition for hypertoric varieties, and a new construction which
includes all hypertoric varieties which come from hyperplane arrangements, as well as
new hypertoric varieties that do not appear in the literature. Among our varieties, those
arising from hyperplane arrangements are precisely the ones that are projective over
their affinizations.
The situation may be understood by analogy to toric varieties. Given a torus T
over a field k, a T -toric variety is a normal variety with an action of T that has a dense
orbit. To each affine toric variety X0, we can associate a cone σ in Lie(T ) and its dual
σ∨ ∈ Lie(T )∗. The cone σ contains precisely those cocharacters that act on X0 with non-
negative weights, while the cone in Lie(T )∗ contains precisely the characters of T that
extend to functions on X0. Given a polyhedron P in Lie(T )
∗ with recession cone σ∨, we
obtain a toric variety XP which is projective over its affinization Spec k[XP ] = X0. The
map P 7→ XP is not a bijection, and the polyhedron in fact determines a presentation
of XP as a GIT quotient An//K. (Equivalently, P determines a toric variety XP along
with a choice of T -equivariant ample line bundle.) In Lie(T ), we instead consider fans
Σ such that |Σ| = σ. Such a fan determines a toric variety XΣ with affinization X0, and
1
in this case the map is proper, but not necessarily projective. In this case, toric varieties
which are proper over X0 are in bijection with fans.
By analogy, the current state of hypertoric varieties is that there is a construction
which takes a central hyperplane arrangement A0 to an affine variety Y0, and which
takes an affinization A of A0 to a variety Y which is projective over its affinization
Spec k[Y ] = Y0. However, there is no abstract definition in the literature, and there
is no equivalent to the fan construction for toric varieties. The natural combinatorial
structure dual to a central arrangement is a zonotope, and the natural structure dual
to an affine arrangement is a zonotopal tiling. Hence, we define a construction which
takes a zonotope Z to an affine variety Y (Z), and a zonotopal tiling to a variety Y (T ).
Additionally, we provide an abstract definition of “hypertoric variety,” and show that
our varieties satisfy this definition if and only if the tiling is full-dimensional.
1.1. Preliminaries
We now give some definitions that build up to the definition of hypertoric variety.
(Definition 1.4)
Fix an algebraically closed field k. We call a variety X over k convex if k[X] is
finitely generated and the natural map pi : X → X0 := Spec k[X] is proper. In this
case, we will also say that X is convex over X0. We call X semiprojective if pi is
projective. We call a line bundle L on X (relatively) very ample if it is very ample
with respect to the map X → X0; thus X is semiprojective if and only if it admits a
very ample line bundle.
Definition 1.1 (Beauville) [Bea00] A symplectic variety X over k is a normal
Poisson variety such that
2
– the Poisson structure on X is induced by a symplectic form ω ∈ Ω2(Xreg) on the
regular locus
– for some (equivalently any) resolution pi : X˜ → X, the form pi∗ω extends to a
2-form on X˜.
We say that an action of Gm on X has nonnegative weights if the induced
grading on k[X] is zero in negative degrees. We say that the action has positive
weights if, in addition, k[X]Gm = k.
Definition 1.2 A symplectic variety X over k is conical if there exists an action of
Gm on X with positive weights, such that the Poisson structure is homogeneous of
negative weight.
Note that the choice of Gm action is not part of the data of a conical symplectic
variety. We call a particular Gm action a weight m action if the Poisson structure is
homogeneous of weight −m.
Definition 1.3 A partial conical symplectic resolution (PCSR) over k is a convex
conical symplectic variety X such that the map X → X0 is an isomorphism over Xreg0 .
We now provide our abstract definition of “hypertoric variety.” Fix a torus T and
d = dimT .
Definition 1.4 A T -hypertoric variety is a PCSR X of dimension 2d equipped with
an effective Hamiltonian action of T . We further require that the conical action of Gm
can be chosen to commute with T .
When we have to choose a moment map µ for the action of T , we always choose the one
which takes the Gm-fixed points to the origin. The level set µ−1(0) which contains these
points is called the extended core of X. We are primarily interested in comparing the
3
Gm and T actions on the extended core, because the extended core is the only level set
of µ which is preserved by Gm.
We use ρ(t, s) denote the automorphism of X induced by (t, s) ∈ T ×Gm.
Definition 1.5 For a cocharacter η of T and a T ×Gm-subvariety X ′ ⊂ X, we say that
Gm acts by η on X ′ if ρ(η(s), 1)|X′ = ρ(1, s)|X′ for all s ∈ Gm. We call η a matched
cocharacter if there exists a T ×Gm-subvariety on which Gm acts by η but does not
act by any other cocharacter.
Definition 1.6 Given a T ×Gm-variety X, we define the η-twisted action to be the
action where (t, s) acts by ρ(η(s)t, s).
In Chapter III, we define a construction Z 7→ Y (Z) which takes as input a weight 2
integral zonotope in the cocharacter lattice N := X∗(T ) and produces an affine Poisson
T ×Gm-variety. In Chapter IV we define T 7→ Y (T ) which takes as input a weight 2
integral zonotopal tiling in the cocharacter lattice N := X∗(T ) and produces a Poisson
T ×Gm-variety. It will be immediately clear from the definitions that if T is the trivial
tiling of the zonotope Z, then Y (Z) = Y (T ) and that the Gm action has weight 2.
Many characteristics of Y (T ) can be determined from the combinatorics of zonotopes
and zonotopal tilings, as defined in Chapter II.
– Y (T ) is a hypertoric variety if and only if T is a full-dimensional tiling. (Theorem
5.3).
– Given two tilings T and T ′, the varieties Y (T ) and Y (T ′) are isomorphic as
Poisson T -varieties if and only if T is a translate of T ′ and are isomorphic as
Poisson T ×Gm-varieties if and only if T = T ′. (Proposition 5.6)
4
– Every refinement of tilings T ≤ T ′ induces a map Y (T ′)→ Y (T ). The affinization
of Y (T ) is the map Y (T )→ Y (|T |). In particular, Y (T ) is affine if and only if T
is the trivial tiling of some zonotope. (Corollary 4.1)
– Strictly convex support functions on T are in bijection with T -equivarant ample
line bundles on Y (T ). In particular, Y (T ) is projective if and only if T is regular.
(Proposition 5.9)
– There is one extended core component E(T )v for each vertex v of T , a toric variety
with associated fan Σv. The intersection of two components E(T )v ∩ E(T )v′ is a
toric variety with fan ΣZ , where Z is the smallest zonotope containing v and v
′,
and is empty if T contains no such zonotope. (Proposition 5.5)
– The inclusion of a face of a zonotope F ⊂ Z corresponds to the inclusion of a
subvariety Y (F ) ⊂ Y (Z). (Note that if a zonotope has positive codimension, then
the associated variety is not a hypertoric variety.) (Corollary 3.12) The variety
Y (T ) is the colimit of the directed system {Y (Z)|Z ∈ T }, where morphisms are
given by face inclusions. (Corollary 4.3)
– For a cocharacter η, the η-twisted Gm action has nonnegative weights if and only
if η ∈ |T | and it has positive weights if and only if η is in the interior of |T |.
(Proposition 5.7)
We also make the conjecture that we have described all hypertoric varieties.
Conjecture 1.7 For every T -hypertoric variety Y , there exists a full-dimensional
weight-two zonotopal tiling T in N (unique up to translation by Theorem 5.3) such that
Y is isomorphic to Y (T ) as a Poisson T -variety.
5
We end this section by giving examples of results on toric varieties and analogous
results on hypertoric varieties. (Table 1.1)
Toric result Hypertoric result
A cone σ in N determines an affine
toric variety X(σ).
A zonotope Z in N determines an
affine hypertoric variety Y (Z).
A cone σ∨ in M := N∗ determines
an affine toric variety X(σ).
A hyperplane arrangement A0 in
M determines an affine hypertoric
variety Y (A).
A polytope P whose recession
cone is σ∨ determines projective
toric partial resolution X(P ) of
X(σ).
An affine hyperplane arrangement
A with associated central
arrangement A0 determines a
projective hypertoric resolution
Y (A) of Y (A0).
A refinement Σ of σ determines
a toric partial resolution X(Σ) of
X(σ)
A tiling T of Z determines a
hypertoric partial resolution Y (T )
of Y (Z).
A support function φ on Σ
determines a T -equivariant line
bundle L(φ) on X(Σ), which is
ample if and only if φ is strictly
convex.
A support function φ on T
determines a T -equivariant line
bundle L(φ) on Y (T ), which is
ample if and only if φ is strictly
convex.
TABLE 1.1. A comparison of results on toric and hypertoric varieties
6
CHAPTER II
COMBINATORICS
We now present standard material on oriented matroids, zonotopes, and hyperplane
arrangements. See for example [BLVS+99] or [Zie95].
Throughout this chapter, a = (a1, . . . , an) is any configuration of vectors in a
lattice N which spans NR
1, and ν is an additional integral vector in N . Given such a
configuration, we describe three related constructions: the oriented matroid M(a), the
linear hyperplane arrangement A(a), and the zonotope Z(a).
2.1. Sign Vectors and Oriented Matroids
We refer to an element of the set {+1,−1, 0}, which we abbreviate {+,−, 0}, as a
sign, and for any index set E, an element of {+,−, 0}E as a sign vector.2 Given a
real number λ, we use sign(λ) to mean +, −, or 0 if λ is positive, negative, or zero, and
given a vector (λi) of real numbers, we define the sign vector sign(λ) componentwise.
Finally, we define a partial order on signs by 0 < + and 0 < −, and define a partial
order on sign vectors componentwise.
Given a sign vector u, we define the support of u to be the set {i ∈ E|ui 6= 0}.
Given two sign vectors u and v, we define the separation set S(u, v) to be the subset
{i|ui = −vi 6= 0} of their mutual support on which they disagree. We define uv as the
componenent wise product (uv)i = uivi, and we say u ⊥ v if uv consists of only 0s, or
has at least one + and at least one −. Finally, we define u ◦ v by (u ◦ v)i = ui if ui 6= 0,
and vi otherwise.
1All of the combinatorial definitions are valid for any configuration, spanning or not. The difference
is only important in Remark 2.6.
2We will almost always use either [n] := {1, . . . , n} or [n] ∪ 0 as index sets.
7
Given a vector configuration a, we can define two sets of sign vectors V =
{sign(λ)|∑λiai = 0} and V∗ = {sign(λ(a))|λ ∈ N∗}. These are our primary examples
of oriented matroids:
Definition 2.1 An oriented matroid M is a set E and a collection F of sign vectors
indexed by E satisfying:
– 000 · · · 0 ∈ F
– If u ∈ F , then −u ∈ F
– If u, v ∈ F , then u ◦ v ∈ F
– If u, v ∈ F and i ∈ S(u, v), then there is w ∈ F such that wi = 0 and wj = (u◦v)j
for all j /∈ S(u, v).
We have the following from [Zie95]:
Proposition 2.2 Let a ∈ Nn. Then M(a) = ([n],V) and M∗(a) = ([n],V∗) are
oriented matroids.
Proposition 2.3 Let (E,F) be an oriented matroid. Then (E,F⊥) is also an oriented
matroid, where F⊥ := {u : u ⊥ v for all v ∈ F}.
The entire set F can be reconstructed from its maximal elements Fmax or its
minimal nonzero elements Fmin with respect to the partial order <.
Proposition 2.4 Let a ∈ Nn. Then V = (V∗)⊥ = (V∗min)⊥ = (V∗max)⊥ and V∗ = V⊥ =
V⊥min = V⊥max.
For any sign vector u on an index set E and set E ′ ⊆ E, we define u|E′ to be the
sign vector obtained by deleting all entries indexed by E \ E ′. If M = (E,F) is a
matroid and E ′ ⊆ E, we define M|E′ to be (E ′, {u|E′ : u ∈ F}).
8
If u and u′ are sign vectors on E and E ′, we use u|u′ to represent the sign vector
on E unionsq E ′ obtained by concatenating their entries (and likewise for a|a′).
2.2. Zonotopes and Linear Hyperplane Arrangements
Given a configuration a in N , we can make two geometric constructions:
Definition 2.5 A (weight 2) zonotope in NR is a polytope which is Minkowski sum
of integral line segments of even length. We define Z(a) :=
∑n
i=1[−1, 1] · ai and more
generally, for any sign vector u, Z(a, u) :=
∑n
i=1 uiai +
∑
ui=0
·ai.
Remark 2.6 Zonotopes of the form Z(a) are precisely the full-dimensional zonotopes
centered at 0. We will always use such a zonotope, and all other zonotopes will be
subsets of the form Z(a, u).
The term “weight 2” is a reference to the length of [−1, 1]. We assume all zonotopes
are weight 2 until Section 5.3.
Alternately, we may view Z(a) as the convex hull of the points
∑n
i=1±ai. It is
clear that the zonotope is unchanged if we permute the ai or replace ai with −ai; we
call such an action a relabeling. It is also unchanged if we replace ` copies of ai with a
single vector `ai or vice versa; we call this a multiplicity operation. Conversely, we
may recover a up to relabeling and multiplicity: each edge is a translate of [−m,m] · ai
for some ai; the vectors ai and −ai appear a total of m times if both length and multiple
appearances are counted.
Figure 2.1 gives an example of a zonotope and a relabeling.
9
origin a1
a3 a2
−+ + + + +
+ +−
+−−−−−
−−+
a1
a2
a3
−+− + +−
+−−
+−+−−+
−+ +
FIGURE 2.1. A zonotope with vertices labeled, and a relabeling of the same zonotope.
To avoid confusion, no axes are drawn, and vectors are based at the origin.
Definition 2.7 We call a zonotope Z a parallelotope if there is an injective affine
map φ : Rn → NR such that φ(Zn) ⊂ N and φ([−1, 1]n) = Z. We call a parallelotope a
cube if φ(Zn) = N .
Definition 2.8 Fix an edge E of a zonotope Z. We define a zone to be all faces of Z
which contain an edge parallel to E.
We note that for any face F of Z(a), there is a unique sign vector u so that
F = Z(a, u).3 Given a zonotope Z, we may construct sign vectors without explicit
reference to a: we number the zones Z1, . . . ,Zn of Z. For each zone Zi, the set
⋃
F /∈Zi F
contains two connected components; we arbitrarily label one the positive side and one
the negative side. Then to each face F , we give it the sign vector whose ith coordinate
is + or − if F is on the positive or negative side of Zi, and 0 if F ∈ Zi.
3In the case of a non-face subzonotope, neither existence nor uniqueness of the sign vector is
guaranteed.
10
Definition 2.9 Let F be a face of a zonotope Z; then we define the cone associated to
F , CF,Z := R+(Z − F ) = {r(n1 − n2) : r ∈ R, n1 ∈ Z, n2 ∈ F}. If there is no ambiguity,
we write CF,Z as CF .
Definition 2.10 Given a ∈ Nn, for each nonzero ai, we define Hi = a⊥i ⊂ M := N∗,
and we define A(a) = {Hi}1≤i≤n. We also define H+i and H−i to be the closed half
spaces which ai maps to [0,∞) and (−∞, 0], respectively.
Given a sign vector u, we define Mu =
⋂
ui 6=0H
ui
i ∩
⋂
ui=0
Hi. We have that Hu is
not empty if and only if u ∈ V∗. For each χ ∈M , we can associate a unique smallest u
so that χ ∈Mu.
There is a bijection between the faces of Z(a) and of A(a) given by sign vectors.
We may describe this bijection more directly by using the fact that elements of M are
linear functionals on N (and thus on Z). Given Mu, Zu consists of the subset of Z
that maximizes all elements of Mu (it suffices to choose an interior element of Mu).
Conversely, given Zu, Mu consists of the elements of M that are maximized at every
point of Zu.
2.3. Zonotopal Tilings
All of the structures in this section are informally “one dimension up” from those
in the previous section. More precisely, an affine oriented matroid is an oriented matroid
on the ground set E ∪ {0}, an affine hyperplane arrangement in MR is equivalent to
a hyperplane arrangement in MR ⊕ R, and many (though crucially, not all) zonotopal
tilings of a zonotope Z ⊂ NR are equivalent to a zonotope Z˜ ⊂ NR ⊕ R.
An affine oriented matroid is an oriented matroid with a distinguished element.
More precisely,
11
Definition 2.11 An affine oriented matroid (E,F , g) is a set E, and element g ∈ E,
and a set F of sign vectors on E such that (E,F) is an oriented matroid and there is
at least one sign vector u ∈ F with ug 6= 0. (In matroid terminology, g is not a loop.)
The positive covectors F+ are the set of sign vectors {u ∈ F : ug = +}. A positive
covector u is called a bounded covector if there is no sign vector v in F with u < v
and vg = 0.
In practice, we will always use 0 as the distinguished element, and conversely if
0 ∈ E then we are viewing the associated matroid as an affine matroid. We say that
(E∪{0}, F ) is an affine matroid over the oriented matroid (E, {u|E : u ∈ F and u0 = 0}).
Note that F = ((F+)⊥)⊥, so that any affine oriented matroid is identified by its
positive covectors.
Definition 2.12 A zonotopal tiling T is a collection of zonotopes such that
– |T | := ⋃Z∈T Z is a zonotope.
– If F is a face of Z ∈ T , then F ∈ T .
– If Z,Z ′ ∈ T , then the intersection Z ∩ Z ′ is a face of both Z and Z ′.
In this case we says that T is a zonotopal tiling of |T |.
Figure 2.2 gives two examples of zonotopal tilings.
Definition 2.13 Let a ∈ Nn be a vector configuration, and M+ be the positive
covectors of some oriented matroid over the matroid M(a). Then T (a,M+) :=
{Z(a, u)|u ∈M+}.
Tilings correspond to oriented matroid extensions. [RGZ94, Theorem 1.7]
12
FIGURE 2.2. Two different tilings of the same zonotope.
Theorem 2.14 (Bohne-Dress) Let a ∈ Nn be a configuration. Then T (a,M+) is a
zonotopal tiling of Z(a). Furthermore, all zonotopal tilings of Z(a) arise this way.
To describe the inverse map, we generalize the notion of zones to tilings.
Definition 2.15 Divide the edges of T into equivalence classes using the relation
generated by E ∼ E ′ if E and E ′ are opposite edges of some zonotope Z ∈ T . Fix one
such equivalence class; the collection of all zonotopes which contain an edge from this
class is called a zone of the tiling. (See Figure 2.3.)
Remark 2.16 If all edges of |T | are edges of T , then E ∼ E ′ if and only if E is parallel
to E ′.
If T is a tiling, then we obtain a sign vector for each zonotope as before: number the
zones and designate one side as “positive” and one as “negative.” From here we obtain
both a configuration a and a set M+ of sign vectors. By the Bohne-Dress Theorem,
these are the sign vectors of an orientation. If we wish to use a designated representation
Z = Z(a′), then we may transform a to a′ by relabeling and multiplicity operations,
provided we apply the same transformations to M+.
13
Definition 2.17 Let Z be a zonotope in T . Then the fan associated to Z is
ΣZ := {CZ,Z′|Z is a face of Z ′ ∈ T }.
We note that a zonotopal tiling can be recovered from the set of vertices and their
associated fans.
Definition 2.18 Given two tilings T and T ′ with |T | = |T ′|, we say that T ′ refines T
if, for every Z ′ ∈ T ′, there is a Z ∈ T such that Z ′ ⊆ Z. In this case, we write T ′ ≤ T .
If T = T (a,M+) and T ′ = T (a,M′+), we say that M+ and M′+ are compatible
if, for u ∈M+ and u′ ∈M′+, Z(a, u) ⊂ Z(a, u′) implies u ≥ u′.
Definition 2.19 We define a support function φ for T to be a piecewise linear
function |T | → R such that
– φ(N ∩ |T |) ⊂ Z
– φ is affine-linear on each zonotope of T , and
– if v′ and v are opposite vertices of |T |, then φ(v′) = −φ(v).
A support function is called strictly convex if it is convex and the maximal domains
of linearity are the maximal zonotopes of T . If there exists a strictly convex support
function for T , then T is called a regular zonotope.
Note that in the case of a zonotope centered at 0, the third condition becomes
φ(−v) = −φ(v). If m and m + ai are both in the ith zone, then φ(m + ai) − φ(m) is
independent of m; we call this integer the slope along ai and refer to it as ri. The points
(ai, ri) and (−ai,−ri) in NR × R are then the vertices of a zonotope Z(φ) (centered at
14
zero by the third condition), and for any a ∈ NR, φ(a) is the maximum value of r such
that (a, r) ∈ Z(φ).
FIGURE 2.3. The bold line segments are an equivalence class. Together with the gray
two-dimensional zonotopes, they form a zone of the tiling. The tiling which includes the
dashed edges and the three parallelotopes which contain them is called the non-Pappus
tiling, and is well-known to be non-regular.
Such a tuple (r1, . . . , rn) can also be used to define an affine arrangement by
translating Hi by ri; more precisely, we define H˜i := {ai + ri = 0}. Then each
hyperplane still has a positive and a negative side as before, and we may again define
chambers Mu. There is of course a bijection between affine arrangements and support
functions given by using the same ri. However, this has a more geometric meaning.
Given a configuration a and a tuple (r1, . . . , rn), define a˜i := (ai, ri) ∈ N ⊕R. Then
the lower faces of Z(a˜) determine a convex support function on a tiling of Z(a), and
the arrangement {H˜i} is the the intersection of the arrangement A(a˜) with NR × {1}.
15
CHAPTER III
VARIETIES FROM ZONOTOPES
Rather than an abstract lattice N , we now assume that T is a torus over a field k,
and a is a vector configuration in N := X∗(T ) which is compatible with k in the sense
that no nonzero ai has a length which is a multiple of the characteristic of k.
A configuration a ⊂ N determines a map from the coordinate torus Gnm to the
torus T . In general, we only have an exact sequence
1→ K → Gnm → T → T ′ → 1
where the connected component K◦ of K is a torus, and K/K◦ is a finite group, and T
and T ′ are tori. If the configuration spans NR, then T ′ is trivial, and if the sublattice
generated by a is saturated, then K = K◦. Unless otherwise noted, we assume for
the rest of this chapter that a spans NR. (We do not assume that the sublattice they
generate is saturated.) If N ′ is a lattice, then we define TN ′ := N ′ ⊗Gm to be the torus
with cocharacter lattice N ′. If N ′ ⊂ N , then TN ′ is a finite quotient of a subtorus of
T = TN . This allows us to write a short exact sequence of tori
1→ K◦ → Gnm → TZa → 1
which is easier to work with. In particular, if we choose isomorphisms K◦ ∼= Gkm
and T ∼= Gdm, then the map from Lie(Gnm) to Lie(T ) is given by an n × d matrix
A = (a1| · · · |an). Then a cocharacter β of Gnm is a cocharacter of K◦ if
∑n
i=1 βiai = 0.
The dependences β are a vector space of dimension k; we choose a basis {bj} of these
to be the rows of a matrix B. Then the map Lie(K) → Lie(Gnm) is given by BT . By
16
choosing this basis carefully, we can make computations easier. Among other things,
throughout this paper it may be assumed that sign(bj) is minimal.
3.1. Varieties from Arrangements and Signed Arrangements
In this section we define the variety Y (Z) for any zonotope Y (Z). First, given a
spanning configuration a, we define Y (a) as the categorical quotient by K of the level
set µ−1K (0). We then define the variety Y (a, u,N) by removing coordinate hyperplanes
from T ∗An before taking the quotient. Finally, we show that the variety obtained by
either construction depends only on the zonotope Z(a) or Z(a, u).
We coordinatize T ∗An = Spec k[x1+, . . . , xn+, x1−, . . . , xn−]. If necessary for clarity,
we may write xi,+ or xi,− instead. For convenience, we often write xi± to mean both
xi+ and xi−; thus we could instead write k[xi± : 1 ≤ i ≤ n]. We use x+ = (x1+, . . . , xn+)
and likewise for x−. Finally, we define xi = xi+xi−.
Given a configuration a ∈ Nn of length n, we define 1
U(a, N) := Spec k[x1+, . . . , xn+, x1−, . . . , xn−] ∼= T ∗An
with the following structures:
– The Poisson structure is given by the symplectic form
∑n
i=1 dxi+ ∧ dxi−.
– The Gnm action is given by t · (x+, x−) = (tx+, t−1x−).
– This action is Hamiltonian with moment map µn(x+, x−) = (x1, . . . , xn) ∈ An ∼=
Lie(Gnm)∗
– The Gm action has weight one on all variables s× (x+, x−) = (sx+, sx−).
1We do not need all the data of (a, N) to define U(a, N), but this keeps U(a, N) consistent with
other notation. In particular, we use the length of a.
17
We will define the associated affine hypertoric variety Y (a, N) as the algebraic
symplectic quotient U(a, N)////K, which will require both taking a level set of the
moment map µK and taking a GIT quotient by K. This may be done in either order,
and it is useful to have notation for the result of either operation done alone.
Definition 3.1 Let a be a configuration of vectors in N that spans NR. Then we
define L(a, N) = V (µK) ⊆ U(a, N) and X(a, N) := U(a, N)//K, and Y (a, N) =
L(a, N)//K ∼= V (µK) ⊂ X(a, N).
Proposition 3.2 Let a′ and a′′ be vector configurations in lattices N ′ and N ′′,
respectively. Then Y (a′|a′′, N ′ ⊕N ′′) ∼= Y (a′, N ′)× Y (a′′, N ′′) as TN ′⊕N ′′ ∼= TN ′ × TN ′′-
Poisson varieties.
Proof: We may choose B to be in block diagonal form. Then we have
k[Y (a′|a′′, N ′ ⊕N ′′)] = k[T ∗An′ × T ∗An′′ ]K′⊕K′′/(µK′ , µK′′)
∼= k[T ∗An′ ]K′/(µK′)⊗ k[T ∗An′′ ]K′′/(µK′′)
∼= k[Y (a′, N ′)]⊗ k[Y (a′′, N ′′)]
2
In particular, we may add or delete 0 entries of a without affecting the variety,
since Y ((0), 0) is a point.
Proposition 3.3 Let a be a configuration of vectors in N that spans NR. Then
Y (a, N) = Y (a,Za)/(K/K◦).
18
Proof: We have that L(a) := L(a, N) = L(a,Za), since it does not depend upon the
ambient lattice. Then k[Y (a, N)] = k[L(a)]K = (k[L(a)]K
◦
)K/K
◦
= k[Y (a,Za)]K/K◦ =
Spec k[Y (a,Za)]//(K/K◦), and the quotient is geometric since K/K◦ is finite. 2
Proposition 3.4 If a and a′ are configurations of vectors in N , compatible with k, with
Z(a) = Z(a′), then Y (a, N) ∼= Y (a′, N).
Proof: It suffices to consider the case where a and a′ differ by a permuation, a sign
change, or a multiplicity operation. In the first two cases, the isomorphisms are obvious.
In the third case, we assume that the operation takes place at the end, so that ai = a
′
i
for i < n, and an = `a
′
i for i ≥ n. We choose B to have the form
B =
 B′′ 0
`C 1

and
B′ =

B′′ 0 0 0
C 1 −1 0
0
. . .
0 0 1 −1

Define a map k[Y (a′)]→ k[Y (a)] by x′i± 7→ xi± for i < n, and x′n± · · ·x′n′± 7→ xn±/` for
i ≥ n, and x′i 7→ xn/`. 2
Note that the compatibility with k is essential; if ` divides chark, then no such
map exists. (There is a map Y (a)→ Y (a′) which is an isomorphism of varieties, but
not of Poisson varieties.)
19
We also may define varieties using sign vectors; if ui 6= 0, then we remove the
locus xi,ui = 0 for each nonzero ui. We use xu :=
∏
xi,ui , where xi,0 = 1. We begin
with U(a, u,N) := Spec k[xi±]xu ⊂ U(a, N), and define the other three analogously:
L(a, u,N) := U(a, u,N) ∩ L(a, N), X(a, u,N) := U(a, u,N)//K, and Y (a, u,N) :=
L(a, u,N)//K ∼= V (µK) ⊂ X(a, u,N). Note that U(a, N) = U(a, 0, N), and thus
likewise for L(a, N), X(a, N), and Y (a, N).
Lemma 3.5 Let (a, u) be a signed configuration and (a′, u′) be the signed configuration
obtained by appending η ∈ N to a and + to u. Then Y (a, u) ∼= Y (a′, u′) as T -varieties,
with Gm actions twisted by η.
Proof: Write B′ by appending a column of 0s to B, and then appending a single
row expressing a′n′ as a linear combination of ai. Then Y (a
′, u′, N) = (Y (a, u,N) ×
k[xn′,±, x−1n′+])
K′/K/(f), where
f = xi− − 1
bn′,d′xi+
n∑
i=1
bi,d′xi
Hence Y (a′, u′, N) = (Y (a, u,N) × k[x±1n′,+])K
′/K , and there is a bijection between K-
invariant monomials in the first n variables, and K ′-invariant monomials in all n′
variables given by the last row of B. 2
In particular, we may append to a any configuration of vectors which sums to 0
and to u the corresponding number of +s without changing Y (a, u,N).
Proposition 3.6 Let (a, u) and (a, u′) be signed configurations in N . If Z(a, u) =
Z(a′, u′), then Y (a, u,N) ∼= Y (a′, u′N) as Poisson T × Gm-varieties. If Z(a, u) =
20
Z(a′, u′) + η for some cocharacter η, then they are isomorphic as Poisson T -varieties,
but the Gm action on Y (a, u,N) is the η-twisted action on Y (a′, u′, N).
The converse of both statements is true; this is proved in the next chapter. In light
of Proposition 3.6, we make the following definition:
Definition 3.7 Let Z = Z(a, u) be a zonotope in N such that a spans NR. Then
Y (Z,N) := Y (a, u,N). If N is understood, we write Y (Z) for Y (Z,N).
Proposition 3.8 There is a TN∩Ra ×Gm-equivariant isomorphism Y (a, u,N ∩ Ra)×
T ∗(TN/TZa) ∼= Y (a, u,N) that is T -equivariant on the second factor.
Proof: Let e = (e1, . . . , e`) be vectors in N that descend to a basis of N/Za. Then
(U(a, u,N)× T × T )////Gnm = U(a|e|e, u|+ · · ·+ | − · · · −, N)////K ′, where K ′ is the
kernel of the map given by the configuration a|e|e. 2
Remark 3.9 All zonotopes are expressible in the form Z(a, u), with many nonzero
signs in the case of full-dimensional zonotopes. We can also define Y (a, u,N) where
a does not span N . Since we wish Proposition 3.6 to hold under this new definition,
we should be able to append pairs of opposite vectors to a, with corresponding +s
appended to u, without changing Y .
Let a be any configuration of n vectors in N . Then we define Y (a, N) =
(U(a, u,N) × T × T )////Gnm, where Gm acts on U(a, u,N) as normal, and on T × T
via (A(t), A(t)−1), where A is the map Gnm → T .
21
3.2. Zonotopes and the Extended Core
Before talking about zonotopes, we need a more concrete description of the action
of Gnm (particularly K) on U(a, N). Let pii be the projection U(a, N) → T ∗A1 =
Spec k[xi±], β ∈ Zn be a character of Gnm, and p ∈ T ∗A. Then there are three possibilities
for lifting the map φβ,i,p : Gm → T ∗A1 given by t 7→ tβ · p to a map A1 → T ∗A1 .
– If the action is nontrivial and p is off the coordinate axes then the orbit is closed,
and φβ,i may not be extended.
– If the action is nontrivial and p is on one coordinate axis, then the origin is in the
closure of tβ · p, and exactly one of φβ,i or φ−β,i may be extended.
– If p is a fixed point, then either p is the origin or βi = 0, and both φβ,i and φ−β,i
may be extended.
We now consider a point p ∈ U(a, N) and the map φβ,p : Gm → U(a, N) given by
t 7→ tβ · p. We define a sign vector indexed by the coordinates of µn(p) which are zero,
where up,i = + if and only if xi+ 6= 0, up,i = − if and only if xi− 6= 0. The map φβ,p may
be extended if and only if it may be extended coordinatewise.
We decompose the set µ−1n (0), where at least one of xi+ or xi− is 0 for every i, as
the union of Gm-orbits O′u where xi,ui 6= 0, and xi,j = 0 otherwise. Clearly, O′u is in the
closure of O′v if and only if u ≤ v, but we can say more about how this closure interacts
with the action of Gnm.
Proposition 3.10 Let β ∈ Zn be a cocharacter of Gnm, let p ∈ O′u. Then tβ · p is closed
if and only if u ⊥ sign(β). The map φβ may be extended if and only if (u sign(β)) has
no −s.
22
The interesting case is where the orbit is not closed and the map may be extended:
in this case the image of 0 under the extension is in the closure of O′u but not in O
′
u.
Proposition 3.11 The K-orbit of a point p ∈ O′u is closed if and only if u ∈M(a). If
u /∈M(a), then the unique closed K-orbit in the closure of K · p is O′v, where u covers
v.
Proof: If u ∈ M(a), then for every cocharacter β of K, sign(β) ∈ M∗(a), so that
β(Gm) · p is closed. If u /∈M(a), then there is some cocharacter β such that sign(β) is
not perpendicular to u. Since sign(β) ⊥ v, it must be that sign(β) and v have disjoint
support. Aassume that wherever sign(β) and u agree, they both have +. Then we have
that lim β(t) ∈ O′w, where wi = 0 if ui = sign(βi) = + and wi = ui otherwise. If w = v,
then we are done; otherwise we may proceed by induction, since u > w ≥ v. 2
Since u ∈M(a) if and only if Z(a, u) is a face of Z(a), this implies:
Corollary 3.12 Let Z be a zonotope and F be a face of Z. Then there is a canonical
inclusion Y (F )→ Y (Z).
3.3. The Extended Core
Definition 3.13 We define the extended core E(a, u,N) of Y (a, u,N) to be the
subvariety2 µ−1(0), where µ is the moment map for the T action.
This is also the subvariety of X(a, u), defined by the ideal (µn). The ideal (µn) is
not prime, but it has an obvious decomposition as (µn) =
⋂
u((x1,u1), . . . , xn,un).
We may now describe the coordinate ring k[Y (Z)] more explicitly:
2That is, we give the reduced scheme structure to the subscheme µ−1(0).
23
By Definition 3.1, Y (Z) ∼= Spec k[xi±]K/(µK) is a subset of X(Z) := Spec k[xi±]K .
The varietyX(Z) is an affine toric variety, called the Lawrence toric variety associated
to Z. It is clear that the components xi = xi+xi− of the map µn are primitive K-invariant
monomials, and therefore all other primitive K-invariant monomials contain at most
one of xi+ or xi− for each i. We call any monomial in k[xi±] which is not in the ideal
generated by µn an extended core monomial, or an EC monomial for short. Thus,
the coordinate ring k[X(Z)] = k[xi±]K is generated by n moment map components
xi+xi−, and some number of invariant EC monomials.
Note that only invariant EC monomials (monomials in k[xi±]K which are not in
the ideal (µ)) are functions on the extended core, but all EC monomials are sections of
line bundles (and thus determine subvarieties).
Note that the extended core is a T ×Gm-variety, whose components are T -toric
varieties. More specifically:
Definition 3.14 For any sign vector u, we define Vu to be the subvariety defined by
xi,−ui = 0, and we define Ou to be the open subset of Vu where xi,ui 6= 0.
Proposition 3.15 The components of the extended core are the Vu for maximal
covectors u. The associated cone is the cone over ua := (uiai)1≤i≤n.
Proof: By relabeling a, it suffices to prove that the statement when u is the sign vector
of all +s, or the subvariety where xi− = 0 for all i. However, this is the T -toric variety
corresponding to the cone over the ai. This toric variety has the same dimension as T if
and only if the cone over the ai is pointed, which occurs exactly when u is a maximal
covector. 2
It is also clear the Ou is the dense T -orbit of Vu.
24
We now use matched cocharacters and positive weight cocharacters to recover Z
from Y (Z).
Proposition 3.16 If η ∈ Z(a, u) for some u ∈ M(a), then Gm acts by η on Vu. In
particular, the matched cocharacters of Y (Z) are the vertices of Z.
Proof: The action of Gm on Vu is given by the cocharacter
∑
aiui. Furthermore, the
action of Gm by ai is trivial if ui = 0 since xi+ = xi− = 0. But every η ∈ Zu is of the
form η =
∑
uiai +
∑
ui=0
ciai for some integers ci. 2
Proposition 3.17 Let η ∈ N . Then the action of Gm on Y (Z) by s ×η p := sη(s)p
has nonnegative weights if and only if η ∈ Z. It has positive weights if and only if η is
in the interior of Z.
Proof: Moment map components always have weight 2, and a cocharacter has positive
weights on Y (Z) if and only if it has positive weights on E(Z), if and only if it has positive
weights on each component of E(Z) where it is nonzero, and similarly for non-negative
weights. Finally, it has positive weight on a component corresponding to a vertex v
if it is in the interior of the cone Σv, and non-negative weights if it lies in the cone at all. 2
3.4. Smoothness
We now give a criterion for Y (Z) to be smooth:
Proposition 3.18 The following statements are equivalent:
1. Z is a cube
25
2. The variety Y (Z) is smooth
3. The variety Y (Z) is smooth at the origin
Proof: If Z is a cube, then we may write it as Z = (e1, . . . , en) for some basis e1, . . . , en
of N . Hence, Y (Z) = T ∗An. If Z is not a cube, note that Y (Z) is a codimension k
subvariety of X(Z), and T ∗0 Y (Z) is a codimension k subspace of T
∗
0X(Z). Since X(Z)
is singular at the origin, so is Y (Z). 2
If Y (Z) is not smooth, then its regular locus is what might combinatorially be
expected.
Proposition 3.19 The regular locus Y (Z)reg =
⋃
F is a cube Y (F ).
Proof: If p ∈ Y (F ) where F is a cube, then p is a regular point since Y (F ) is smooth.
For the converse, let F be the smallest face such that p ∈ Y (F ) and suppose that F is
not a cube. Then p maps to (0, p′) ∈ Y (F )× T ∗(TN/N ′) under the isomorphism of 3.8,
and hence is a singular point. 2
Proposition 3.20 If Z is a parallelotope, then Y (Z) is an orbifold.
Proof: In this case Z = Z(a) for some basis of NR. 2
26
CHAPTER IV
VARIETIES FROM TILINGS
We now construct varieties of the form Y (T ) for zonotopal tilings T . We are
now removing a set of codimension at least 2 from T ∗An, although the definition does
not take this form; instead of describing U(a,M+, N) as the complement of a set of
codimension 2, we describe it as the union of U(a, u,N) for u ∈M+.
Let M+ be the positive sign vectors of an affine oriented matroid over M(a).
Then we define U(a,M+, N) =
⋃
u∈M+ U(a, u,N) ⊆ U(a, N). As before, we define
X(a,M+, N) := U(a,M+, N)//K, L(a,M+, N) = U(a,M+, N) ∩ L(a, N), and
Y (a,M+, N) = L(a,M+, N).
The variety X(a,M+, N) is called a Lawrence toric variety, with affinization
X(a, N). Its fan is necessarily a subdivision of the cone associated to X(a, N). We
denote the associated cone ΣZ , and refer to the rays of ΣZ as ρi+ if they are associated
to xi+ or ρi− if they are associated to xi−. To a sign vector u, we associate the cone
σu := R+({ρi,j|j ≤ ui}). For any tiling T of Z, the map φ : Zu 7→ σu then associates
a collection of cones {φ(Z)}Z∈T to T . This collection determines a fan1, and U(T ) is
the open set associated to it; hence X(T ) := U(T )//K is a partial toric resolution of
X(|T |).
Because L(a,M+, N) ⊂ L(a, N) with complement at least codimension 2, we have
that their coordinate rings are equal, and thus so are K-invariant functions. The induced
map Y (a,M+, N)→ Y (a, N) is thus the affinization map. On the level of tilings, we
have the following proposition.
1The collection is not itself a fan, but it does determine one; it includes all maximal cones of the fan.
27
Proposition 4.1 Let T be any zonotopal tiling. Then Y (T )0 ∼= Y (|T |).
Note that if T consists of a zonotope Z and its faces, then U(Z) = U(T ), so that
we we may consider Z as a tiling.
4.1. Refinements and Partial Resolutions
For any refinement of tilings T ≤ T ′ of Z, we have an inclusion U(T ′) ⊂ U(T ),
which induces a T -equivariant Poisson map Y (T ′) → Y (T ). For each zonotope Z =
Z(a, u) ∈ T , the K-orbits in U(a, u,N) which are in U(a, T ′, N), are precisely the
K-orbits indexed by v ≥ u.
The next lemma says that these maps are all partial resolutions.
Proposition 4.2 The map Y (T ′) → Y (T ) is proper and an isomorphism over the
regular locus of Y (T ).
Proof: Choose compatible representations T ′ = T (a,M′+) and T = T (a,M+). On the
level of Lawrence toric varieties, X(a,M′+, N)→ X(a,M+, N) is proper; this remains
when taking a closed subvariety. The map is an isomorphism over the regular locus
because cubes cannot be subdivided. 2
The following description of Y (T ) is now evident:
Proposition 4.3 The variety Y (T ) is the colimit of the directed system {Y (Z)|Z ∈ T }
with morphisms given by the inclusions of faces as in Corollary 3.12.
28
CHAPTER V
THE MAIN THEOREM
We now prove that Y (T ) is a hypertoric variety for any full-dimensional tiling T .
It is clear that there is a positive weight Gm action, that the action of T is effective. By
Proposition 4.2 we have that Y (T ) is convex and a partial resolution. In the case that
Z is a parallelotope, we have symplecticness by the following result of Beauville:
Lemma 5.1 If Z is a full-dimensional parallelotope, then Y (Z) is a hypertoric variety.
Proof: In this case K is finite, so Y (Z) = A2n/K, which is symplectic by [Bea00, 2.4]. 2
We prove that Y (T ) is symplectic in general by using the maps φT ′,T and the
following lemma:
Lemma 5.2 Suppose that X → Y is a Poisson map of normal varieties that is an
isomorphism over Y reg. Then X is symplectic if and only if Y is.
Proof: Choose any resolution X˜ → X. If either X or Y is symplectic, then so is X˜
since it is a resolution of both. And if X˜ is symplectic, then both X and Y are, since
the map to each is an isomorphism over the regular locus. 2
Theorem 5.3 Let T be any full-dimensional zonotopal tiling. Then Y (T ) is a hypertoric
variety.
Proof: Choose a refinement T ′ of T by parallelotopes. To prove normality, let Z ∈ T .
Then Y (T ′|Z) is normal, hence Y (Z) = Y (T ′|Z)0 is too. Since the Y (Z) cover Y (T ),
29
Y (T ) is normal. To prove symplecticness, we note that Y (T ′) is symplectic, since by
Proposition 5.1 it has a cover by symplectic varieties; hence by Lemma 5.2 Y (T ) is
symplectic too. 2
5.1. The Extended Core
We now state the analogues of Propositions 3.16 and 3.17 for tilings.
Proposition 5.4 If η ∈ Z(a, u), then Gm acts by η on Vu.
Proof: The proof of 3.16 applies without modification. 2
We also note that Vu ⊂ Vv if and only if u ≥ v. Hence we can describe the extended
core more completely:
Proposition 5.5 There is one extended core component E(T )v for each vertex v of T , a
toric variety with associated fan Σv. The intersection of two components E(T )v∩E(T )v′
is a toric variety with fan ΣZ, where Z is the smallest zonotope containing v and v
′,
and is empty if T contains no such zonotope.
Proposition 5.6 Given two tilings T and T ′, Y (T ) and Y (T ′) are isomorphic as
Poisson T -varieties if and only if T = T ′ + η. In this case, the × action on Y (T ) is
the ×η action on Y (T ′).
Proof: If Y (T ) and Y (T ′) are isomorphic as Poisson T -varieties, then they have the
same number of extended core components, and hence T and T ′ have the same number
of vertices. Furthermore, the fans associated to corresponding vertices are the same.
This means that there are bijections between the zones of T and T ′ as well. If T and
30
T ′ are not translation-equivalent, then at least one zone is of different widths on the
two tilings. But then the two varieties do not have the same resolutions, and hence are
not isomorphic. 2
Because of the map φT ,T ′ , we can extend Proposition 3.17 to all varieties of the
form Y (T ).
Corollary 5.7 Let η ∈ N . Then the action of Gm on Y (T ) by s ×η p := sη(s)p has
nonnegative weights if and only if η ∈ |T |. It has positive weights if and only if η is in
the interior of |T |.
5.2. Line Bundles
For line bundles, we confine our discussion to the smooth case. Suppose that we
have a zonotopal tiling T consisting of cubes, and a support function φ. This support
function determines n integers ri, which we may interpret as a character (r1, . . . , rn) on
T n, which descends to a character α on K. Conversely, a character of K may be lifted
to a character of T n, which provides the ri to determine a support function.
Proposition 5.8 Let T be a tiling of cubes. Then strictly convex support functions on
T are in bijection with T -equivariant ample line bundles on Y (T ).
Proof: We have a commutative diagram
Zn H2Tn(T ∗An) H2T (Y )
X∗(K) H2K(T
∗An) H2(Y )
≈
≈
31
of Kirwan maps. Given r ∈ Zn, we have a T -equivariant line bundle in H2T (Y ). The
cone of strictly convex support functions is the GIT cone under this bijection; by [BPW,
2.22], the cone of strictly convex support functions in Zn corresponds to the ample cone
in H2(Y ). 2
Proposition 5.9 If φ is a strictly convex support function for T , then the set U(T ) is
precisely the semistable set for the character α.
Proof: We may lift φ to a support function on the fan of X(T ); by the theory of toric
varieties (for example, Section 14.2 of [CLS11]), X(T ) = T ∗An//αK. 2
Proposition 5.10 Let T be a zonotopal tiling. Then Y (T ) is a projective hypertoric
variety if and only if T is regular.
Proof: Let T ′ be a regular tiling of |T |. Then by the ample cone of any conical
symplectic resolution of Y (|T |) is a projective GIT quotient at some character of Gnm.
But we already know these quotients; they are the hypertoric varieties associated to
regular tilings. 2
5.3. Alternate Weights
We have chosen to focus on weight 2 actions of Gm. However, we can produce the
same results with any weight.
Definition 5.11 A weight m zonotope is a set of the form Zm(a) :=
∑n
i=0[0,m] · ai
or a translate Zm(a, ν) := ν + Z(a).
32
Note that although Z(a) 6= Z2(a), the two are translates of each other; hence this
definition of “weight 2 zonotope” agrees with our earlier one.
We define Ym(a), Ym(Z), and Ym(T ) to be the same as before, except that we
begin by equipping T ∗An with the action s× (z, w) = (smz, w). All of our proofs follow,
except the proof of conicality; we must translate Zm(a) so that the origin is an interior
point; this may not be possible in the case m = 1.
33
CHAPTER VI
RECOVERING THE TILING
We have two methods for recovering the tiling T from the variety Y (T ). Under
conjecture 1.7, this would provide classification of all hypertoric varieties.
First, we can focus on the extended core. In this case, we decompose the core into
its components, each of which is a T -toric variety. To each component, we thus may
associate a fan, and we may determine by which cocharacter Gm acts. In this case,
Conjecture 1.7 is then a statement that these fans are compatible, in the sense that they
do come from a zonotopal tiling.
Conjecture 6.1 Let Y be any T -hypertoric variety. Then:
1. For any matched cocharacter η, there is a unique orbit Oη on which it acts.
2. There is a zonotopal tiling T in N with vertex set equal to the matched cocharacters
of Y , and with Ση equal to the fan of the toric variety Vη.
3. Y is isomorphic to Y (T ) as a Poisson T ×Gm-variety.
Second, given a T -hypertoric variety Y , we can find a cover by affine hypertoric
varieties. For each of these affine patches, we can recover the associated zonotope by
finding the cocharacters which act with positive weights. In this case, Conjecture 1.7
implies that each affine patch has an associated zonotope, and also that these zonotopes
and their faces will then form a zonotopal tiling.
Conjecture 6.2 1. Let Y be an affine T -hypertoric variety. Then there is a zonotope
Z(Y ) such that a cocharacter η has positive weights on Y if and only if η ∈ Z(Y ).
34
2. Let Y be any hypertoric variety. Then Y is covered by affine hypertoric varieties
Y1, . . . , Y`, and Z(Y1), . . . , Z(Y`) are the maximial zonotopes of a zonotopal tiling
T , and Y ∼= Y (T ) as a Poisson T ×Gm-variety.
Proposition 6.3 Conjecture 1.7 implies Conjectures 6.1 and 6.2
We note that both Proposition 6.1 and 6.2 are true in the case of varieties of the
form Y (T ).
35
CHAPTER VII
A WORKED EXAMPLE
In this chapter we work through an extended example by constructing T ∗CP 2 from
a zonotopal tiling, then recovering the tiling from the variety.
7.1. Constructing T ∗CP 2
We begin with the zonotopal tiling
which we center at the origin. First, we write the underlying zonotope as |T | = Z(a),
where a1 = (0, 1), a2 = (1, 0), and a3 = (−1,−1), which determines a sign vector for
each face of T . (Edges are not labeled, but their labels can be inferred.)
−+− + +−
+−−
+−+−−+
−+ + −−−
00−
0− 0
−00
We also choose the matrix B = (111) so that BTA = 0.
We are now ready to begin the geometric construction; we start with the affine
space T ∗C3 = SpecC[z1, z2, z3, w1, w2, w3] and define an open set for each maximal
36
zonotope Z ∈ T . The open set U(T , Z+00) = SpecC[z1, . . . , w3]z1 , and we have a similar
description for each of the other two. The union is U(T ) = T ∗C3 \ V (z1, z2, z3). We
then have that Z(a) is the familiar description of T ∗CP 2: all 6-tuples of numbers such
that µK(z, w) := z1w1 + z2w2 + z3w3 = 0, and zi are not all 0, modulo the torus K.
7.2. Recovering the Tiling
We now describe how to recover the tiling, using only intrinsic properties of T ∗CP 2.
The orbit O−−− is where zi 6= 0 = wi for all i. Its closure is V−−− = (C3 \
(0, 0, 0))//K, which we recognize as CP 2. Coordinates for O−−− are z1/z3 and z2/z3,
on which Gm acts with weight 0, so the matched cocharacter for O−−− is (0, 0). Finally,
we have the fan which is associated to V−−− as a toric variety in Figure 7.1 (a).
FIGURE 7.1. The fans for the toric varieties V−−−, V+−−, and V++−.
The oribit O+−− is where z1 = w2 = w3 = 0 and w1, z2, z3 6= 0. Its closure is
V+−− = (C3 \ C × (0, 0))//K, which is the blowup of A2 at a point. Coordinates
for O+−− are z2w1 and z3w1. On the one hand, we have (t1, t2) · z2w1 = t−11 t2z2w1
and (t1, t2) · z3w1 = t−11 z3w1; on the other hand, we have s × z2w1 = s2z2w1 and
s × z3w1 = s2z3w1. Thus s2 = t−11 t2 and s2 = t−11 , so t1 = s−2 and t2 = 1, and the
matched cocharacter is (−2, 0). Finally, the fan for V+−− is shown in Figure 7.1 (b).
37
The last type of orbit is given by O++−, which has closure V++− = (C3 \ C× C×
0)//K = A2. Coordinates on O++− are z1w3 and z2w3. Using the same method as
before, we find that the matched cocharacter is (−2,−2), and the fan is Figure 7.1 (c).
After performing a similar analysis on the other sign vectors, we can place a fan at
each matched cocharacter and recover the tiling:
We finish by considering some twisted Gm actions. To illustrate all the possibilities,
we choose an interior point of Z, a point in the relative interior of an edge, a vertex,
and a point not in Z.
Gm ×(1,1) (z, w) = (s2z1, s2z2, sz3, w1, w2, sw3)
Gm ×(1,2) (z, w) = (s2z1, s3z2, sz3, w1, s−1w2, sw3)
Gm ×(1,3) (z, w) = (s2z1, s4z2, sz3, w1, s−2w2, sw3)
Gm ×(2,2) (z, w) = (s3z1, s3z2, sz3, s−1w1, s−1w2, sw3)
Since the invariant monomials are precisely the ziwj , it is easy to see that the minimum
weight monomials have weights 1, 0, -1, and 0. We also note that (2, 2), which lies
on a codimension 2 face, has 2 weight zero monomials, while (2, 1), which lies on a
codimension 1 face, has 1 weight zero monomial.
38
Finally, we choose the support function which takes the following values on vertices:
−1 −1
−1
11
1
−1
The slopes along a1, a2, and a3 are 0, 0, and 1, so this corresponds to the character
of T n which takes (t1, t2, t3) to t3. But this restricts to the usual character α of K to
write T ∗CP 2 as a projective GIT quotient T ∗C3//αK.
39
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