Gluing affine torus actions via divisorial fans

0.34 MB·

by  Klaus Altmann

GLUING AFFINE TORUS ACTIONS VIA DIVISORIAL FANS
KLAUSALTMANN,JU¨RGENHAUSEN,ANDHENDRIKSU¨SS
Abstract. Generalizingthepassagefromafantoatoricvariety,weprovide
8 acombinatorialapproachtoconstructarbitraryeffectivetorusactionsonnor-
0 mal,algebraicvarieties. Basedonthenotionofa“properpolyhedraldivisor”
0 introduced in earlier work, we develop the concept of a “divisorial fan” and
2 show that these objects encode the equivariant gluing of affine varieties with
n torus action. We characterize separateness and completeness of the resulting
a varieties in terms of divisorial fans, and we study examples like C∗-surfaces
J andprojectivizations of(non-split)vector bundlesovertoricvarieties.

8
2

]
G 1. Introduction
A This paper continues work of the first two authors [AH06], where the concept
. of “proper polyhedral divisors (pp-divisors)” was introduced in order to provide
h
a complete description of normal affine varieties X that come with an effective
t
a action of an algebraic torus T. Recall that such a pp-divisor lives on a normal
m
semiprojective (e.g., affine or projective) variety Y, and, at first glance, is just a
[ finite linear combination

2 D = X∆D⊗D,
v
D
2
7 where D runs over the prime divisors of Y and the coefficients ∆D are convex
7 polyhedra,alllivinginacommonrationalvectorspaceN andallhavingthesame
Q
6 pointed cone σ ⊆ N as their tail. To see an example, let Y be the projective
Q
0
line, and take the points 0,1 and ∞ as prime divisors on Y. Then one obtains a
6
pp-divisor D on Y by prescribing polyhedral coefficients as follows.
0
/
h
t
a ∆
0
m ∆ ∆
1 ∞
:
v
i
X
r
a
0 1 ∞
TheaffineT-varietyX associatedtoDisthespectrumofamultigradedalgebraA
arising from D. Evaluating the polyhedral coefficients turns the pp-divisor into a
piecewise linear map from the dual cone σ∨ ⊆ M of the common tail to the
Q
rational Cartier divisors on Y: it sends u ∈ σ∨ to the divisor D(u) = α D,
P D
where α = minhu,∆ i. The global sections of these evaluations fit together to
D D
the desired multigraded algebra:
A := M Γ(Y,D(u)).
u∈M∩σ∨
1991 Mathematics Subject Classification. 14L24,14M17,14M25.
1
2 K.ALTMANN,J.HAUSEN,ANDH.SU¨SS
In the present paper, we pass from the affine case to the general one. In the
setting of toric varieties, the general case is obtained from the affine one by gluing
cones to a fan. This is also our approach; applying Sumihiro’s Theorem, we glue
pp-divisors to a “divisorial fan”. There is an immediate naive idea of how such a
divisorialfanshouldlook: allitsdivisorsDi liveonthesamesemiprojectivevariety
Y,theirpolyhedralcoefficients∆i liveinthesamevectorspaceN ,and,forevery
D Q
prime divisor D, the ∆i should form a polyhedral subdivision. For example, the
D
single pp-divisor on Y =P1 discussed before could fit as a D1 into a divisorial fan
comprising five further pp-divisors as indicated below.
D1
D6 D1 D1
D2 D6 D2 D6 D2
D5
D3 D5 D3 D5 D3
D4 D4
D4
0 1 ∞
Todescribe the gluing ofaffineT-varietiesamounts tounderstanding their open
subsetsintermsofpp-divisors;adetailedstudy isgiveninSections3and4. Based
on this, in Section 5 we introduce a concept of a divisorial fan. We show that each
such divisorial fan canonically defines a normal variety with torus action (Theo-
rem5.3), andit turns outthat everynormalvariety with effective torus actioncan
be obtainedin this way(Theorem 5.6). In Section 6, we discuss “coherent” diviso-
rial fans–a special concept, which is much closer to the intuition than the general
one. For example, the figure just drawn fits into this framework: it describes the
projectivization of the cotangent bundle over the projective plane.
Following the philosophy of toric geometry that geometric properties of a toric
varietyshouldbereadofffromitsdefiningcombinatorialdata,inSection7westudy
separateness and completeness and provide a complete characterization of these
properties in terms of divisorial fans (Theorem 7.5). The last section is devoted
to examples. We give the divisorial fans of Danilov-Gizatullin compactifications of
affine K∗-surfaces,recently discussed using different methods by Flenner, Kaliman
and Zaidenberg. This example indicates that our constructions may be used for
finding compactification of varieties with torus actions in a rather intuitive way.
In our last example we provide a translation of Klyachko’s description of vector
bundles on toric varieties into the picture of divisorial fans.
We expect applications of our constructions in all fields where toric varieties
have proved their usefulness. Following the toric program, next steps will be the
description of divisors, bundles and equivariant maps as well as the understanding
of intersection products and cohomology on divisorial fans.
Contents
1. Introduction 1
2. The affine case 3
3. Open embeddings 5
4. Patchworking 8
5. Divisorial Fans 11
6. Coherent fans 14
7. Separateness and Completeness 17
8. Further examples 21
ALGEBRAIC TORUS ACTIONS 3
References 23
2. The affine case
In this section, we briefly recall the basic concepts and results from [AH06],
and we introduce some notions needed later. We begin with fixing our notation in
convex geometry.
Throughoutthis paper, N denotes alattice, i.e. afinitely generatedfree abelian
group, and M := Hom(N,Z) is the associated dual lattice. The rational vector
space associated to N is N := Q⊗ N. Given a homomorphism F: N → N′ of
Q Z
lattices, we write F: N → N′ for the corresponding linear map. For two convex
Q Q
polyhedra ∆,∆′ ⊆N , we write ∆(cid:22)∆′ if ∆ is a face of ∆′.
Q
Let σ ⊆N be a pointed, convex, polyhedral cone. A σ-polyhedron is a convex
Q
polyhedron ∆ ⊆ N having σ as its tail cone (also called recession cone). With
Q
respect to Minkowski addition, the set Pol+(N) of all σ-polyhedra is a semigroup
σ
with cancellation law; we write Pol (N) for the associated Grothendieck group.
σ
Now, let Y be a normal, algebraic variety defined over an algebraically closed
fieldK. ExceptinSection7, weunderstandpoints alwaystobe closedpointsofY.
The group of polyhedral divisors on Y is defined to be
WDiv (Y,σ) := Pol (N)⊗ WDiv(Y),
Q σ Z
whereWDiv (Y)denotesthegroupofrationalWeildivisorsonY. Anypolyhedral
Q
divisor D = ∆ ⊗D with ∆ ∈Pol+(N) evaluates to a piecewise linear convex
P D D σ
map on the dual cone σ∨ ⊆M of σ ⊆N , namely
Q Q
D: σ∨ →WDivQ(Y), u7→Xevalu(∆D)D, where evalu(∆D) := minhu,vi.
v∈∆D
Here, convexity is understood in the setting of divisors, this means that we always
haveD(u+u′)≥D(u)+D(u′). Aproperpolyhedraldivisor (abbreviatedpp-divisor)
is a polyhedral divisor D∈WDiv (Y,σ) such that
Q
(i) there is a representation D = ∆ ⊗ D with effective divisors D ∈
P D
WDiv (Y) and ∆ ∈Pol+(N),
Q D σ
(ii) eachevaluationD(u), where u∈σ∨, is a semiample Q-Cartierdivisor,i.e.
has a base point free multiple,
(iii) for any u in the relative interior of σ∨, some multiple of D(u) is a big
divisor, i.e. admits a section with affine complement.
¿Fromnowon,wesupposethatY is,additionally,semiprojective,i.e. projective
over some affine variety. Every pp-divisor D = ∆ ⊗D on Y defines a sheaf of
P D
graded O -algebras,and we have the corresponding relative spectrum:
Y
A := M O(D(u)), X := SpecY(A).
e
u∈M∩σ∨
The grading of A gives rise to an effective action of the torus T := Spec(K[M])
on X, the canonical map π: X →Y is a good quotient for this action, and for the
fieldeof invariant rational funcetions, we have
K(X)T = K(Y).
e
By [AH06, Theorem 3.1], the ring of global sections A := Γ(X,O) = Γ(Y,A) is
e
finitely generated and normal, and there is a T-equivariant, projective, birational
morphism r: X → X onto the normal, affine T-variety X := X(D) := Spec(A).
e
Conversely, [AH06, Theorem 3.4] shows that every normal, affine variety with an
effective torus action arises in this way.
4 K.ALTMANN,J.HAUSEN,ANDH.SU¨SS
The assignment from pp-divisors to normal, affine varieties with torus action is
even functorial, see [AH06, Sec. 8]. Consider two pp-divisors,
D′ = X∆′D′ ⊗D′ ∈ PPDivQ(Y′,σ′), D = X∆D⊗D ∈ PPDivQ(Y,σ).
If ψ: Y′ → Y is a morphism such that none of the supports of the D’s contains
ψ(Y), and if F: N′ →N is a linear map with F(σ′)⊆σ, then we set
ψ∗(D) := X∆D⊗ψ∗(D), F∗(D′) := X(cid:0)F(∆′D′)+σ(cid:1)⊗D′.
D D′
Suppose that for some “polyhedral principal divisor” div(f)= (v +σ)⊗div(f )
P i i
with v ∈N and f ∈K(Y′), we have inside WDiv (Y′,σ) the relation
i i Q
ψ∗(D) ≤ F (D′)+div(f)
∗
afterevaluatingwitharbitraryu∈σ∨. Thenthetriple(ψ,F,f)iscalledamapfrom
the pp-divisor D′ to the pp-divisor D. It induces homomorphisms of O -modules:
Y
O(D(u)) → ψ O(D′(F∗u)), h 7→ f(u)ψ∗(h).
∗
These maps fit together to a gradedhomomorphismA→ψ A′. This in turn gives
∗
rise to a commutative diagram of equivariant morphisms, where the rows contain
the geometric data associated to the pp-divisors D and D′ respectively:
π r
Y oo X //X
OO OO OO
e
ψ ϕe ϕ
Y′ oo X′ // X′.
π′ e r′
In particular, the map D′ → D defines an equivariant morphism X′ → X with
respect to T′ →T defined by F: N′ →N.
Here,wewillfrequentlyconsideraspecialcaseoftheaboveone. Namely,suppose
that Y′ = Y and ∆′ ⊆ ∆ holds for every prime divisor D ∈ WDiv(Y). Then
D D
σ∨ ⊆(σ′)∨ holdsforthe dualizedtailcones. Moreover,foreveryu∈σ∨,weobtain
D(u) = X minhu,viD ≤ X minhu,viD = D′(u).
v∈∆D v∈∆′D
Consequently, we have a graded inclusion morphism A ֒→ A′ of the associated
sheavesof O -algebras,and hence a monomorphismA֒→A′ on the level of global
Y
sections, which in turn determines a T-equivariantmorphism X′ →X.
In[AH06, Prop.7.8andCor.7.9],we tooka closerlook atthe fibers ofthe map
π: X → Y arising from a pp-divisor D = ∆ ⊗D. Suppose that all D’s are
P D
primee. For a point y ∈Y, its fiber polyhedron is the Minkowski sum
∆y := X∆D ∈ Pol+σ(N).
y∈D
Let Λ denote the normal fan of the fiber polyhedron ∆ . Then Λ subdivides the
y y y
cone σ∨, and the faces of ∆ are in order reversingbijection to the cones of Λ via
y y
F 7→ λ(F) := {u∈M ; hu,v−v′i≥0 for all v ∈∆, v′ ∈F}.
Q
Now, for z ∈π−1(y), let ω(z) denote its orbit cone, i.e. the convex cone generated
byallweightsu∈M admittingau-homogeneousfunctiononπ−1(y)withf(u)6=0.
Then there is a bijection:
{T-orbits in π−1(y)} → Λ T·x 7→ ω(x).
y
e e
This does eventually provide an order and dimension preserving bijection between
the T-orbits of π−1(y) and the faces of ∆ .
y
ALGEBRAIC TORUS ACTIONS 5
Now, for the gluing of pp-divisors performed later, it is necessary to relax our
notation: We will allow ∅ as an element of Pol (N). This new element is subject
σ
to the rules ∅+∆ := ∅ and 0·∅ := σ. Moreover, if ∅ occurs as a coefficient of a
pp-divisor D= ∆ ⊗D, then we will always assume that suppD is the
P D S∆D=∅
support of an effective, semiample divisor, and we understand D ∈ PPDiv (Y,σ)
Q
as D| ∈PPDiv (Loc(D),σ) with
Loc(D) Q
Loc(D):=Y \ [ suppD.
∆D=∅
This new convention is compatible with the following evaluation of the coefficients
of a polyhedral divisor.
Definition 2.1. Let N be a lattice, σ ⊆ N a pointed polyhedral cone, and
Q
D= ∆ ⊗D a polyhedral divisor on a normal variety Y. If
P D
µ: {prime divisors on Y} → R
isanymap,thenwedefinetheassociatedweighted sumofthepolyhedral coefficients
to be
∆µ := Dµ := µ(D) := Xµ(D)·∆D ∈ Polσ(N).
Example 2.2. (i) For the trivial map µ≡0, the weighted sum ∆ gives the
0
common tail cone tail(D) of the coefficients of D.
(ii) Fixing a prime divisor P ∈ WDiv(Y), we may consider µ (D) := δ .
P D,P
The corresponding D =µ (D) recovers the coefficient ∆ of P.
P P P
(iii) Given a point y ∈Y, set µ (D) := 1 if y ∈D and µ (D) := 0 else. Then
y y
µ (D) is precisely the fiber polyhedron ∆ of the point y ∈Y.
y y
(iv) If C ⊆ Y is a curve, then µ (D) := (C ·D) leads to µ (D) =: (C ·D) ∈
C C
Pol (N). InthecaseofY =C,orY =Pn andC beingtheline,wedenote
σ
(C·D) also by degD.
3. Open embeddings
In this section, we begin the study of open embeddings of affine T-varieties in
terms of pp-divisors. The first statement is a description of the equivariant basic
open sets obtained by homogeneous localization. Recall that in toric geometry
equivariant localization corresponds to passing to a face of a given cone. The
generalization to pp-divisors involves also operations on the base variety; here is
the precise procedure. Fix a lattice N and a normal, semiprojective variety Y.
Moreover,let σ ⊆N be a pointed cone and consider a pp-divisor
Q
D = X∆D⊗D ∈ PPDivQ(Y,σ).
As usual, A denotes the associated sheaf of M-graded O -algebras, A := Γ(Y,A)
Y
is the algebra of global sections, and we set X :=Spec (A), and X :=Spec(A).
Y
e
Definition 3.1. Let w∈σ∨∩M and f ∈A =Γ(Y,O(D(w))).
w
(i) The face of ∆∈Pol+(N) defined by w is
σ
face(∆,w):={v ∈∆; hw,vi≤hw,v′i for all v′ ∈∆}∈Pol+ (N).
σ∩w⊥
(ii) The zero set of f and the principal set associated to f are
Z(f) := Supp(div(f)+D(w)), Y := Y \Z(f).
f
(iii) The localization of the pp-divisor D by f is
Df :=Xface(∆D,w)⊗D|Yf =∅⊗(div(f)+D(w))+Xface(∆D,w)⊗D.
6 K.ALTMANN,J.HAUSEN,ANDH.SU¨SS
Lemma 3.2. Let w ∈σ∨∩M and f ∈A =Γ(Y,O(D(w))) as in Definition 3.1.
w
Then, for u∈σ∩w∨ and k≫0, one has u+kw∈σ∨ and
D (u) = D(u+kw)| −D(kw)| .
f Yf Yf
Proof. Set σ := σ∩w⊥ and ∆ := face(∆,w). The first part of the assertion is
w w
clearbyσ =σ∨−Q w. Thesecondpartisobtainedbycomparingthenon-empty
w ≥0
coefficients of the prime divisors. For D (u), they are of the form minh∆ ,ui. If
f w
u attains this minimum at v ∈∆ , then v provides a minimal value for u+kw on
w
the whole ∆. Thus, the claim follows from
minh∆ ,ui = hv,ui = hv,u+kwi−hv,kwi = minh∆,u+kwi−minh∆,kwi.
w
(cid:3)
Proposition 3.3. For a pp-divisor D on a normal, semiprojective variety Y, let
D be the localization of D by a homogeneous f ∈A . Then D is a pp-divisor on
f w f
Y , and the canonical map D →D describes the open embedding X →X.
f f f
Proof. We may assume that D has non-empty coefficients. Recall that Y is ob-
f
tainedbyremovingthe supportofD =div(f)+D(w) fromY. Inparticular,D(w)
is principal on Y , and thus, for k ≫0, Lemma 3.2 gives
f
O (D (u)) ∼= O (D(u+kw)).
Yf f Yf
Using this, one sees that the assignment u 7→ D (u) inherits from u 7→ D(u) the
f
properties (i) to (iii) of a pp-divisor formulated in Section 2.
To see that D → D defines an open embedding X → X, it suffices to verify
f f
that, for any linear form u∈(σ∩w⊥)∨∩M =(σ∨∩M)−N·w, we have
[ Γ(cid:0)Y, D(u+kw)(cid:1)/fk = Γ(cid:0)Yf, D(u+kw)−kD(w)(cid:1) where k ≫0.
k≫0
Consider an element g/fk of the left hand side. Then div(g)+D(u+kw) ≥ 0
holds. Hence, still on Y, we have
div(g/fk)+D(u+kw)−kD(w) ≥ −div(fk)−kD(w) = −kZ(f).
Thus, div(g/fk)+D(u+kw)−kD(w) is effective on Y , which means that g/fk
f
belongs to the right hand side.
For the reverse inclusion, take any element from the right hand side; we may
write this element as g/fk with k ≫0. From the relation
div(g/fk)+D(u+kw)−kD(w) ≥ 0
on Y , we obtain the existence of anℓ∈Z suchthat the same divisoris ≥−ℓZ(f)
f
on Y. Moreover,we may assume that ℓ≥k. Then,
div(g/fk)+D(u+kw)−kD(w) ≥ −div(fℓ)−ℓD(w)
holds on Y. Using the convexity property of the assignment u 7→ D(u), we can
conclude
div(gfℓ−k)+D(u+ℓw) ≥ div(gfℓ−k)+D(u+kw)+(ℓ−k)D(w) ≥ 0.
However,thisshowsthatg/fk =gfℓ−k/fℓ belongstothebigunionofthelefthand
side. (cid:3)
Whereas in toric geometry every equivariant open embedding of affine toric va-
rieties is a localization, this needs no longer hold for general T-varieties. Thus, in
viewofequivariantgluing,wehaveto takecareofmoregeneralaffineopenembed-
dings. We consider the following situation. By N, we denote again a lattice, and
ALGEBRAIC TORUS ACTIONS 7
Y is a normal variety. Moreover, σ′ ⊆ σ ⊆ N are pointed polyhedral cones, and
Q
we consider two pp-divisors
D′ = X∆′D⊗D ∈ PPDivQ(Y,σ′), D = X∆D⊗D ∈ PPDivQ(Y,σ).
We suppose that ∆′ ⊆ ∆ holds for every prime divisor D ∈ WDiv(Y). For the
D D
respective loci of these divisors, we then obtain
V′ := Loc(D′) = {y ∈Y; ∆′ 6=∅} ⊆ {y ∈Y; ∆ 6=∅} = Loc(D) =: V.
y y
NotethatwehaveanaturalmapD′ →Dofpp-divisors. AsmentionedinSection2,
this gives rise to a commutative diagram of T-equivariant morphisms, where the
rows contain the geometric data associated to D and D′ respectively:
π r
V oo X // X
OO OO OO
e
V′ oo X′ //X′
π′ e r′
Proposition 3.4. The morphism X′ → X associated to D′ → D is an open
embedding if and only if any y ∈V′ admits w ∈σ∨∩M and f ∈A with
w
y ∈V ⊆V′, ∆′ =face(∆ ,w), face(∆′,w)=face(∆ ,w) for every v ∈V .
f y y v v f
Proof. Suppose that X′ → X is an open embedding. For short we write X′ ⊆ X.
Given y ∈V′, let T·z′ ⊆(π′)−1(y) the (unique) closed T-orbitand choose f ∈A ,
w
where w ∈σ∨∩M, such that
f(r′(z′))6=0, f =0.
|X\X′
Then we always have X = X′. Since the maps r: X → X and r′: X′ → X′ are
f f
e e
birational and proper, this implies
B := Γ(X ,O) = Γ(X ,O) = Γ(X′,O) = Γ(X′,O) =: B′.
f f f f
e e
Considering the invariant parts, we obtain that Γ(V ,O) equals Γ(V′,O). Since
f f
both V′ ⊆V are semiprojective,this gives V′ =V . Using [AH06, Thm. 3.1 (iii)],
f f f f
we arrive at
y ∈ π(r−1(X )) = V = V′ ⊆ V′.
f f f
Moreover,B and B′ are the algebras of global sections of the localized pp-divisors
D and D′ living on V =V′. By [AH06, Lemma. 9.1], B = B′ implies D = D′.
f f f f f f
Thus, we obtain
face(∆′,w) = face(∆ ,w)
v v
for every v ∈ V . Finally, f(r′(z′)) 6= 0 implies ∆′ = face(∆′,w), which, together
f y y
with the preceding obervation, shows ∆′ =face(∆ ,w).
y y
NowsupposethatD′andDsatisfytheassumptionsoftheproposition. Forevery
y ∈V′ choose w and f ∈A as in the assertion. Then we have ∆′ =face(∆′,w).
w y y
From this, we can conclude (π′)−1(y) ⊆ X′. Consequently, the sets X′, where
f f
e e
y ∈V′, cover X′.
Moreover, thee assumption implies that the localized pp-divisors D′ and D co-
f f
incide. Hence, the canonicalmaps X′ →X are isomorphisms,andthis also holds
f f
e e
for the canonical maps X′ → X . Since X′ is covered by the sets X′, we obtain
f f f
that X′ →X is an open embedding. e e (cid:3)
Remark 3.5. Suppose we are in the situation of Proposition 3.4.
8 K.ALTMANN,J.HAUSEN,ANDH.SU¨SS
(i) The condition face(∆′,w)=face(∆ ,w) for every v ∈V is equivalent to
v v f
the following one: If face(∆′ ,w) 6=face(∆ ,w), then D ∈ WDiv(Y) is a
D D
prime divisor supported in Z(f).
(ii) The conditionof the previousProposition3.4implies that ∆′ (cid:22)∆ holds
y y
forally ∈Y. Thisweakerconditionturnsouttobeequivalenttothemap
X′ →X being an open embedding.
e e
Example 3.6. LetY =P1 andN =Z. Thepp-divisorD =[0,∞)⊗{0}+[1,∞)⊗
{∞}describesK2withitsstandardK∗-action. Ontheotherhand,wemayconsider
D′ :=[0,∞)⊗{0}+∅⊗{∞}. The morphism D′ →D describes the blowing up of
the origin in K2, hence, it is not an open embedding.
4. Patchworking
In this section, we continue the study of equivariant open embeddings. Given a
pp-divisoranditsassociatedaffineT-varietyX,ouraimistoconstructapp-divisor
foraninvariant,affine,opensubsetX′ ⊆X. Clearly,X′ isaunionofhomogeneous
localizations of X. We need the following setting.
Definition 4.1. Let X be an affine T-variety, and let X′ ⊆ X be an invariant,
open, affine subset. We say that f ,...,f ∈Γ(X,O) reduce X to X′ if
1 r
(i) each f is homogeneous and X′ = r X holds,
i Si=1 fi
(ii) each f is invertible on some orbit closure in X′.
i
Remark 4.2. For any invariant,affine, open subset X′ ⊆X of anaffine T-variety
X, there exist homogeneous functions f ,...,f ∈ Γ(X,O) that reduce X to X′.
1 r
If X′ ֒→X is an open embedding that arises from a map of pp-divisorsD′ →D as
in Proposition 3.4, then the functions f ∈A mentioned there will do.
w
By[AH06,Thm.8.8],theopenembeddingX′ ֒→X mayberepresentedbysome
map of pp-divisors D′ →D. In the following, we will show that D′ may be chosen
to live on the same base Y as D does.
Proposition 4.3. Consider a pp-divisor D = ∆ ⊗D on a normal semipro-
PD D
jective variety Y, denote the associated geometric data by
π r
Y oo X // X ,
e
and let X′ ⊆ X be an invariant, affine, open subset. Then Y′ := π(r−1(X′)) ⊆ Y
is open and semiprojective. Moreover, if f ∈A reduce X to X′, then
i wi
D′ := [Dfi := X∆′D⊗D|Y′, where ∆′D := [ face(∆D,wi) (cid:22) ∆D
D∩Y′ 6=∅
fi
is a pp-divisor on Y′ = Y , and the canonical map D′ → D defines an open
S fi
embedding of affine varieties having X′ as its image.
Corollary4.4. LetX betheaffinevarietyarisingfromapp-divisor D onanormal
variety Y, and let the pp-divisors
D′ = X∆′D⊗D = [Dfi, D′′ = X∆′D′ ⊗D = [Dgj
with loci Y′ ⊆Y and Y′′ ⊆Y, respectively, describe open subsets X′ = X and
S fi
X′′ = X as in Proposition 4.3. Then we have
S gj
D′∩D′′ := X(∆′D ∩∆′D′)⊗D = [Dfigj.
In particular, D′ ∩D′′ is a pp-divisor with locus Y′∩Y′′ ⊆ Y, and the canonical
map D′∩D′′ →D describes an open embedding having X′∩X′′ as its image.
ALGEBRAIC TORUS ACTIONS 9
For the proof of Proposition 4.3, we need two preparatory lemmas. Let Y, Y′′
be normal semiprojective varieties, N, N′′ lattices, σ ⊂ N and σ′′ ⊂ N′′ pointed
Q Q
cones, and consider pp-divisors
D = X∆D⊗D ∈ PPDivQ(Y,σ), D′′ = X∆′D′ ⊗D′′ ∈ PPDivQ(Y′′,σ′′)
with non-empty coefficients. Moreover, let (ψ,F,f) be a map from D′′ to D. As
indicated in Section 2, the map (ψ,F,f) gives rise to a commutative diagram of
equivariantmorphisms,wherethe rowscontainthe geometricdata associatedtoD
and D′′ respectively:
π r
Y oo X //X
OO OO OO
e
ψ ϕe ϕ
Y′′ oo X′′ // X′′.
π′′ e r′′
Lemma 4.5. In the above notation, suppose that the morphism ϕ: X′′ →X is an
open embedding. Then the following holds.
(i) We have ϕ(X′′) = r−1(ϕ(X′′)), and the induced morphism ϕ: X′′ →
ϕ(X′′) is peropeer and birational. e e
(ii) Teheeimage ψ(Y′′)⊆Y is open and semiprojective, and ψ: Y′′ →ψ(Y′′) is
a projective birational morphism.
(iii) For every y ∈ ψ(Y′′), the intersection U := π−1(y)∩ϕ(X′′) contains a
y
unique T-orbit that is closed in U . e e e
y
e
Proof. Consider the open subset U := ϕ(X′′), its inverse image U := r−1(U) and
e
V :=π(U). By [Ha05, Lemma 2.1], the latter set is open in Y. These data fit into
e
the commutative diagram
π r
V oo U //U
OO OO OO
e
ψ ϕe ∼= ϕ
Y′′ oo X′′ // X′′.
π′′ e r′′
The map ϕ is birational, because r′′, ϕ and r are birational. Moreover, since r′′
and henceeϕ◦r′′ are proper, we infer from the diagram that ϕ is proper, and thus
surjective. Consequently, we obtain ψ(Y′′) = V; in particulaer, this set is open in
Y. In order to see that V is a semiprojective variety, note first that for its global
functions, we have
Γ(V,O) ∼= Γ(U,O) ∼= Γ(X′′,O) ∼= Γ(Y′′,O);
0 0
e e
thefirstequalityisguaranteedby[Ha05,Lemma2.1]. Thus,settingA′′ :=Γ(Y′′,O)
0
and Y′′ :=Spec(A′′), we obtain a commutative diagram
0 0
ψ
Y′′ // V
BBBBBBBB!! ~~~~~~~~~~
Y′′
0
Since ψ issurjective,V →Y′′ is proper. SinceV isquasiprojective,weevenobtain
0
that V → Y′′ is projective, and so is ψ: Y′′ → V. The map ψ is also birational,
0
10 K.ALTMANN,J.HAUSEN,ANDH.SU¨SS
because we have the commutative diagram
ψ∗
K(Y) // K(Y′′)
= =
(cid:15)(cid:15) (cid:15)(cid:15)
∼=
K(X) // K(X′′)
e 0 ϕe∗ e 0
Now, consider y ∈ V and the intersection U := π−1(y)∩ϕ(X′′). Since π−1(y)
y
contains only finitely many T-orbits, the samee holds for U .eLeet T·z ,...T·z be
y 1 r
e
the closed T-orbits of U . We claim
y
e
r
ψ−1(y) = π′′(ϕ−1(Uy)) = [π′′(ϕ−1(T·zi)).
e e i=1 e
The first equality is clear by surjectivity of ϕ and the quotient maps π,π′′. The
second one is verified below; it uses properneess of ϕ: Given y′′ ∈π′′(ϕ−1(U )), we
y
have y′′ = π′′(z′′) for some z′′ ∈ ϕ−1(U ). Since πe′′ is constant on oerbit celosures,
y
we may assume that T′′·z′′ is cloesed ien ϕ−1(U ). By properness of ϕ, the image
y
ϕ(T′′·z′′) = T·ϕ(z′′) is closed in U . It feollowes that y′′ belongs to thee right hand
y
seide. e e
Having verified the claim, we may proceed as follows. The closed invariant
subsets ϕ−1(T ·z ) ⊆ X′′ are pairwise disjoint. By the properties of the good
i
quotienteπ′′: X′′ →Y′′,ethe images π′′(ϕ−1(T·z )) are pairwise disjoint as well. In
i
particular,ψ−e1(y)isdisconnectedifr >e1. Thelatterisimpossiblebecauseψ,asa
birationalprojectivemorphismbetween normalvarieties,has connectedfibers. (cid:3)
Lemma 4.6. For the functions f ∈A of Proposition 4.3, we always have
i wi
∆′D := [ face(∆D,wi) (cid:22) ∆D.
D∩Y′ 6=∅
fi
In particular, for every f with D∩Y′ 6=∅, we have face(∆ ,w )(cid:22)∆′ .
i fi D i D
Proof. Let D be a prime divisor intersecting Y′, and consider a point y ∈ D∩Y′
such that y ∈Y holds for all the f of Proposition 4.3 with D∩Y 6=∅ and D is
fi i fi
the only prime divisor with ∆ 6=0 containing y. Then we have
D
∆ = ∆ .
y D
According to [AH06, Thm. 8.8], the inclusion X′ ⊆ X is described by a map
of pp-divisors. Thus, we may apply Lemma 4.5 and obtain that there is a unique
closedT-orbitT·z inπ−1(y)∩r−1(X′). ThisorbitcorrespondstoafaceF(z)(cid:22)∆
y
via
T·z 7→ ω(z) 7→ face(∆ ,u) with u∈intω(z),
y
where intω(z) denotes the relative interior of the cone ω(z). Since z ∈ r−1(X′)
holds,someofthef ofProposition4.3satisfiesf (z)6=0butvanishesonT·z\T·z,
i i
where the closure is taken in π−1(y). This means w ∈ intω(z). To conclude the
i
proof, it suffices to show
[ face(∆y,wj) = face(∆y,wi).
D∩Y′ 6=∅
fj
For any f with D ∩Y′ 6= ∅, we have y ∈ Y . Thus, there is a point z ∈
j fj fj j
π−1(y)∩X′ with f(z ) 6= 0. We may choose z such that ω(z ) is minimal; this
j j j
means w ∈intω(z ). Since z ∈T·z holds, we obtain ω(z)(cid:22)ω(z ). This in turn
j j j j
implies face(∆ ,w )(cid:22)face(∆ ,w ), and the above equation follows. (cid:3)
y j y i