Quotients of an Affine Variety by an Action of a Torus

Cent. Eur. J. Math. • 11(11) • 2013 • 1863-1880 DOI: 10.2478/s11533-013-0295-8

Central European Journal of Mathematics

Quotients of an aﬃne variety by an action of a torus

Research Article

Olga V. Chuvashova1∗, Nikolay A. Pechenkin2†

1 Department of Higher Algebra, Faculty of Mechanics and Mathematics, Moscow State University, 1 Leninskie Gory, Moscow, 119991, Russia

2 Algebraic Geometry Section, Steklov Mathematical Institute, 8 Gubkin str., Moscow, 119991, Russia

Received 6 April 2012; accepted 7 January 2013

Abstract: Let X be an affine T -variety. We study two different quotients for the action of T on X: the toric Chow quotient X/ T and the toric Hilbert scheme H. We introduce a notion of the main component H of H, which parameterizes C 0 general T -orbit closures in X and their flat limits. The main component U of the universal family U over H is a 0 preimage of H . We define an analogue of a universal family W over the main component of X/ T . We show 0 X C that the toric Chow morphism restricted on the main components lifts to a birational projective morphism from U to W . The variety W also provides a geometric realization of the Altmann–Hausen family. In particular, the 0 X X notion of WX allows us to provide an explicit description of the fan of the Altmann–Hausen family in the toric case.

MSC: 14L24, 14C05, 14M25

Keywords: Torus action • Toric variety • Toric Chow quotient • Toric Hilbert scheme © Versita Sp. z o.o.

1. Introduction

An important problem in algebraic geometry is to introduce a good notion of a quotient for an action of a reductive algebraic group G on a variety X. For many actions there exists an open subset U ⊂ X where G acts freely, such that a variety U/G exists as a geometric quotient. Constructing a quotient X/G is, thus, choosing a compactiﬁcation of U/G. In the case when X is projective, one approach to this problem is provided by geometric invariant theory (GIT) developed by Mumford [16]. Given a G-equivariant embedding of X in the projectivisation of a G-module, the GIT-quotient is the projective spectrum of the subring of G-invariants in the homogeneous coordinate ring on X. There are two other natural compactiﬁcations, provided by appropriate Chow varieties of algebraic cycles and Hilbert schemes. The Chow quotient

∗ E-mail: [email protected] † E-mail: [email protected]

1863 Quotients of an aﬃne variety by an action of a torus

of a projective variety X parameterizes the closures of G-orbits in X having the same dimension and degree and their limits in the Chow variety of all algebraic cycles having these parameters. The Chow quotient of a toric projective variety by a subtorus action was studied in [15]. The invariant Hilbert scheme classifies closed G-invariant subschemes Z ⊂ X such that the G-module O(Z) has prescribed multiplicities [6]. The G-Hilbert scheme is a particular case of the invariant Hilbert scheme. It arises in the case of a finite group G and is considered in [5]. The main component of the G-Hilbert scheme parameterizes regular G-orbits in X and their flat limits. Another particular case of an invariant Hilbert scheme is the toric Hilbert scheme which will be considered below. We are interested in the following case. Let X be an affine variety and G = T be an algebraic torus. Denote by X s the subset of stable points in X under the torus action which is an open subset in X where T acts freely (see subsection 3.2.2 for the precise definition). The toric Hilbert scheme H is defined as the invariant Hilbert scheme parameterizing T -invariant ideals in k[X] having the same Hilbert function as the toric variety T x, where x ∈ X s. The main component H H X s/T H 0 of is the irreducible component obtained by closing the image of in . In the case of the subtorus action on the affine toric variety X the toric Hilbert scheme and its main component were studied in [7]. X/ T X T X/ T The Chow quotient ( C )0 of by the -action is by definition the main component of the inverse limit C of GIT- X s/T X/ T quotients (see subsection 3.1.2). It is obtained by closing the image of in C , so there is a canonical rational q X X/ T Y X/ T norm X/ T map : 99K ( C )0. Denote by = ( C )0 the normalization of ( C )0. In [2] Altmann and Hausen introduced a certain family ψ : Xe → Y of T -varieties, where the T -variety Xe can be interpreted as a resolution of X improving the quotient behavior. The morphism ψ is a good quotient under the action of T and there is a proper and birational morphism ' : Xe → X such that the following diagram is commutative:

' Xe // X

ψ qnorm Y:

The family ψ is closely related to the combinatorial-geometrical datum used in [2] for the description of affine normal T -varieties. In subsection 3.1.3 we recall the construction of ψ and explain how one can easily pass from ψ to this datum and back. A first result of the paper, Proposition 3.9, realizes ψ as the normalization of an elementarily constructed p W → X/ T W q p family C : X ( C )0. The variety X is defined as the closure of the graph of the rational map , and C is the X/ T X projection on ( C )0. In Theorem 4.2 we then use this result to construct the fan of variety e in the toric case. U → H Our main result, Theorem 3.18, relates the universal family 0 0 over the main component of the toric Hilbert scheme W → X/ T H X/ T via a toric Chow morphism to the family X ( C )0. The toric Chow morphism from to C was constructed by Haiman and Sturmfels in [13] in the case when X is a finite-dimensional T -module. We generalize their construction and include this morphism into a commutative square

U0 // WX

p C (1) H X/ T : 0 / ( C )0

We state that horizontal morphisms here are projective and birational. The paper is organized as follows. In Section 2 we fix notation concerning toric geometry and T -varieties and recall the definition of the functor of points which is necessary to define the invariant Hilbert scheme. In subsection 3.1 we provide an exposition of the results of [2]. We recall the notion of the Altmann–Hausen family and then prove our first result, Proposition 3.9, about its simple geometrical realization. In subsection 3.2 we recall the definition of the toric Hilbert scheme and introduce the notion of its main component in the T -variety case. In subsection 3.3 we investigate the toric Chow morphism, lift it to the universal families and prove our main result, Theorem 3.18. Section 4 is, in fact, a repetition of Section 3 in the case of a subtorus action on an affine toric variety. In this case almost all the statements H ; X/ T ;U of Section 3 can be interpreted in terms of fans. We provide the construction of the fans of toric varieties 0 ( C )0 0 and WX and compute them in concrete examples.

1864 O.V. Chuvashova, N.A. Pechenkin

2. Preliminaries

We consider the category of schemes over an algebraically closed ﬁeld k of characteristic zero. Our main references on schemes are [10, 14]. A variety is a separated reduced scheme of ﬁnite type. We denote by OZ the structure sheaf of a scheme Z, and k[Z] denotes the algebra of sections of OZ over Z.

2.1. Basic facts from toric geometry

An n-dimensional torus T is an algebraic group isomorphic to the direct product of n copies of the multiplicative × × × group k . We use the notation X(T ) = Hom(T ; k ) and Λ(T ) = Hom(k ;T ) for the lattices of characters and one- X X parameter subgroups of T , respectively. Denote by (T )Q = (T )⊗Z Q and Λ(T )Q = Λ(T )⊗Z Q the corresponding Q-vector spaces. For any aﬃne scheme X with an action of a torus T its algebra of regular functions k[X] is graded by the group X(T ) T of characters of , M k[X] = k[X]χ ; χ∈X(T ) where k[X]χ = {f ∈ k[X]: t ·f = χ(t)f; t ∈ T } is the subspace of T -semi-invariant functions of weight χ. Let

Σ = {χ ∈ X(T ): k[X]χ =6 0}:

If X is an irreducible variety, then Σ is a finitely generated monoid called the weight monoid. If T acts on X faithfully, then Σ generates X(T ). A T -variety is a normal variety endowed with a faithful regular action of T . Let X be a T -variety. A morphism π : X → Y T is called a good quotient for this action if π is affine, T -invariant, and the canonical map π# : OY → π∗(OX ) is an isomorphism. Given a scheme S, a family of affine T -schemes over S is a scheme X equipped with an action of T and with a morphism p: X → S such that p is affine, of finite type and T -invariant. Then the sheaf of OS-algebras p∗(OX ) is equipped with a compatible grading by X(T ). A toric variety under the torus T is an irreducible T -variety X that contains an open orbit isomorphic to T . We do not require X to be normal. We will consider only those toric varieties X that admit an open covering by affine T -invariant charts (all normal toric varieties satisfy this condition). Our main references on toric varieties are [8, 11, 17]. Given a C toric variety X, we denote by X ⊂ Λ(T )Q the associated fan. The T -orbits on X are in order-reversing one-to-one correspondence with the cones of CX . If σ(Z) is the cone in CX corresponding to a T -orbit Z, then a one-parameter λ ∈ T σ Z λ s Z subgroup Λ( ) lies in the interior of ( ) if and only if lims→0 ( ) exists and lies in . A toric variety is determined by its fan up to normalization. X T X T ' X → X toric ' Let 1 under a torus 1 and 2 under a torus 2 be toric varieties. A morphism : 1 2 is if maps the torus T T ' T T → T ' 1 into 2 and 1 : 1 2 is a group homomorphism. For the toric morphism we have the following commutative diagram:

T1 ×X1 / X1

'T ×' ' 1 T2 ×X2 // X2: ' T → T ' T → T ' T → The morphism of algebraic groups : 1 2 induces a Z-linear map : Λ( 1) Λ( 2) and Q-linear map Q : Λ( 1)Q T Λ( 2)Q. X X In the following two propositions 1 and 2 are normal toric varieties.

1865 Quotients of an aﬃne variety by an action of a torus

Proposition 2.1 ([8, Theorem 3.3.4]). A morphism ' X → X is toric if and only if the corresponding -linear map ' is compatible with fans CX and CX . : 1 2 Z 1 2 It means that for every cone ς ∈ CX there exists a cone ς ∈ CX such that ' ς ⊂ ς. 1 e 2 Q( ) e

Denote by |C| the support of the fan C.

Proposition 2.2 ([8, Theorem 3.4.11]). − A toric morphism ' X → X is proper if and only if ' 1 |CX | |CX |. : 1 2 Q ( 2 ) = 1

If X is toric variety under a factor torus T/T , then we can consider the fan of this variety as a quasifan in Λ(T)Q whose cones include the subspace Λ(T )Q ⊂ Λ(T)Q.

Proposition 2.3. Let X = T x, Y = T y be toric varieties. Then the fan associated to the toric variety T (x; y) ⊂ X ×Y is the coarsest common reﬁnement of fans CX and CY (in particular, its support is equal to |CX | ∩ |CY |).

Note that the action of the torus T on the toric varieties X and Y is not required to be faithful here. So the fans CX and CY should be considered as quasifans in Λ(T ).

λ ∈ T λ t x; y λ t x λ t y Proof. Let Λ( ). The limit limt→0 ( )( ) exists if and only if both limits limt→0 ( ) and limt→0 ( ) exist. x; y λ λ The limits of the point ( ) with respect to one-parameter subgroups 1 and 2 coincide if and only if the limits with λ λ x y λ λ C C respect to 1 and 2 of both points and coincide, i.e. 1 and 2 lie in the same cones of the fans X and Y .

2.2. Basic facts on the functor of points

Recall that any scheme Z is characterized by its functor of points that is the contravariant functor from the category of schemes to the category of sets: ◦ Z : (Sch) → (Set);Z(X) = Mor(X;Z);

where Mor(X;Z) is the set of morphisms of schemes from X to Z over k (we denote the functor of points of a scheme by the corresponding underlined letter). Each f ∈ Mor(X;Y ) defines a morphism of sets Z(f): Z(Y ) → Z(X). For g ∈ Mor(Y;Z) we have Z(f)(g) = g◦f ∈ Mor(X;Z). ◦ Let F : (Sch) → (Set) be an arbitrary functor. We say that a scheme Z represents the functor F if there exists an ∼ isomorphism of functors Z = F (the scheme Z is then also called the fine moduli space of the functor F). Denote by ◦ Fun ((Sch) ; (Set)) the category of contravariant functors from the category of schemes to the category of sets. ◦ The covariant functor ∗: (Sch) → Fun ((Sch) ; (Set)) is defined by X 7→ X. By Yoneda’s lemma, the functor ∗ is an equivalence between the category of schemes and the full subcategory of the category of functors. In particular, it defines a natural bijection between the sets Mor(X;Y ) and Mor(X;Y ).

3. General constructions

3.1. The toric Chow quotient and the Altmann–Hausen theory

3.1.1. Combinatorial description of aﬃne normal T -varieties

In this section we recall the description of normal aﬃne T -varieties in terms of proper polyhedral divisors on a normal semiprojective variety given in [2].

1866 O.V. Chuvashova, N.A. Pechenkin

Let M be a lattice, N be its dual lattice, MQ = M ⊗Z Q, NQ = N ⊗Z Q. Denote by h·; ·i: MQ ×NQ → Q the natural ∨ duality pairing, by σ a pointed polyhedral cone in NQ, by σ ⊂ MQ its dual cone, and by Polσ (NQ) the set of all σ-polyhedra, i.e. polyhedra in NQ with the recession cone σ (see [21, Definition 1.11] for the definition of the recession cone). Let T = Spec k[M] be an algebraic torus with its lattice of characters equal to M. To a σ-polyhedron ∆ ∈ Polσ (NQ) h σ ∨ → we associate its support function ∆ : Q, defined by

h m hm; i hm; pi: ∆( ) = min ∆ = min p∈∆

Definition 3.1. A variety is called semiprojective if it is projective over an affine variety. Let Y be a normal semiprojective variety. D P A σ-polyhedral divisor on Y is a formal sum = Z ∆Z ·Z, where Z runs over all prime divisors on Y , ∆Z ∈ Polσ (NQ), σ Z m ∈ σ ∨ D m P h m ·Z Y σ and ∆Z = for all but finitely many . For we define the Q-divisor ( ) = Z ∆Z ( ) on .A -polyhedral divisor D is called proper if the following two conditions hold:

∨ • D(m) is semiample and Q-Cartier for all m ∈ σ , ∨ • D(m) is big for all m ∈ relint σ .

A Q-Cartier divisor D is called semiample if there exists r > 0 such that the linear system |rD| is base point free, and big D ∈ |rD| r > Y \ D if there exists a divisor 0 , for some 0, such that the complement supp 0 is aﬃne. P P For a Q-divisor D = Z aZ ·Z, let bDc = Z baZ c·Z be the round-down divisor of D, and O(D) = O(bDc) be the corresponding sheaf of OY -modules. Any σ-polyhedral divisor D deﬁnes an M-graded quasicoherent sheaf of algebras Y on as follows: M A[Y; D] = O(D(m)): m∈σ∨∩M

Let A[Y; D] = Γ(Y; A[Y; D]) be the M-graded algebra corresponding to D.

Theorem 3.2. To any proper σ-polyhedral divisor D on a normal semiprojective variety Y one can associate a normal affine T -variety of dimension dim Y + dim T defined by X [Y; D] = Spec A[Y; D]. Conversely, any normal affine T -variety is isomorphic to X [Y; D] for some semiprojective variety Y and some proper σ-polyhedral divisor D on it.

We will discuss the converse correspondence in subsection 3.1.3.

3.1.2. Ingredients from GIT

Let X be an aﬃne T -variety. Let us recall the construction of the inverse limit of GIT-quotients. X X The cone ω generated by the weight monoid Σ in the vector space (T )Q = (T )⊗Z Q is called the weight cone. For any χ ∈ Σ, consider the set of its semistable points [ [ ss Xχ = Xf : r>0 f∈k[X]rχ

ss ss χ ∈ χ ∈ equivalent Xχ Xχ Two characters 1 Σ and 2 Σ are called if 1 = 2 . Under this equivalence Σ decomposes into Q X Q ﬁnitely many equivalence classes which are forming the GIT-fan ⊂ (T )Q with supp = ω [4, Section 2]. Recall some deﬁnitions giving us the construction of the GIT-fan.

Deﬁnition 3.3 ([4, Deﬁnition 2.1]). For any x ∈ X, the orbit cone wx associated to x is the following convex cone in X(T ): wx = cone χ ∈ Σ: there exists f ∈ k[X]χ such that f(x) =6 0 :

1867 Quotients of an aﬃne variety by an action of a torus

Deﬁnition 3.4 ([4, Deﬁnition 2.8]). For any character χ ∈ Σ, the associated GIT-cone σχ is the intersection of all orbit cones containing χ:

\ σχ = wx : {x∈X:χ∈wx }

By [4, Theorem 2.11], the collection of GIT-cones forms a fan Q having ω as its support. Moreover, the following statement holds.

Proposition 3.5 ([4, Proposition 2.9]). ss ss Let χ ; χ ∈ . Then Xχ ⊆ Xχ if and only if σχ ⊇ σχ . 1 2 Σ 1 2 1 2

ss ss ss For any cone λ ∈ Q, denote Xλ = Xχ , where χ is an arbitrary character in relint λ. Let X/λ T = Xλ //T be the good χ quotient under the action of T . Varieties X/λ T are called GIT-quotients. Notice also that X/λ T = Proj k[X]( ) for any χ ∈ relint λ, where ∞ χ M k[X]( ) = k[X]rχ : r=0 X/ T X//T k X T In particular, 0 = = Spec [ ] . ss Denote by qλ : Xλ → X/λ T the quotient map. We consider natural morphisms between GIT-quotients. Namely, if λ ⊃ λ λ ; λ ∈ Q 1 2, where 1 2 , then we have the following commutative diagram: ss ss Xλ / Xλ 1 2

qλ qλ 1 2

pλ λ 1 2 X/λ T / X/λ T: 1 2

q X ss → X/ T q X → X//T So the quotient maps λ : λ λ form a ﬁnite inverse system with 0 : sitting at the end. We have a canonical morphism \ ss ss q X Xχ → X/χ T X/ T: : = lim←− = C χ∈Σ X/ T GIT-limit The variety C is called the .

Deﬁnition 3.6. main component X/ T X/ T q X ss ⊂ X/ T The ( C )0 of the GIT-limit C is the closure of the image ( ) C .

The main component of the GIT-limit is also called the toric Chow quotient (see e.g. [9]). This terminology corresponds to results of [15], where it was proved that in the case of a projective toric variety X the main component of GIT-limit is, indeed, isomorphic to the Chow quotient by the action of a subtorus.

3.1.3. The Altmann–Hausen family

Now we give a construction of the Altmann–Hausen family of an aﬃne normal T -variety X following [2, Section 6]. Y X/ T norm qnorm X ss → Y q Let = ( C )0 be the normalization of the main component of the GIT-limit, : be the lifting of to the normalization, and pλ : Y → X/λ T be the composition of the normalization map and canonical morphisms from GIT-limit λ ; λ ∈ Q λ ⊃ λ to the elements of the inverse system restricted on the main component. For 1 2 , 1 2, we have the following commutative diagram:

1868 O.V. Chuvashova, N.A. Pechenkin

ss ss ss X / Xλ / Xλ / X 1 2

qnorm qλ qλ 1 2

pλ pλ λ 1 1 2 Y // X/λ T // X/λ T 1 2 q 0 p p λ 0 pλ 0 1 20

%% ( X//T

pλ pλ λ As it was shown in [2, Lemma 6.1], the morphisms and 1 2 are projective surjections with connected ﬁbers. In T particular, Y is projective over the aﬃne variety X//T = Spec k[X] . Moreover, the morphisms pλ are birational when T λ ∩ relint σ =6 ∅, and, consequently, dim Y = dim X − dim T and k(Y ) = k(X) . χ A character χ ∈ Σ is called saturated if the algebra k[X]( ) is generated by elements of degree one. It is well known that for any χ ∈ Σ there exists an integral nχ > 0 such that knχ χ is saturated for all integers k > 0. For multigraded version of this notion and an algebraic characterization of GIT-fan see [3].

λ ∈ Q χ ∈ λ O A q O ss For any and relint , we have a sheaf of X/λT -modules λ;χ = ( λ)∗( Xλ )χ . It is easy to see that the χ defined sheaf coincides with the twisting sheaf of Serre, if the variety X/λ T is considered as Proj k[X]( ). In particular, A χ χ A p ∗ A the sheaf λ;χ is invertible if is saturated. For a saturated , denote by χ = ( λ(χ)) ( λ(χ);χ ) the invertible sheaves of OY -modules. Firstly, assume that χ is saturated. If f ∈ k[X]χ , λ = σχ ∈ Q, then Xf //T is an open affine subset in X/λ T and A f ·O Y p−1 X //T Y Y λ;χ Xf //T = Xf //T (see [2, Lemma 6.3 (ii)]). Denote f = λ ( f ). The open sets f cover , and by definition of the inverse image we have (see [2, Lemma 6.4 (ii)])

A f ·O ⊂ f ·k Y f ·k X T k X : χ Yf = Yf ( ) = ( ) = ( )χ

Consequently, we can consider the sheaves Aχ as subsheaves of the constant sheaf k(X) on Y . A nχ A For unsaturated χ, we define χ (U) = {f ∈ k(X): f ∈ nχ χ (U)}. The defined sheaves together constitute the X T OY ( )-graded sheaf of -algebras M A = Aχ : χ∈Σ From [2, Lemma 6.4] we can see that the multiplication is defined correctly and preserves the grading.

Deﬁnition 3.7. The Altmann–Hausen family is the family ψ : Xe → Y of T -schemes, where Xe = SpecY A is the relative spectrum, ψ is A → A T X X T A the morphism given by 0 and the -action on e is given by the ( )-grading of .

O By [2, Lemma 6.4 (ii)], we have X = Spec Γ(X;e Xe ). Summarizing all that we have done in this section, note that from a T -variety X we constructed a semiprojective variety Y equipped with a X(T )-graded sheaf of OY -algebras A such that X can be restored from this data. ∨ A Denote Λ(T )Q = Λ(T )⊗Z Q and σ = ω ⊂ Λ(T )Q. One can easily pass from the sheaf of algebras to the proper σ-polyhedral divisor D on Y such that A = A[Y; D]: (2) To do this, let us choose a (non-canonical) homomorphism s: X(T ) → k(X) such that for every χ ∈ X(T ) the function s(χ) is homogeneous of degree χ. Such homeomorphisms always exist since T acts on X faithfully. Next, for a saturated χ ∈ ω ∩ X(T ) there exists an unique Cartier divisor D(χ) on Y such that

1 O(D(χ)) = · Aχ ⊂ k(Y ): s(χ)

1869 Quotients of an aﬃne variety by an action of a torus

For unsaturated χ, deﬁne 1 D(χ) = · D(nχ χ): nχ Now one can check that Q-Cartier divisors D(χ) can be “glued” together to form a proper σ-polyhedral divisor D on Y satisfying condition (2) (see [2, Section 6]). The divisor D constructed in such way is called minimal. We see that the variety Xe appears as a middle step in the correspondence between normal aﬃne T -varieties and combinatorial data of this action (Y; D) from Theorem 3.2 with an additional minimality condition on D:

2 T y Xe l

× T y X {(Y; D)};

where one of the arrows is dashed, because the reconstruction of D from Xe is not canonical. The main result illustrating the role of the Altmann–Hausen family in this assignment is the following one.

Proposition 3.8 ([2, Theorem 3.1]). The morphism ψ is a good quotient for the T -action. The canonical morphism ' : Xe → X is T -equivariant, birational and proper. We have the following commutative diagram:

' X ss // X o / Xe '−1

qnorm ψ !! ~~ Y:

3.1.4. A geometric description of the Altmann–Hausen family

Deﬁne W { x; q x x ∈ X ss} ⊂ X × X/ T ; X = ( ( )) : ( C )0 p and let C be projection on the second component. The following result modulo normalization morphism was stated in [20, Lemma 1], where actions of complexity one were considered, but the same proof is valid for a general case.

Proposition 3.9. The Altmann–Hausen family ψ X → Y is isomorphic to the normalization W norm → X/ T norm of the family of T -schemes : e X ( C )0 p W → X/ T . C : X ( C )0

Proof. θ X → X/ T ι ◦ ψ ψ X → X/ T norm Deﬁne a morphism : e ( C )0 as a composition of the morphism : e ( C )0 and the ι X/ T norm → X/ T normalization morphism :( C )0 ( C )0. Due to commutativity of the diagram from Proposition 3.8, the mappings ' X → X θ X → X/ T α '; θ X → W : e and : e ( C )0 deﬁne a morphism = ( ): e X .

α '

WX s

θ % " X × X/ T X ( C )0 //

$ X/ T ( C )0

1870 O.V. Chuvashova, N.A. Pechenkin

The morphism ι is affine as a normalization morphism, ψ is affine as a good quotient, so θ is affine as a composition of affine morphisms. The morphism ' is proper, so α is proper. Further, α is finite, because it is both proper and affine. But α is also birational, hence it is the normalization morphism.

Note that the variety WX is equipped with the natural faithful action of the torus T , which lifts to an action of T on norm X O A O X Xe = WX and defines a (T )-grading of the sheaf ψ∗( Xe ). By formula χ = ψ∗( Xe )χ , one can recover the (T )-graded sheaf of OY -algebras A, and, as well, the proper polyhedral divisor on Y (see subsection 3.1.3). p W → X/ T p p Denote by C the projection X ( C )0. Note that C is the good quotient. Indeed, it is clear that C is affine, T T p# O X/ T → p ∗ OW p# -invariant, and C : ( C )0 ( C) ( X ) is injective. To prove the surjectiveness of C , use the fact that functions which are constant on the fibers come from the base. W We have the following commutative diagram, where pX denotes the projection on X:

W pX WX // X

p C π X/ T X//T : ( C )0 W / pX //T

Example 3.10. X n T k× T X t · x ; : : : ; x tx ; : : : ; tx Let = A , = and act on by scalar multiplication, i.e. ( 1 n) = ( 1 n). Then the Z-grading of k X k x ; : : : ; x x i ; : : : ; n λ the algebra [ ] = [ 1 n] is given by deg i = 1 for = 1 . For this action, GIT-fan consists of two cones 0 λ and 1:

λ0 λ1

−1 0r 1

We have

ss T Xλ X;X/λ T X//T k X {pt} 0 = 0 = = Spec [ ] = ; ss n−1 Xλ X \{x ::: xn };X/λ T k x ; : : : ; xn P : 1 = 1 = = = 0 1 = Proj [ 1 ] =

X/ T X/λ T The inverse limit C of GIT-system is equal to the variety 1 , so

n−1 ss ss Y X/ T X/ T X/λ T P ;X Xλ X \{x ::: xn}: = ( C )0 = C = 1 = = 1 = 1 = =

qnorm X ss → Y qnorm x ; : : : ; x x ::: x W ⊂ X × X/ T In coordinates the map : is given by ( 1 n) = [ 1 : : n]. The variety X ( C )0 = n × n−1 x y x y x ; : : : ; x n y ::: y A P is deﬁned by the equations i j = j i, where ( 1 n) are coordinates on A , and [ 1 : : n] are n− n homogeneous coordinates on P 1, i.e. WX is the blowing-up of A at the point (0;:::; 0). On the other hand,

M M M Xe A An OY ⊕ Aλ ;n O n O − = SpecY = SpecY = SpecY 1 = SpecY ( ) = Tot( ( 1)) n∈ n∈ n∈ Z>0 Z>0 Z>0

n− is the total space of the tautological line bundle on P 1. It is well known that these varieties are isomorphic.

3.2. The toric Hilbert scheme and its main component

Let X be an irreducible aﬃne T -variety. In the following subsection we give an exposition of basic properties on multigraded Hilbert schemes and study some particular cases of this notion. See [6, Section 3] for some generalizations of these results on the case of G-variety X with a reductive algebraic group G.

1871 Quotients of an aﬃne variety by an action of a torus

3.2.1. Definitions and basic facts on multigraded Hilbert schemes Definition 3.11. Given a function h: X(T ) → N, a family p: F → S of affine T -schemes has Hilbert function h if for every χ ∈ X(T ) the sheaf of OS-modules p∗(OF )χ is locally free of rank h(χ).

Note that the morphism p is ﬂat. If h(0) = 1, then p is the good quotient of X by the action of T .

Definition 3.12 ([13]). h Given a function h: X(T ) → N, the Hilbert functor is the contravariant functor HX;T from the category of schemes to the category of sets assigning to any scheme S the set of all closed T -stable subschemes Z ⊆ S ×X such that the projection p: Z → S is a (flat) family of affine T -schemes with Hilbert function h.

h In [13, Theorem 1.1] it was proved that there exists a quasiprojective scheme HV ;T which represents this functor in the h case when X is a finite-dimensional T -module V ; the scheme HV ;T is called the multigraded Hilbert scheme. In the case of an arbitrary X there exists a T -equivariant closed immersion X,→ V , where V is a finite-dimensional T -module. h h h h Then the Hilbert functor HX;T is represented by a closed subscheme HX;T of HV ;T [1, Lemma 1.6]. The scheme HX;T is called the invariant Hilbert scheme. h h h h Recall that the universal family UX;T is the element of HX;T (HX;T ) corresponding to the identity map {Id: HX;T → h h h h h h h h HX;T } ∈ HX;T (HX;T ) = Mor(HX;T ;HX;T ). So UX;T is the closed subscheme of HX;T ×X such that for any Z ∈ HX;T (S) h Z U × h S T X T we have = X;T HX;T . In particular, for any -equivariant closed immersion of in a finite-dimensional -module V , we have the following Cartesian diagram: h h UX;T / UV ;T

h h HX;T / HV ;T :

Lemma 3.13. Assume that h(0) = 1. Then we have the following commutative diagram:

p h X UX;T / X

pH π

p //T h X HX;T // X//T ;

where pX is the projection, and pX //T assigns to any family its quotient by T . The morphisms pX and pX //T are projective.

Proof. The commutativity of the diagram can be seen easily by considering the corresponding morphisms of functors h h of points. The morphism pX //T is projective by [7, Lemma 3.3]. The morphism UX;T → HX;T ×X//T X is a closed embedding, so the morphism pX is projective. See also [6, Proposition 3.15].

3.2.2. Deﬁnition of main component

Let \ [ X s = Xf : χ∈Σ f∈k[X]χ

1872 O.V. Chuvashova, N.A. Pechenkin

Note that, if x ∈ X s, then T x has the following Hilbert function:

( χ ∈ ; h χ 1 if Σ Σ( ) = 0 otherwise.

X T T x ' X x ∈ X s H Let Σ denote the aﬃne toric -variety with the weight monoid Σ; then Σ for . We denote by the h Σ s invariant Hilbert scheme HX;T ; it is called the toric Hilbert scheme [18]. So for any x ∈ X we have the k-rational point h h Σ Σ T x ∈ H. Also denote U = UX;T , H = HX;T . X s ×T X X s ×X T t · x; y t−1x; ty Let Σ be the quotient of the variety Σ by the action of given by ( ) = ( ).

Lemma 3.14. There exists the geometric quotient πs : X s → X s/T and an open embedding of X s/T in the toric Hilbert scheme H. Moreover, we have the following Cartesian diagram:

s T X × XΣ / U

pH X s/T / H:

Proof. Step 1. There exists the categorical quotient πs : X s → X s/T . Moreover, πs is a locally trivial bundle. χ ; : : : ; χ ∈ k X k X i ; : : : ; r X s Indeed, let 1 r Σ be such that the algebra [ ] is generated by [ ]χi , = 1 . Then is covered by T -invariant open aﬃne subschemes Xf ·:::·f k X f ·:::·f ; 1 r = Spec( [ ] 1 r ) f ∈ k X f 6 e ; : : : ; e ∈ X T χ ; : : : ; χ where i [ ]χi , i = 0. Fix a basis 1 d ( ). Since the characters 1 r generate Σ and, consequently, X T hl ∈ k Xf ·:::·f e ( ), we can choose non-zero elements [ 1 r ] l . Then we have

T ± ± T k Xf ·:::·f ' k Xf ·:::·f ⊗ k h 1; : : : ; h 1 ' k Xf ·:::·f ⊗ k T : [ 1 r ] [ 1 r ] 1 d [ 1 r ] [ ]

Note that these isomorphisms satisfy the compatibility conditions and the variety X s/T obtained by gluing aﬃne charts Xf /T is separated. Indeed, we have to show that the diagonal morphism

Xfg/T → Xf /T ×Xg/T

X /T k X T X /T k X T f f · ::: · f g g · ::: · g f ; g ∈ k X is closed for any f = Spec( [ f ] ) and g = Spec( [ g] ), where = 1 r, = 1 r and i i [ ]χi . This is equivalent to showing that the corresponding homomorphism of algebras

T T T k[Xf ] ⊗ k[Xg] → k[Xfg]

k T k k k k is surjective. But this is clear since for any element h/(fg) ∈ k[Xfg] we have h/(fg) = (h/f2 )(f /g ). Step 2. Let us prove that X s/T represents an open subfunctor of H. Consider the family

ps X s ×T X → X s/T: : Σ

X s/T X s ×T X X s/T We shall show that this is the universal family over . Indeed, Σ is a locally trivial bundle over with X X s ×T X ⊂ X × X s/T ﬁber Σ. Further, there is a canonical closed embedding Σ ( ), which is locally given by the surjective homomorphisms of algebras T M k X ⊗ k Xf ·:::·f → k Xf ·:::·f χ [ ] [ 1 r ] [ 1 r ] χ∈Σ

1873 Quotients of an aﬃne variety by an action of a torus

T L s X s × X k Xf ·:::·f χ H X /T ( Σ is covered by the open aﬃne subschemes Spec χ∈Σ [ 1 r ] ). This gives us an element in ( ), i.e., we have the following Cartesian diagram: s T X × XΣ / U

pH X s/T / H:

Step 3. It is clear that the image of X s/T lies in the locus Hs of H where the ﬁbers of the universal family are −1 s −1 s irreducible and reduced. By [12, Theorem 12.1.1], it follows that pH (H ) is an open subscheme of U. Since pH (H ) is T -invariant and pH maps closed T -invariant subsets to closed subsets, it follows that Hs is an open subscheme of H. We shall show that the morphism Φ: X s/T → Hs is an isomorphism. Given a morphism φ: S → X s/T , we have the following commutative diagram: s T s Z / X × XΣ / U

S / X s/T // Hs;

s s s s where U = H ×H U and Z = S ×Hs U is the image of φ in H (S). Note that all the squares of this diagram are Z Z s ×T X Z s S× X s Z ∩ S ×X s Cartesian. In particular, it follows that = Σ, where = Xs/T = ( ). 0 Conversely, let us construct the inverse morphism Φ : Hs → X s/T . Let Z ∈ Hs(S). Consider Z s = Z ∩ (S ×X s); then Z s/T = S. So the projection Z s → X s deﬁnes a morphism of quotients {S → X s/T } ∈ X s/T (S). Moreover, Z s S × X s Z Z s ×T X ◦ 0 0 ◦ = Xs/T and = Σ. Thus we see that Φ Φ = IdHs and Φ Φ = IdXs/T .

Deﬁnition 3.15. main component H H H X s/T The 0 of the toric Hilbert scheme is the closure in of the image of .

H H main By Lemma 3.14, it follows that the main component 0 is an irreducible component of . Consider also the component U p−1 H p p U of the universal family 0 = H ( 0). Denote by 0 the restriction of H on 0.

Lemma 3.16.

The main component U0 is an irreducible component of U.

U Proof. It is sufficient to show that 0 is irreducible. By [7, Lemma 3.5], the dimension of any irreducible component of p−1 x T x ∈ H p the fibre ( ) equals dim for any . Since 0 is flat, this implies that the dimension of any irreducible component Z U X Z H Z ∩ X s ×T X of 0 is equal to dim [14, Corollary 9.6]. It follows that dominates 0. Consequently, the intersection ( Σ) Z Z ∩ X s ×T X U X s ×T X is non-empty and = ( Σ). This implies that 0 = Σ is irreducible.

3.3. The toric Chow morphism

H → X/ T First of all, let us recall the construction of the toric Chow morphism C . This construction was given in [13, Section 5] for the case when X is a ﬁnite-dimensional T -module, but it is almost the same in the case of an aﬃne T -variety. nχ For any character χ ∈ Σ, there exists n > 0 such that the Z-graded algebra Rnχ = k[X]( ) is generated by its elements h X/σ T Rnχ H × of degree one. Note that the scheme χ = Proj represents the functor Spec Rnχ ;k , where

( 1 if r ≥ 0; h(m) = 0 otherwise,

1874 O.V. Chuvashova, N.A. Pechenkin

h H → H × (see [7, Corollary 3.5]). The natural morphism of functors Spec Rnχ ;k induces the canonical morphism of the schemes representing this functors H → X/σχ T and it does not depend on the choice of n. Moreover, by [13, Lemma 7.5], these H → X/ T morphisms commute with the morphisms of the inverse system, so they deﬁne a canonical morphism Ψ: C .

Lemma 3.17. There exists an open embedding X s/T ⊂ X/ T . Moreover, we have the following Cartesian diagram: ( C )0

s T X × XΣ / WX

p C X s/T X/ T : // ( C )0

Proof. Note that in the inverse limit of GIT-quotients X/λT , λ ∈ Q, it is suﬃcient to take the limit over λ intersecting s ss s −1 s ss relint ω. Moreover, for any such λ the quotient X /T is an open subscheme in Xλ //T and X = qλ (X /T ) ⊂ Xλ . X s T Xf ·:::·f Indeed, is covered by the open aﬃne -stable subschemes 1 r (with the notation of the proof of Lemma 3.14). P c cr ci > ciχi χ n > χ ∈ λ f f 1 · ::: · f ∈ k X χ There exist 0 such that = for some 0 and relint . Then = 1 r [ ] , and − Xf ·:::·f Xf q 1 Xf /T ⊂ X ss 1 r = = λ ( ) λ . This implies that we have a Cartesian diagram X s / X ss

s X /T // X/C T:

X s X ss ∩ X ×X s/T ⊂ X ×X/ T X s ×T X X s ⊂ X ×X s/T This means that = ( ) C . Note also that Σ = . Finally, this implies X s ×T X W ∩ X ×X s/T ⊂ X ×X/ T that Σ = X ( ) C .

In the following theorem we show that the toric Chow morphism Ψ restricted on the main component lifts to a birational projective morphism U0 → WX .

Theorem 3.18. We have the following commutative diagram:

Φ U0 / WX

0 W pX pX "" X z

p p H C

H X/ T 0 / ( C )0 Ψ0

p0 //T pW //T X "" zz X X//T ; where 0 is the restriction of on H and is the restriction of the morphism 0 × on U ⊂ H ×X. The morphisms Ψ Ψ 0 Φ Ψ IdX 0 0 0 W W Ψ ; Φ; pX and pX //T are projective ant the ﬁrst three of them are, moreover, birational.

1875 Quotients of an aﬃne variety by an action of a torus

0 X s/T ⊂ H X s/T ⊂ X/ T 0 Proof. The restriction of Ψ on 0 is the identity map on ( C )0, thus Ψ is a birational morphism. 0 × X s ⊂ U X s ⊂ W 0 × U W Also, the restriction of Ψ IdX on 0 is the identity map on X;T , so Ψ IdX maps 0 birationally on X . 0 W 0 W 0 By construction of the morphisms Ψ and Φ, it follows that pX = pX ◦Φ and pX //T = pX //T ◦Ψ . Further, by Lemma 3.13, 0 0 0 the morphisms pX and pX //T are projective. It follows that Φ and Ψ are projective. The other assertions are obvious.

Corollary 3.19. There exists a canonical birational projective morphism Unorm → X. 0 e

4. Subtorus actions on aﬃne toric varieties

In this section X = Xσ is a normal aﬃne toric variety under the torus T, σ ⊂ Λ(T)Q is a corresponding cone. We consider the action on X of the subtorus T ⊂ T. The natural homomorphism T → T/T induces the surjective map of vector spaces that are generated by the corresponding lattices of one-parameter subgroups α : Λ(T)Q → Λ(T/T )Q. The X X ∨ embedding of torus T,→ T induces the surjective map of vector spaces β : (T)Q → (T )Q such that β(σ ) = ω, where X ω ⊂ (T )Q is the weight cone.

4.1. The fan representation of the Altmann–Hausen family

Let ψ : Xe → Y be the Altmann–Hausen family of the T -action on X, and let ' : Xe → X be the canonical morphism. We use the notation of subsection 3.1.3 related to the construction of ψ. x X X ss ⊃ x x ; x ; y x Let T be the open orbit of the T-action on . It can be easily seen that T . Denote by λ C the images of X/ T X/ T Y q X/ T in λ , ( C )0 and respectively. The quotient maps λ induce actions of the torus T on the GIT-quotients λ . So X/ T x / λ ω λ = T λ is the toric variety under the torus T Txλ , where Txλ is the stabilizer. If lies in the interior of , general qλ T x T pλ λ ﬁbers of the morphism contain a unique dense -orbit, and consequently T λ = . Note that the morphisms 1 2 in the GIT-system are T-equivariant, so Y = Ty is a normal toric variety under the torus T/T [9, Proposition 3.8], and the p W x; x ⊂ X × X/ T morphisms λ are T-equivariant. Also X = T( C) ( C )0 is a toric (not necessarily normal) variety with the torus T.

Lemma 4.1. The fan C of the toric variety W x; x ⊂ X × X/ T is the coarsest common reﬁnement of the cone σ and the WX X = T( C) ( C )0 − quasifan α 1 C X/ T . ( ( C )0 )

Proof. Note that we can consider a fan of the toric variety WX (which is not necessarily normal), because there W X ×U X/ T S U exists a T-invariant affine covering X = i, where ( C )0 = i is an affine T-invariant covering of the variety X/ T ( C )0. The existence of the last one immediately follows from the existence of a T-equivariant closed embedding of X/ T variety ( C )0 in the direct product of GIT-factors. Lemma follows now from Proposition 2.3 and the definition of the variety WX .

Q X Q0 Q Let ⊂ (T )Q be the GIT-fan. Denote by the set of λ ∈ satisfying the condition λ ∩ relint ω =6 ∅. We have already seen that for λ ∈ Q0 the corresponding GIT-quotients X/λT are toric varieties under the torus T/T . The variety Y is also toric under the torus T/T . The construction of the fans of these varieties is described in [9]. In the framework of that paper a more general case of a semiprojective toric variety X corresponding to the normal fan of some polyhedra X ∨ P ⊂ (T)Q. The case of an aﬃne toric variety X = Xσ considered in [7, Section 5] (in this case P = σ ). We shall continue our exposition following [7]. − ∨ For all χ ∈ ω, denote Pχ = β 1(χ) ∩ σ . Let λ ∈ Q0. Then, for all χ ∈ relint λ, the polyhedra Pχ have the same normal fan Nλ ⊂ Λ(T/T ) [7, Remark 4.12] which coincides with the fan of the toric variety X/λT . The fan of the toric variety Y is the coarsest common reﬁnement of all the fans Nλ, where λ ∈ Q0, so it is the normal fan of the Minkowski sum X Pχλ ; λ∈Λ0

1876 O.V. Chuvashova, N.A. Pechenkin

where χλ ∈ relint λ.

CY C X/ T Having the description of the fan = ( C )0 , the following result allows us to describe the Altmann–Hausen family in the toric case.

Theorem 4.2. Let X = Xσ be the toric variety with a big torus T, and let T ⊂ T be a subtorus. Then the variety Xe constructed for the C action of T on X is a normal toric variety with the torus T. Its fan Xe ⊂ Λ(T) is the coarsest common reﬁnement of the − quasifan α 1 C X/ T and the cone σ, where α → /T is the natural map. ( ( C )0 ) : Λ(T) Λ(T )

C Proof. By Proposition 3.9, the toric variety Xe is isomorphic to the normalization of the variety WX , so their fans Xe C and WX coincide. Now our statement follows directly from Lemma 4.1.

C C The fact that the fans Xe and WX coincide (and so Theorem 4.2) could be proved by methods of toric geometry without using Proposition 3.9. We shall give this proof below. Moreover, one can prove Proposition 3.9 in the toric case using Theorem 4.2 and then pass on to general case.

Another proof of Theorem 4.2. The morphism ' : Xe → X is proper, so, by Proposition 2.2, the support of the fan C CXe coincides with σ. The morphism ψ : Xe → Y is toric, hence, by Proposition 2.1, the image αQ(τ) of any cone τ ∈ Xe is contained in some cone δ ∈ CY . Also the morphism ψ is a good quotient, so, by [19, Theorem 4.1], we have that −1 δ = α(τ) and τ = α (δ) ∩ σ, which exactly means that the fan CXe is the coarsest common reﬁnement of the quasifan − α 1 C X/ T σ CW C ( ( C )0 ) and the cone . By Lemma 4.1, it follows that the fans X and Xe coincide.

Example 4.3. Let us consider the action of the one-dimensional torus T on the four-dimensional aﬃne space X = A4 given by t · x ; x ; x ; x tx ; tx ; t−1x ; t−1x C X ( 1 2 3 4) = ( 1 2 3 4). Our main purpose is to construct the fan Xe of the toric variety e. Along the way, we will construct the fans of the toric varieties X/λT and Y , where λ ∈ Q0. Q λ ; λ λ The GIT-fan consists of the three cones −1 0, and 1:

λ−1 λ0 λ1

−1 0r 1

We will consider the torus T as the subtorus of the four-dimensional torus T that acts on X by the rescaling of X X σ ⊂ e ; e ; e ; e coordinates. Then = σ , where Λ(T)Q is the cone generated by the standard basis 1 2 3 4 of Λ(T)Q.

There is a natural immersion Λ(T )Q ,→ Λ(T)Q. We will identify the space Λ(T/T )Q with the orthogonal complement of Λ(T )Q in the space Λ(T)Q. Then the natural map α : Λ(T)Q → Λ(T/T )Q will be the orthogonal projection.

GIT-system

1877 Quotients of an aﬃne variety by an action of a torus

ρ ; − ; ; ρ ; ; − ; ρ − ; ; ; ρ In the coordinates of the standard basis of Λ(T)Q denote 1 = (3 1 1 1), 2 = (1 1 1 3), 3 = ( 1 3 1 1), 4 = ; ; ; − ρ ; ; ; X/λ T X/λ T X/λ T (1 1 3 1) and = (1 1 1 1). The fans of the toric varieties 0 , −1 and 1 are the normal fans of the P P− P CX/λ T Nλ polyhedra 0, 1 and 1 respectively. After some calculations one can verify that the fan 0 = 0 consists of one hρ ; ρ ; ρ ; ρ iQ≥ CX/λ T Nλ− hρ ; ρ ; ρ iQ≥ hρ ; ρ ; ρ iQ≥ maximal cone 1 2 3 4 0 , the fan −1 = 1 consists of two maximal cones 1 2 3 0 and 3 4 1 0 , CX/λ T Nλ hρ ; ρ ; ρ iQ≥ hρ ; ρ ; ρ iQ≥ CY and the fan 1 = 1 consists of two maximal cones 4 1 2 0 and 2 3 4 0 . The fan , as the coarsest CX/λ T CX/λ T CX/λ T hρ ; ρ ; ρiQ≥ hρ ; ρ ; ρiQ≥ hρ ; ρ ; ρiQ≥ common reﬁnement of 0 , −1 and 1 , consists of four maximal cones 1 2 0 , 2 3 0 , 3 4 0 , hρ ; ρ ; ρi 4 1 Q≥0 . C α−1 C σ µ ; ; ; The fan Xe is the coarsest common reﬁnement of the quasifan ( Y ) and the cone . Denote 1 = (1 1 0 0), µ ; ; ; C he ; e ; µ ; µ i he ; e ; µ ; µ i 2 = (0 0 1 1). Calculations show that Xe consists of four maximal cones 1 3 1 2 Q≥0 , 1 4 1 2 Q≥0 , he ; e ; µ ; µ i he ; e ; µ ; µ i 2 3 1 2 Q≥0 , and 2 4 1 2 Q≥0 . On the picture below you can see polyhedral slice of this fan by the hy- x x x x perplane 1 + 2 + 3 + 4 = 1.

e2 ¡HBS H H ¡ BS HH ¡ B S H ¡ B S HH H µ ¡ B S He 1 H 3 ¡S H B S ¡ ¡ S HB S ¡ ¡ S B H S ¡ ¡ S B H S ¡ e ¡ S B HS¡ 1 H µ2 HH S B ¡ H S B ¡ HH S B ¡ H HHSB ¡ SBB¡H e4

4.2. The fan of the main component of the toric Hilbert scheme

Finally, we complete a description of all the fans of the toric varieties from the commutative diagram (1) given in Section 1 by recalling some results from [7]. H U Consider the main component 0 of the toric Hilbert scheme and the main component 0 of the universal family in the toric situation with their structure of reduced schemes. As shown in [7, Proposition 3.6], these varieties are toric under the tori T/T and T respectively. Our purpose is to recall the construction of the fans corresponding to toric varieties H U 0 and 0. I −1 For all χ ∈ ω ∩ X(T ) denote Pχ = conv β (χ) ∩ X(T) .

Remark 4.4. I The equality Pχ = Pχ holds if and only if the polyhedra Pχ have integer vertices. Such characters χ are called integer.

We call two polyhedra equivalent if they have the same normal fan.

Proposition 4.5 ([7, Proposition 4.9 (2)]). I There are only ﬁnitely many non-equivalent polyhedra P for χ ∈ ω. The fan CH corresponding to the variety H is χ 0 0 the normal fan of the Minkowski sum over a system of representatives of the equivalent classes of these polyhedra.

CH Pχ Of course, it is again equivalent to say that 0 is the coarsest common reﬁnement of the normal fans of all . U y; x ⊂ H ×X x ∈ X y T x ∈ H The variety 0 coincides with T( ) 0 , where lies in the open T-orbit and = 0. From CU CH σ Proposition 2.3 it follows that the fan 0 is the coarsest common reﬁnement of the fan 0 and .

1878 O.V. Chuvashova, N.A. Pechenkin

CH Now we know the fans of all varieties from the commutative diagram (1). By Remark 4.4 it follows that the fan 0 C X/ T CU CW 0 reﬁnes the fan ( C )0 and the fan 0 reﬁnes the fan X , what illustrates that morphisms Ψ and Φ are toric.

Remark 4.6. 0 W From construction of all these fans or straightforwardly from the properties of morphisms Ψ ; Φ; pX it follows that |CH | |C X/ T | |CU | |CW | σ 0 = ( C )0 and 0 = X = .

χ ∈ ω∩X T CH C X/ T In Example 4.3 all characters ( ) are integer, since the fans of the toric varieties 0 and ( C )0 coincide (and CU CW so, certainly, the fans 0 and X also coincide). But this is not true in general. The example below was considered CH C X/ T in [7]. There were constructed the fans 0 and ( C )0 , and it was shown that they do not coincide in the considered CU CW C situation. We shall complete this example constructing the fans 0 and X = Xe .

Example 4.7 ([7, Example 5.3]). X 3 k× 3 T k× t · x ; x ; x tx ; tx ; t2x Let σ = A , T = ( ) act by rescaling of coordinates, and = act by ( 1 2 3) = ( 1 2 3). Denote by e ; ; e ; ; e ; ; ν ; ν ; ν X 1 = (1 0 0), 2 = (0 1 0), 3 = (0 0 1) the standard basis of Λ(T)Q, and let 1 2 3 be its dual basis in (T)Q.

CH P C X/ T It is easy to see that the fan 0 is the normal fan of the trapezoid 3, whereas the fan ( C )0 is the normal fan of P /T T the triangle 2. As in Example 4.3, we will identify the space Λ(T )Q with the orthogonal complement of Λ( )Q in the space Λ(T)Q. ρ ; ; − ρ − ; ; ρ ; − ; Denote 1 = (1 1 1), 2 = ( 5 1 2), 3 = (1 5 2) in the coordinates of the standard basis of Λ(T)Q. All these /T C X/ T hρ ; ρ i hρ ; ρ i vectors are lying in Λ(T )Q, and it is easy to verify that ( C )0 consists of three maximal cones 1 2 Q≥0 , 2 3 Q≥0 , hρ ; ρ i CH hρ ; ρ i hρ ; ρ i h−ρ ; ρ i h−ρ ; ρ i 1 3 Q≥0 , and 0 consists of four maximal cones 1 2 Q≥0 , 1 3 Q≥0 , 1 2 Q≥0 , 1 3 Q≥0 .

−ρ 1 ρ 6 ρ 2 3 HYH ¨¨* ¡H ¡A ¨A ¡ H¡ ¨A A ¡ ¡ A P A ¡ 2 A P ? 3 ρ 1

CH C X/ T We see that the fan 0 subdivides the fan ( C )0 , and, in particular, they do not coincide. If we intersect the σ he ; e ; e i CU CW corresponding quasifans in Λ(T)Q with the cone = 1 2 3 Q≥0 , we get the fans 0 and X respectively. Denote κ ; ; µ ; ; µ ; ; µ ; ; CW hκ; µ ; e ; µ i = (1 1 2), 1 = (1 1 0), 2 = (1 0 2), and 3 = (0 1 2). Then X consists of three maximal cones 1 1 2 Q≥0 , hκ; µ ; e ; µ i hκ; µ ; e ; µ i CU hκ; µ ; e ; µ i hκ; µ ; e ; µ i hκ; e ; µ i 1 2 3 Q≥0 , 2 3 3 Q≥0 , and 0 consists of four maximal cones 1 1 2 Q≥0 , 1 2 3 Q≥0 , 3 2 Q≥0 , hκ; e ; µ i 3 3 Q≥0 .

Acknowledgements

The authors are grateful to Ivan V. Arzhantsev for his attention to this work and fruitful discussions. We also thank Dmitri A. Timashev for many important comments. The ﬁrst author is thankful to Michel Brion for stimulating ideas and useful discussions. The second author thanks Roman Gayduk and Mikhail Gorsky for helpful comments. The authors are grateful to the referee for important comments which led to improvement of the article. The second author was partially supported by the grants NSh-5139.2012.1, MK-6612.2012.1, RFFI 12-01-31506, RFFI 12-01-33024.

1879 Quotients of an aﬃne variety by an action of a torus

e µ e 3 3 2 HH ¨¨ @ H ¨ @ H ¨¨ @ H κ ¨ µ H ¨ 2@ H ¨ ¨H¨ @ ¨ HH ¨@¨ H ¨ @ HH ¨ H ¨ ¨ @ H ¨ H H @ H µ H 1 O H @ HH @ HH @ HH @ H @ HH H @ HH@ HH@ H e @ 1

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