On orbits of the automorphism group on a complete toric variety

0.97 MB·English

by  Ivan Bazhov

ON ORBITS OF THE AUTOMORPHISM GROUP ON
A COMPLETE TORIC VARIETY
IVAN BAZHOV
Abstract. LetX beacompletetoricvarietyandAut(X)bethe
automorphism group. We give an explit description of Aut(X)-
orbits on X. In particular, we show that Aut(X) acts on X tran-
2
sitively if and only if X is a product of projective spaces.
1
0
2

n
a
J 1. Introduction
2
1
Let us recall that a toric variety is a normal algebraic variety X with

] a generically transitive and effective action of an algebraic torus T.
G The complement X(cid:114)O of the open T-orbit O is a union of prime T-
0 0
A invariant divisors D ,...,D . Every T-orbit on X is uniquely defined
1 r
.
h by the divisors D which contain the orbit. In particular, the number
i
t of T-orbits is finite.
a
m It is known that the group of algebraic automorphisms Aut(X) of
[ a projective or, more generally, complete toric variety X is an affine

2 algebraic group. The group Aut(X) contains T as a maximal torus.
v Hence, the connected component of identity Aut0(X) is generated by
5
the torus T and root subgroups, i.e., one dimensional unipotent sub-
7
2 groups normalized by T. For smooth X a combinatorial description
4
of root subgroups was found by M.Demazure [8]. A generalization of
.
0 this description to complete simplicial toric varieties in terms of total
1
coordinates was given in [6]. The case of an arbitrary complete toric
1
1 variety was studied in [5], see also [10]. The paper [4] worked out an-
:
v other method to describe the automorphism group of a projective toric
Xi variety.
It is clear that any Aut0(X)- as well as any Aut(X)-orbit on X is
r
a a union of T-orbits. Our goal is to describe T-orbits which lie in the
same Aut0(X)- or Aut(X)-orbit.
Let us associate with a T-orbit O the set of all prime T-invariant
divisors D(O) := {D ,...,D } which do not contain O. Let Γ(O) be
i1 il
the monoid in the divisor class group Cl(X) generated by classes of the
divisors in D(O).
2010 Mathematics Subject Classification. 14M25, 14L30.
Key words and phrases. toric variety, automorphism, invariant divisor.
1
2 IVAN BAZHOV
Theorem 1. Let X be a complete toric variety. Then two T-orbits O
and O(cid:48) on X lie in the same Aut0(X)-orbit if and only if
Γ(O) = Γ(O(cid:48)).
The proof uses techniques of paper [2]. It is shown there that for
an affine toric variety any non-trivial orbit of a root subgroup meets
exactly two T-orbits and a combinatorial description of such pairs of
orbits is given. We adapt these results to a complete toric variety.
Paper [3] gives a systematic treatment of toric varieties in term of
bunches of cones. These cones are images of Γ(O) in Cl(X)⊗Q and
they are Gale dual to cones in ∆.
Theorem 3.6 brings a description of the closures of Aut0(X)-orbits
on X. Theorem 3.7 gives a description of Aut(X)-orbits. As a corol-
lary we show that the action of Aut(X) on X is transitive if and only
if X is a product of projective spaces. A related result obtained in
[1, Proposition 4.1] states that every complete toric variety which ad-
mits a transitive action of a semisimple algebraic group is a product of
projective spaces.
The base field k is assumed to be algebraically closed and of charac-
teristic zero.
The author is grateful to his supervisor I.V.Arzhantsev for stating
the problem, the idea to apply Gale duality and useful references, and
to the referee for his/her careful reading of the manuscript and many
helpful comments.
2. Preliminaries
Toric varieties and fans. Let N ∼= Zd be the lattice of one-
parameter subgroups of a torus T, M be the dual lattice of characters,
and (cid:104)·,·(cid:105) : M × N → Z be the natural pairing. Also we put N =
Q
N ⊗ Q, M = M ⊗ Q.
Z Q Z
With any toric variety X one associates a fan ∆ = ∆ of rational
X
polyhedral cones in N , e.g., see [9, 7, 11]. Let ∆(1) = {τ ,...,τ }
Q 1 r
be the set of one-dimensional cones in ∆ and σ(1) be the set of one-
dimensional faces of a cone σ. The primitive lattice generator of a ray
τ is denoted as v and v = v .
τ i τi
There exists a one-to-one correspondence σ ↔ O between cones σ
σ
in ∆ and T-orbits O on X such that dimO = dim∆−dimσ. A toric
σ σ
variety X is complete if and only if the fan ∆ is complete.
Every prime T-invariant divisor D is a closure of an orbit O for
i τi
someτ ∈ ∆(1). AT-orbitO iscontainedintheintersectionofdivisors
i σ
D corresponding to rays τ ∈ σ(1).
i i
There is a natural map from the free abelian group Div (X) gener-
T
ated by all prime T-invariant divisors to the divisor class group Cl(X):
D (cid:55)→ [D]. If X is complete then the following sequence is exact [7,
AUTOMORPHISMS OF A TORIC VARIETY 3
Theorem 4.1.3]:
(1) 0 → M → Div (X) → Cl(X) → 0,
T
r
(cid:80)
where m ∈ M goes to (cid:104)m,v (cid:105)D .
i i
i=1
Example 2.1. Let us give some examples of complete toric surfaces.
(i) The projective plane P2, the fan ∆ is shown in Figure 1(i),
P2
∆ (1) is generated by (1,0), (0,−1), (−1,1), Cl(P2) ∼= Z.
P2
(ii) The Hirtzebruch surface H , its fan is shown in Figure 1(ii),
s
∆ (1), s (cid:62) 1, is generated by (1,0), (0,−1), (−1,s), (0,1),
H
s
Cl(H ) ∼= Z2.
s
(iii) The weighted projective plane P(1,1,s), s (cid:62) 2, is defined
by the fan in Figure 1(iii), ∆ (1) is generated by (1,0),
P(1,1,s)
(0,−1), (−1,s), Cl(P(1,1,s)) ∼= Z.
(iv) The variety B , s (cid:62) 1, is defined by the fan in Figure 1(iv),
s
∆ (1) is generated by (s,1), (s,−1), (−s,−1), (−s,1),
B
s
Cl(B ) ∼= Z2 ⊕Z .
s 2s
Figure 1. Fans of P2, H , P(1,1,s), and B .
s s
Root subgroups and the automorphism group. Let X be a
complete toric variety and ∆ be the corresponding fan.
Definition 2.2. An element m ∈ M is called a Demazure root of ∆ if
the following conditions hold:
∃τ ∈ ∆(1) : (cid:104)m,v (cid:105) = −1 and ∀τ(cid:48) ∈ ∆(1)(cid:114){τ} : (cid:104)m,v (cid:105) (cid:62) 0.
τ τ(cid:48)
4 IVAN BAZHOV
Let us denote the set of all roots of the fan ∆ as R. Denote by η ,
m
m ∈ R, the unique ray generator v such that (cid:104)m,v (cid:105) = −1. A root
τ τ
m ∈ R∩(−R) is called semisimple.
Example 2.3. The sets of roots for the varieties of Example 2.1 are
easy to picture.
(i) Roots for P2 are pictured in Figure 2(i).
(ii) Roots for H are pictured in Figure 2(ii).
s
(iii) The set of roots for P(1,1,s) and for H coincide.
s
(iv) The set of roots for B is empty.
s
Figure 2. Demazure roots for P2, H , and P(1,1,s).
s
A root m ∈ R defines a root subgroup H ⊂ Aut(X), see [2, 6,
m
11]. The root subgroups H are unipotent one-parameter subgroups
m
normalized by T.
Lemma 2.4. For every point Q ∈ X (cid:114) XHm the orbit H · Q meets
m
exactly two T-orbits O and O on X and dimO = 1+dimO .
1 2 σ1 σ2
Proof. Let U , i ∈ I, be affine charts corresponding to cones in ∆ which
i
contain η . Due to the proof of [11, Proposition 3.14], the charts U are
m i
preserved by H and the complement V = X (cid:114) (cid:83) U is fixed by H
m i m
i∈I
pointwise. Applying [2, Proposition 2.1] to affine charts U completes
i
the proof. (cid:3)
Definition 2.5. ApairofT-orbits(O ,O )fromLemma2.4iscalled
σ1 σ2
H -connected.
m
Lemma 2.6. ([2, Lemma 2.2]) A pair (O ,O ) is H -connected if
σ1 σ2 m
and only if
(2) m| (cid:54) 0 and σ = σ ∩m⊥ is a facet of cone σ .
σ2 1 2 2
(cid:3)
Every automorphism ψ ∈ Aut(N,∆) of the lattice N preserving the
fan ∆ defines an automophism of X denoted by P .
ψ
AUTOMORPHISMS OF A TORIC VARIETY 5
Theorem 2.7. ([6, Corollary 4.7], [8], [11, page 140]) Let X be a com-
plete toric variety. Then Aut(X) is a linear algebraic group generated
by T, H for m ∈ R, and P for ψ ∈ Aut(N,∆). Moreover, the group
m ψ
Aut0(X) is generated by T and H for m ∈ R. (cid:3)
m
3. Main results
Let X be a complete toric variety defined by the fan ∆ with ∆(1) =
{τ ,...,τ }, v = v , D = D . Let us consider the collection ∆(cid:101) of all
1 r i τi i τi
cones generated by some rays in ∆(1). For a cone σ ∈ ∆(cid:101) let us define
a monoid
(cid:88)
Γ(σ) = Z [D ].
(cid:62)0 i
τi∈∆(1)(cid:114)σ(1)
For a T-orbit O the monoids Γ(O )(see Introduction) and Γ(σ) coin-
σ σ
cide. Let us put
Υ(∆) = {Γ(σ) : σ ∈ ∆}.
Example 3.1. Let us find Υ(∆) for every fan of Example 2.1.
(i) For P2 there is one monoid Z , see Figure 3(i).
(cid:62)0
(ii) For H there are two monoids of rank 2, see Figure 3(ii).
s
(iii) ForP(1,1,s)there aremonoidsZ andsZ , seeFigure3(iii).
(cid:62)0 (cid:62)0
(iv) For B there are four monoids generated by 2 elements, four
s
monoids generated by 3 elements and a monoid generated by
all 4 elements [D ], [D ], [D ], [D ]. We picture monoids gen-
1 2 3 4
erated by 2 elements in 3-dimensional space remembering that
2s[D −D ] = 0, see Figure 3(iv).
3 1
Lemma 3.2. Let σ , σ ∈ ∆(cid:101). If there exists a root m ∈ R such that
1 2
(2) holds, then
σ ∈ ∆ ⇔ σ ∈ ∆.
1 2
Proof. Let us assume that σ ∈ ∆. There exists a T-orbit O on X.
1 σ1
By Lemmas 2.4 and 2.6, there exists an orbit O which meets H ·O .
2 m σ1
By conditions (2), the cone σ , and the root m define σ uniquely and
1 2
the cone corresponding to O is σ . Hence, σ ∈ ∆.
2 2 2
Conversely, by (2), σ is a face of σ and σ ∈ ∆. (cid:3)
1 2 1
Lemma 3.3. Let σ , σ ∈ ∆(cid:101) with σ (1) = σ (1) ∪ {τ } for some
1 2 2 1 k
1 ≤ k ≤ r. Then conditions (2) hold for some root m ∈ R if and only
if
Γ(σ ) = Γ(σ ).
1 2
Proof. We have
(cid:88) (cid:88)
Γ(σ ) = Z [D ] = Z [D ]+ Z [D ],
1 (cid:62)0 i (cid:62)0 k (cid:62)0 i
τi∈/σ1(1) τi∈/σ2(1)
(cid:88)
Γ(σ ) = Z [D ],
2 (cid:62)0 i
τi∈/σ2(1)
6 IVAN BAZHOV
Figure 3. Collections Υ(∆) for P2, H , P(1,1,s), and B .
s s
and the condition Γ(σ ) = Γ(σ ) can be written as
1 2
(cid:88)
(3) [D ] = a [D ]
k i i
τi∈/σ2(1)
for some a ∈ Z , τ ∈/ σ (1). Let us put a = −1 and a = 0 for
i (cid:62)0 i 2 k i
τ ∈ σ (1). Due to exact sequence (1), condition (3) is equivalent to
i 1
existence of m ∈ M such that (cid:104)m,v (cid:105) = a for all 1 (cid:54) i (cid:54) r. It means
i i
that m is a root and (2) is satisfied for σ , σ . (cid:3)
1 2
The following lemma shows how we can reconstruct the fan ∆
from Υ(∆) and ∆(1).
Lemma 3.4. Let σ be in ∆(cid:101). Then Γ(σ) ∈ Υ(∆) if and only if σ ∈ ∆.
Proof. The implication σ ∈ ∆ ⇒ Γ(σ) ∈ Υ(∆) is trivial.
Let us assume that Γ(σ) ∈ Υ(∆). According to the definition of
Υ(∆) there exists a cone σ(cid:48) ∈ ∆ such that Γ(σ(cid:48)) = Γ(σ). With-
out loss of generality we may assume σ(1) = {τ ,...,τ }, σ(cid:48)(1) =
1 p
{τ ,τ ...,τ }. If σ(1) ∩ σ(cid:48)(1) (cid:54)= ∅, then σ(1) ∩ σ(cid:48)(1) = {τ ,τ ,
s s+1 p+q s s+1
...,τ }, otherwise we may assume s = p+1. Consider a sequence of
p
cones in ∆(cid:101)
(4) σ = ς ,ς ,...,ς = σ(cid:48),
1 2 s+q
ς = cone (τ ,...,τ ), 1 (cid:54) i (cid:54) s−1,
i Q i p
ς = cone (τ ,...,τ ) if s (cid:54) p, ς = cone (0) otherwise,
s Q s p s Q
AUTOMORPHISMS OF A TORIC VARIETY 7
ς = cone (τ ,...,τ ), 1 (cid:54) i (cid:54) q.
s+i Q s p+i
We have
Γ(ς ) = Γ(σ)+Γ(σ(cid:48)) = Γ(σ) = Γ(σ(cid:48)).
s
Moreover,
Γ(σ) ⊂ Γ(ς ) ⊂ Γ(ς ), hence, Γ(ς ) = Γ(σ) for 1 (cid:54) i (cid:54) s,
i s i
Γ(σ(cid:48)) ⊂ Γ(ς ) ⊂ Γ(ς ), hence, Γ(ς ) = Γ(σ(cid:48)) for 1 (cid:54) i (cid:54) q.
s+i s s+i
By Lemma 3.3, for ς ,ς and some roots m ∈ M conditions (2)
i i+1 i
are satisfied for all 1 (cid:54) i (cid:54) s + q − 1. By Lemma 3.2, ς ∈ ∆ for
i
s+q −1 (cid:62) i (cid:62) 1. Hence, σ ∈ ∆. (cid:3)
Proof of Theorem 1. Since Aut0(X) is generated by T and H , m ∈ R,
m
orbits O and O lie in the same Aut0(X)-orbit if and only if there
σ σ(cid:48)
exists a sequence
(5) σ = ς ,ς ,...,ς = σ(cid:48),
1 2 l
of cones from ∆ such that (O , O ) is H -connected for some m ∈
ςi ςi+1 mi i
R, 1 (cid:54) i (cid:54) l−1.
If Γ(σ) = Γ(σ(cid:48)) we can take sequence (4) as the required one.
Conversely, let us assume there is sequence (5). By Lemma 3.3, we
have Γ(σ) = Γ(ς ) = ... = Γ(ς ) = Γ(σ(cid:48)) and Γ(σ) = Γ(σ(cid:48)). (cid:3)
1 l
Example 3.5. For every toric variety X an Aut0(X)-orbit on X con-
sists of T-orbits and can be pictured as a set of cones. We describe
these orbits for the varieties from previous examples.
(i) The action of Aut0(P2) on P2 is transitive, see Figure 4(i).
(ii) There are two orbits on H , see Figure 4(ii).
s
(iii) There are two orbits on P(1,1,s): the regular locus and the
singular point, see Figure 4(iii).
(iv) Any orbit of Aut0(B ) on B is a T-orbit.
s s
Theorem 3.6. Let O and O(cid:48) be T-orbits. Then
Aut0(X)·O ⊂ Aut0(X)·O(cid:48) ⇔ Γ(O) ⊂ Γ(O(cid:48)).
Proof. Let us put σ = cone (v : [D ] ∈/ Γ(O)). If O corresponds
max Q i i
to a cone σ ∈ ∆, then σ ⊆ σ. Hence, σ (1) = σ ∩ ∆(1)
max max max
and Γ(σ ) = Γ(O). By Lemma 3.4, there exists a T-orbit O
max max
corresponding to σ . It is easy to see that σ ≺ σ and O ⊆ O .
max max max
Since O is an arbitrary T-orbit with Γ(O) = Γ(O ), we have
max
O ⊂ Aut0(X)·O and O = Aut0(X)·O.
max max
Without loss of generality we may assume O = Aut0(X)·O and
O(cid:48) = Aut0(X)·O(cid:48). It means that
σ(1) = {v : [D ] ∈/ Γ(O)} and σ(cid:48)(1) = {v : [D ] ∈/ Γ(O(cid:48))}.
i i i i
8 IVAN BAZHOV
Figure 4. Orbits on P2, H , P(1,1,s), and B .
s s
We have
Γ(O) ⊆ Γ(O(cid:48)) ⇔ ∆(1)(cid:114)σ(1) ⊆ ∆(1)(cid:114)σ(cid:48)(1)
⇔ σ(cid:48) ≺ σ
⇔ O ⊆ O(cid:48)
⇔ Aut0(X)·O ⊆ Aut0(X)·O(cid:48).
(cid:3)
Theorem 3.7. Torus orbits O and O on X lie in the same Aut(X)-
σ σ(cid:48)
orbit if and only if there exists an automorphism φ : Cl(X) → Cl(X)
with the following properties
• φ(Γ(O )) = Γ(O ),
σ σ(cid:48)
• φ(Υ(∆)) = Υ(∆),
• there exists a permutation f of elements in {1,...,r} such that
φ([D ]) = [D ].
i f(i)
Proof. Let us assume O and O lie in the same Aut(X)-orbit. By
σ σ(cid:48)
Theorem2.7,thereexistsP ∈ Aut(X)definedbysomeψ ∈ Aut(N,∆)
ψ
thatmapsAut0(X)·O toAut0(X)·O . TheautomorphismP defines
σ σ(cid:48) ψ
an automorphism φ : Cl(X) → Cl(X) with the following properties:
• φ([D ]) = [P (D )] = [D ];
τ ψ τ ψ(τ)
• φ(Γ(σ )) = Γ(σ );
1 2
• φ(Υ(∆)) = Υ(∆).
AUTOMORPHISMS OF A TORIC VARIETY 9
Conversely, for every φ let us find P ∈ Aut(X) mapping
ψ
Aut0(X)-orbit corresponding to Γ(σ) to Aut0(X)-orbit corresponding
to Γ(σ(cid:48)) = φ(Γ(σ)). There is a commutative diagram:
(cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47)
0 M Div (X) Cl(X) 0
T
ψ∗ f˜ φ
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
(cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47)
0 M Div (X) Cl(X) 0
T
where f˜is defined by f. There exists an automorphism ψ∗ : M → M
making the diagram commutative. Let us define ψ : N → N as dual
to ψ∗. One can check that ψ preserves ∆(1). By Lemma 3.4, the cone
ψ(ς), ς ∈ ∆, belongs to ∆, since Γ(ψ(ς)) = φ(Γ(ς)) ∈ Υ(∆). The
automorphism ψ defines the desired P . (cid:3)
ψ
Example 3.8. It is easy to see that Aut0- and Aut-orbits on P2, H ,
s
and P(1,1,s) coincide.
LetusdescribetheAut(B )-orbitsonB . Fors > 1thegroupZ ⊕Z
s s 2 2
acts on Υ(∆) and defines orbits on B in the following way: the open
s
Aut(B )-orbit coincides with the open T-orbit, one dimensional torus
s
orbits belong to the same Aut(B )-orbit, and four T-stable points form
s
two Aut(B )-orbits, see Figure 4(iv). For s = 1 all T-fixed points on
s
B are in the same Aut(B )-orbit.
1 1
Theorem 3.9. Let X be a complete toric variety. The group Aut(X)
acts on X transitively if and only if
s
(cid:89)
X ∼= Pki
i=1
for some k ∈ N.
i
Proof. Let us assume that the action of Aut(X) is transitive. Then X
is smooth, in particular, all cones of ∆ are simplicial. It is easy to see
that open Aut0(X)- and open Aut(X)-orbits coincide. Hence, Υ(∆)
consists of one monoid.
Let σ ∈ ∆ be a cone of maximal dimension d. We may assume that
σ(1) = {τ ,...,τ }. There is a cone σ(cid:48) ∈ ∆ of dimension d which meets
1 d
σ along the facet without τ : σ(cid:48)(1) = {τ ,...,τ ,τ }.
1 2 d d+1
Since Υ(∆) contains only one monoid, the following monoids coin-
cide:
(cid:88)
Γ(σ(cid:48)) = Z [D ]+ Z [D ],
(cid:62)0 1 (cid:62)0 i
i>d+1
(cid:88)
Γ(σ) = Z [D ]+ Z [D ].
(cid:62)0 d+1 (cid:62)0 i
i>d+1
10 IVAN BAZHOV
We have
(cid:88)
[D ] = a [D ]+ a [D ],
1 d+1 d+1 i i
i>d+1
(6)
(cid:88)
[D ] = b [D ]+ b [D ],
d+1 1 1 i i
i>d+1
where a ,b ∈ Z and a ,b ∈ Z for d + 2 ≤ i ≤ r. Let us put
d+1 1 (cid:62)0 i i (cid:62)0
a = −1, b = −1 and a = 0, b = 0 for 2 ≤ i ≤ d.
1 d+1 i i
Due to equations (6) there exist m ,m(cid:48) ∈ M such that
1 1
(cid:104)m ,v (cid:105) = a for 1 ≤ i ≤ r,
1 i i
(cid:104)m(cid:48),v (cid:105) = b for 1 ≤ i ≤ r.
1 i i
Since (cid:104)m ,v (cid:105) = (cid:104)m(cid:48),v (cid:105) = 0, 2 ≤ i ≤ d, the characters m and m(cid:48) are
1 i 1 i 1 1
proportional. Hence, a = −b for all i? in particular, a = b = 0 for
i i i i
i > d+1. We have [D ] = [D ] and (cid:104)m ,v (cid:105) = −1, (cid:104)m ,v (cid:105) = 1,
1 d+1 1 1 1 d+1
(cid:104)m ,v (cid:105) = 0, i (cid:54)= 1,d+1. It means that m ∈ R is a semisimple root.
1 i 1
Since we can take any element of σ(1) instead of τ there exists a
1
set of semisimple roots m ,...,m such that (cid:104)m ,v (cid:105) = −δ for all
1 d i j ij
1 (cid:54) i,j (cid:54) d. Hence, m ,...,m are linearly independent.
1 d
In [10, Theorem 3.18] it is shown that if X is a complete toric variety
of dimension d and there are d linearly independent semisimple roots
m ,...,m , then
1 d
s
(cid:89)
X ∼= Pki
i=1
for some k ∈ N. Applying this theorem completes the proof in one
i
direction.
Conversely, the group
s
(cid:89)
PGL(k +1),
i
i=1
s
acts on (cid:81) Pki transitively. (cid:3)
i=1
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