arXiv:1704.02043v2 [math.AG] 9 Nov 2017

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS AND
GROUPS OF PSEUDO-AUTOMORPHISMS
SERGE CANTAT AND YVES DE CORNULIER
A BSTRACT. Pseudo-automorphisms are birational transformations acting as regular automorphisms in codimension 1. We import ideas from geometric group
theory to study groups of birational transformations, and prove that a group of birational transformations that satisfies a fixed point property on CAT(0) cubical complexes is birationally conjugate to a group acting by pseudo-automorphisms on
some non-empty Zariski-open subset. We apply this argument to classify groups
of birational transformations of surfaces with this fixed point property up to birational conjugacy.

1. I NTRODUCTION
1.1. Birational transformations and pseudo-automorphisms. Let X be a quasiprojective variety, over an algebraically closed field k. Denote by Bir(X) the group
of birational transformations of X and by Aut(X) the subgroup of (regular) automorphisms of X. For the affine space of dimension n, automorphisms are invertible
transformations f : Ank → Ank such that both f and f −1 are defined by polynomial
formulas in affine coordinates:
f (x1 , . . . , xn ) = (f1 , . . . , fn ), f −1 (x1 , . . . , xn ) = (g1 , . . . , gn )
with fi , gi ∈ k[x1 , . . . , xn ]. Similarly, birational transformations of Ank are given by
rational formulas, i.e. fi , gi ∈ k(x1 , . . . , xn ).
Birational transformations may contract hypersurfaces. Roughly speaking, pseudo-automorphisms are birational transformations that act as automorphisms in
codimension 1. Precisely, a birational transformation f : X 99K X is a pseudoautomorphism if there exist Zariski-open subsets U and V in X such that X r U
and X r V have codimension ≥ 2 and f induces an isomorphism from U to V.
The pseudo-automorphisms of X form a group, which we denote by Psaut(X).
For instance, all birational transformations of Calabi-Yau manifolds are pseudoautomorphisms; and there are examples of such manifolds for which Psaut(X) is
Date: November 9, 2017.
2010 Mathematics Subject Classification. Primary 14E07, Secondary 14J50, 20F65.
1

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infinite while Aut(X) is trivial (see [10]). Pseudo-automorphisms are studied in
Section 2.
Definition 1.1. Let Γ ⊂ Bir(X) be a group of birational transformations of an
irreducible projective variety X. We say that Γ is pseudo-regularizable if there
exists a triple (Y, U, ϕ) where
(1) Y is a projective variety and ϕ : Y 99K X is a birational map;
(2) U is a dense Zariski open subset of Y ;
(3) ϕ−1 ◦ Γ ◦ ϕ yields an action of Γ by pseudo-automorphisms on U.
More generally if α : Γ → Bir(X) is a homomorphism, we say that it is pseudoregularizable if α(Γ) is pseudo-regularizable.
One goal of this article is to use rigidity properties of commensurating actions,
a purely group-theoretic concept, to show that many group actions are pseudoregularizable. In particular, we exhibit a class of groups for which all actions by
birational transformations on projective varieties are pseudo-regularizable.
1.2. Property (FW). The class of groups we shall be mainly interested in is characterized by a fixed point property appearing in several related situations, for instance for actions on CAT(0) cubical complexes. Here, we adopt the viewpoint of
commensurated subsets. Let Γ be a group, and Γ × S → S an action of Γ on a set
S. Let A be a subset of S. One says that Γ commensurates A if the symmetric
difference
γ(A)4A = (γ(A) r A) ∪ (A r γ(A))
is finite for every element γ of Γ. One says that Γ transfixes A if there is a subset
B of S such that A4B is finite and B is Γ-invariant: γ(B) = B, for every γ in Γ.
A group Γ has Property (FW) if, given any action of Γ on any set S, all com√
mensurated subsets of S are automatically transfixed. For instance, SL 2 (Z[ 5]) and
SL 3 (Z) have Property (FW), but non-trivial free groups do not share this property.
Property (FW) is discussed in Section 3.
Let us mention that among various characterizations of Property (FW) (see [11]),
one is: every combinatorial action of Γ on a CAT(0) cube complex fixes some cube.
Another, for Γ finitely generated, is that all its infinite Schreier graphs are oneended.
1.3. Pseudo-regularizations. Let X be a projective variety. The group Bir(X)
does not really act on X, because there are indeterminacy points; it does not act
on the set of hypersurfaces either, because some of them may be contracted. As

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

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˜
we shall explain, one can introduce the set Hyp(X)
of all irreducible and reduced
0
hypersurfaces in all birational models X 99K X (up to a natural identification).
Then there is a natural action of the group Bir(X) on this set, given by strict transforms. The rigorous construction of this action follows from a general categorical
framework, which is developed in Section 4. Moreover, this action commensurates
the subset Hyp(X) of hypersurfaces of X. This construction leads to the following
result.
Theorem A. Let X be a projective variety over an algebraically closed field. Let Γ
be a subgroup of Bir(X). If Γ has Property (FW), then Γ is pseudo-regularizable.
There is also a relative version of Property (FW) for pairs of groups Λ ≤ Γ,
which leads to a similar pseudo-regularization theorem for the subgroup Λ: this is
discussed in Section 6.4, with applications to distorted birational transformations.
Remark 1.2. Theorem A provides a triple (Y, U, ϕ) such that ϕ conjugates Γ to
a group of pseudo-automorphisms on the open subset U ⊂ Y . There are two extreme cases for the pair (Y, U) depending on the size of the boundary Y r U. If
this boundary is empty, Γ acts by pseudo-automorphisms on a projective variety Y .
If the boundary is ample, its complement U is an affine variety, and then Γ actually acts by regular automorphisms on U (see Section 2.4). Thus, in the study of
groups of birational transformations, pseudo-automorphisms of projective varieties
and regular automorphisms of affine varieties deserve specific attention.
1.4. Classification in dimension 2. In dimension 2, pseudo-automorphisms do not
differ much from automorphisms; for instance, Psaut(X) coincides with Aut(X) if
X is a smooth projective surface. Thus, for groups with Property (FW), Theorem A can be used to reduce the study of birational transformations to the study
of automorphisms of quasi-projective surfaces. Combining results of Danilov and
Gizatullin on automorphisms of affine surfaces with a theorem of Farley on groups
of piecewise affine transformations of the circle, we will be able to prove the following theorem.
Theorem B. Let X be a smooth, projective, and irreducible surface, over an algebraically closed field. Let Γ be an infinite subgroup of Bir(X). If Γ has Property
(FW), there is a birational map ϕ : Y 99K X such that
(1) Y is the projective plane P2 , a Hirzebruch surface Fm with m ≥ 1, or the
product of a curve C by the projective line P1 . If the characteristic of the
field is positive, Y is the projective plane P2k .

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(2) ϕ−1 ◦ Γ ◦ ϕ is contained in Aut(Y ).
Remark 1.3. The group Aut(Y ) has finitely many connected components for all
surfaces Y listed in Assertion (1) of Theorem B. Thus, changing Γ into a finite
index subgroup Γ0 , one gets a subgroup of Aut(Y )0 . Here Aut(Y )0 denotes the
connected component of the identity of Aut(Y ); this is an algebraic group, acting
algebraically on Y .
Example 1.4. Groups with Kazhdan Property (T) satisfy Property (FW). Thus, Theorem B extends Theorem A of [8] and the present article offers a new proof of that
result.
√
Theorem B can also be applied to the group SL 2 (Z[ d]), where d ≥ 2 is a nonsquare positive integer. Thus, every action of this group on a projective surface by
birational transformations is conjugate to an action by regular automorphisms on
P2k , the product of a curve C by the projective line P1k , or a Hirzebruch surface.
Moreover, in this case, Margulis’ superrigidity theorem can be combined with Theorem B to get a more precise result, see §10.
Remark 1.5. In general, for a variety X one can ask whether Bir(X) transfixes
Hyp(X), or equivalently is pseudo-regularizable. For a surface X, this holds precisely when X is not birationally equivalent to the product of the projective line
with a curve. See §7.1 for more precise results.
1.5. Acknowledgement. This work benefited from interesting discussions with
Jérémy Blanc, Vincent Guirardel, Vaughan Jones, Christian Urech, and Junyi Xie.
2. P SEUDO - AUTOMORPHISMS
This preliminary section introduces useful notation for birational transformations
and pseudo-automorphisms, and presents a few basic results.
2.1. Birational transformations. Let X and Y be two irreducible and reduced
algebraic varieties over an algebraically closed field k. Let f : X 99K Y be a
birational map. Choose dense Zariski open subsets U ⊂ X and V ⊂ Y such
that f induces an isomorphism fU,V : U → V . Then the graph Gf of f is defined
as the Zariski closure of {(x, fU,V (x)) : x ∈ U } in X × Y ; it does not depend on
the choice of U and V . The graph Gf is an irreducible variety; both projections
u : Gf → X and v : Gf → Y are birational morphisms and f = v ◦ u−1 .
We shall denote by Ind(f ) the indeterminacy set of the birational map f .

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Theorem 2.1 (Theorem 2.17 in [23]). Let f : X 99K Y be a rational map, with X
a normal variety and Y a projective variety. Then the indeterminacy set of f has
codimension ≥ 2.
Example 2.2. The transformation of the affine plane (x, y) 7→ (x, y/x) is birational,
and its indeterminacy locus is the line {x = 0}: this set of co-dimension 1 is
mapped “to infinity”. If the affine plane is compactified by the projective plane,
the transformation becomes [x : y : z] 7→ [x2 : yz : xz], with two indeterminacy
points.
Assume that X is normal; in particular, it is smooth in codimension 1. The jacobian determinant Jac(f )(x) is defined in local coordinates, on the smooth locus
of X, as the determinant of the differential dfx ; Jac(f ) depends on the coordinates,
but its zero locus does not. The zeroes of Jac(f ) form a hypersurface of the smooth
part of X; the zero locus of Jac(f ) will be defined as the Zariski closure of this
hypersurface in X. The exceptional set of f is the subset of X along which f is
not a local isomorphism onto its image; by a corollary of Zariski’s main theorem,
it coincides with the union of Ind(f ), the zero locus of Jac(f ), and additional parts
which are contained in the singular locus of X and have therefore codimension ≥ 2.
Its complement is the largest open subset on which f is a local isomorphism (see
[33, 36], for instance).
The total transform of a subset Z ⊂ X is denoted by f∗ (Z). If Z is not contained in Ind(f ), we denote by f◦ (Z) its strict transform, defined as the Zariski
closure of f (Z r Ind(f )). We say that a hypersurface W ⊂ Z is contracted if it is
not contained in the indeterminacy set and the codimension of its strict transform is
larger than 1.
2.2. Pseudo-isomorphisms. A birational map f : X 99K Y is a pseudo-isomorphism if one can find Zariski open subsets U ⊂ X and V ⊂ Y such that
(i) f realizes a regular isomorphism from U to V and
(ii) X r U and Y r V have codimension ≥ 2.
Pseudo-isomorphisms from X to itself are called pseudo-automorphisms (see
§ 1.2). The set of pseudo-automorphisms of X is a subgroup Psaut(X) of Bir(X).
Example 2.3. Start with the standard birational involution σn : Pnk 99K Pnk which
is defined in homogeneous coordinates by σn [x0 : . . . : xn ] = [x−1
: . . . : x−1
].
0
Q n
Blow-up the (n + 1) vertices of the simplex ∆n = {[x0 : . . . : xn ];
xi =
0}; this provides a smooth rational variety Xn together with a birational morphism

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

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π : Xn → Pnk . Then, π −1 ◦ σn ◦ π is a pseudo-automorphism of Xn , and is an
automorphism if n ≤ 2.
Proposition 2.4. Let f : X 99K Y be a birational map between two (irreducible,
reduced) normal algebraic varieties. Assume that the codimension of the indeterminacy sets of f and f −1 is at least 2. Then, the following properties are equivalent:
(1) The birational maps f and f −1 do not contract any hypersurface.
(2) The jacobian determinants of f and f −1 do not vanish on the regular loci
of X r Ind(f ) and Y r Ind(f −1 ) respectively.
(3) For every smooth point q ∈ X r Ind(f ), f is a local isomorphism from a
neighborhood of q to a neighborhood of f (q), and the same holds for f −1 .
(4) The birational map f is a pseudo-isomorphism from X to Y .
Proof. Denote by g be the inverse of f . If the Jacobian determinant of f vanishes at
some (smooth) point of X r Ind(f ), then it vanishes along a hypersurface V ⊂ X.
If (1) is satisfied, the image of V is a hypersurface W in Y , and we can find a point
p ∈ V r Ind(f ) such that f (p) is not an indeterminacy point of g. Since the product
of the jacobian determinant of f at p and of g at f (p) must be equal to 1, we get
a contradiction. Thus (1) implies (2), and (2) is equivalent to (1). Now, assume
that (2) is satisfied. Then f does not contract any positive dimensional subset of
X reg r Ind(f ): f is a quasi-finite map from X reg r Ind(f ) to its image, and so is g.
Zariski’s main theorem implies that f realizes an isomorphism from X reg r Ind(f )
to Y r Ind(g) (see [33], Prop. 8.57). Thus, (2) implies (4) and (3). By assumption,
Ind(f ) and Ind(g) have codimension ≥ 2; thus, (3) implies (2). Since (4) implies
(1), this concludes the proof.

Example 2.5. Let X be a smooth projective variety with trivial canonical bundle
KX . Let Ω be a non-vanishing section of KX , and let f be a birational transformation of X. Then, f ∗ Ω extends from X r Ind(f ) to X and determines a new section
of KX ; this section does not vanish identically because f is dominant, hence it does
not vanish at all because KX is trivial. As a consequence, Jac(f ) does not vanish, f
is a pseudo-automorphism of X, and Bir(X) = Psaut(X). We refer to [10, 16] for
families of Calabi-Yau varieties with an infinite group of pseudo-automorphisms.
2.3. Projective varieties.
Proposition 2.6 (see [5]). Let f : X 99K Y be a pseudo-isomorphism between two
normal projective varieties. Then
(1) the total transform of Ind(f ) by f is equal to Ind(f −1 );

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(2) f has no isolated indeterminacy point;
(3) if dim(X) = 2, then f is a regular isomorphism.
Proof. Let p ∈ X be an indeterminacy point of the pseudo-isomorphism f : X 99K
Y . Then f −1 contracts a subset C ⊂ Y of positive dimension on p. Since f and
f −1 are local isomorphisms on the complement of their indeterminacy sets, C is
contained in Ind(f −1 ). The total transform of a point q ∈ C by f −1 is a connected
subset of X that contains p and has dimension ≥ 1. This set Dq is contained in
Ind(f ) because f is a local isomorphism on the complement of Ind(f ); since p ∈
Dq ⊂ Ind(f ), p is not an isolated indeterminacy point. This proves Assertions (1)
and (2). The third assertion follows from the second one because indeterminacy
sets of birational transformations of projective surfaces are finite sets.

Let W be a hypersurface of X, and let f : X 99K Y be a pseudo-isomorphism.
The divisorial part of the total transform f∗ (W ) coincides with the strict transform
f◦ (W ). Indeed, f∗ (W ) and f◦ (W ) coincide on the open subset of Y on which f −1
is a local isomorphism, and this open subset has codimension ≥ 2.
Recall that the Néron-Severi group NS(X) is the free abelian group of codimension 1 cycles modulo cycles which are numerically equivalent to 0. Its rank is finite
and is called the Picard number of X.
Theorem 2.7. The action of pseudo-isomorphisms on Néron-Severi groups is functorial: (g ◦ f )∗ = g∗ ◦ f∗ for all pairs of pseudo-isomorphisms f : X 99K Y and
g : Y 99K Z. If X is a normal projective variety, the group Psaut(X) acts linearly
on the Néron-Severi group NS(X); this provides a morphism
Psaut(X) → GL (NS(X)).
The kernel of this morphism is contained in Aut(X) and contains Aut(X)0 as a
finite index subgroup.
As a consequence, if X is projective the group Psaut(X) is an extension of a
discrete linear subgroup of GL (NS(X)) by an algebraic group.
Proof. The first statement follows from the equality f∗ = f◦ on divisors. The
second follows from the first. To study the kernel K of the linear representation
Psaut(X) → GL (NS(X)), fix an embedding ϕ : X → Pm
k and denote by H the
m
polarization given by hyperplane sections in Pk . For every f in K, f∗ (H) is an
ample divisor, because its class in NS(X) coincides with the class of H. Now, a
theorem of Matsusaka and Mumford implies that f is an automorphism of X (see

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[25] exercise 5.6, and [32]). To conclude, note that Aut(X)0 has finite index in the
kernel of the action of Aut(X) on NS(X) (see [31, 27]).

2.4. Affine varieties. The group Psaut(Ank ) coincides with the group Aut(Ank ) of
polynomial automorphisms of the affine space Ank : this is a special case of the
following proposition.
Proposition 2.8. Let Z be an affine variety. If Z is factorial, the group Psaut(Z)
coincides with the group Aut(Z).
Proof. Fix an embedding Z → Am
k . Rational functions on Z are restrictions of
m
rational functions on Ak . Thus, every birational transformation f : Z → Z is
given by rational formulas f (x1 , . . . , xm ) = (f1 , . . . , fm ) where each fi is a rational
function
pi
∈ k(x1 , . . . , xm );
fi =
qi
here, pi and qi are relatively prime polynomial functions. Since the local rings OZ,x
are unique factorization domains, we may assume that the hypersurfaces WZ (pi ) =
{x ∈ Z; pi (z) = 0} and WZ (qi ) = {x ∈ Z; qi (z) = 0} have no common
components. Then, the generic point of WZ (qi ) is mapped to infinity by f . Since f
is a pseudo-isomorphism, WZ (qi ) is in fact empty; but if qi does not vanish on Z, f
is a regular map.

3. G ROUPS WITH P ROPERTY (FW)
3.1. Commensurated subsets and cardinal definite length functions (see [11]).
Let G be a group, and G × S → S an action of G on a set S. Let A be a subset
of S. As in the Introduction, one says that G commensurates A if the symmetric
difference A4gA is finite for every element g ∈ G. One says that G transfixes A if
there is a subset B of S such that A4B is finite and B is G-invariant: gB = B for
every g in G. If A is transfixed, then it is commensurated. Actually, A is transfixed
if and only if the function g 7→ #(A4gA) is bounded on G.
A group G has Property (FW) if, given any action of G on a set S, all commensurated subsets of S are automatically transfixed. More generally, if H is a
subgroup of G, then (G, H) has relative Property (FW) if every commensurating action of G is transfixing in restriction to H. This means that, if G acts on a
set S and commensurates a subset A, then H transfixes automatically A. The case
H = G is Property (FW) for G.

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We refer to [11] for a detailed study of Property (FW). The next paragraphs
present the two main sources of examples for groups with Property (FW) or its
relative version, namely Property (T) and distorted subgroups.
Remark 3.1. Property (FW) should be thought of as a rigidity property. To illustrate
this idea, consider a group K with Property (PW); by definition, this means that K
admits a commensurating action on a set S, with a commensurating subset C such
that the function g 7→ #(C4gC) has finite fibers. If G is a group with Property
(FW), then, every homomorphism G → K has finite image.
3.2. Property (FW) and Property (T). One can rephrase Property (FW) as follows: G has Property (FW) if and only if every isometric action on an “integral
Hilbert space” `2 (X, Z) has bounded orbits, where X is any discrete set.
A group has Property (FH) if all its isometric actions on Hilbert spaces have
fixed points. More generally, a pair (G, H) of a group G and a subgroup H ⊂ G
has relative Property (FH) if every isometric G-action on a Hilbert space has an
H-fixed point. Thus, the relative Property (FH) implies the relative Property (FW).
By a theorem of Delorme and Guichardet, Property (FH) is equivalent to Kazhdan’s Property (T) for countable groups (see [13]). Thus, Property (T) implies Property (FW).
Kazhdan’s Property (T) is satisfied by lattices in semisimple Lie groups all of
whose simple factors have Property (T), for instance if all simple factors have real
rank ≥ 2. For example, SL 3 (Z) satisfies Property (T).
Property (FW) is actually conjectured to hold for all irreducible lattices in semisimple Lie groups of real rank ≥ 2, such as SL 2 (R)k for k ≥ 2. (here, irreducible
means that the projection of the lattice modulo every simple factor is dense.) This
is known in the case of a semisimple Lie group admitting at least one noncompact
simple factor with Kazhdan’s Property (T), for instance in SO (2, 3) × SO (1, 4),
which admits irreducible lattices (see [12]).
3.3. Distortion. Let G be a group. An element g of G is distorted in G if there
exists a finite subset Σ of G generating a subgroup hΣi containing g, such that
limn→∞ n1 |g n |Σ = 0; here, |g|Σ is the length of g with respect to the set Σ. If G is
finitely generated, this condition holds for some Σ if and only if it holds for every
finite generating subset of G. For example, every finite order element is distorted.
Example 3.2. Let K be a field. The distorted elements of SL n (K) are exactly the
virtually unipotent elements, that is, those elements whose eigenvalues are all roots
of unity; in positive characteristic, these are elements of finite order. By results of

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Lubotzky, Mozes, and Raghunathan (see [29, 28]), the same characterization
holds
√
in the group SL n (Z), as soon as n ≥ 3; it also holds in SL n (Z[ d]) when n ≥ 2
and d ≥ 2 is not a perfect square. In contrast, in SL 2 (Z), every element of infinite
order is undistorted.
Lemma 3.3 (see [11]). Let G be a group, and H a finitely generated abelian subgroup of G consisting of distorted elements. Then, the pair (G, H) has relative
Property (FW).
This lemma provides many examples. For instance, if G is any finitely generated
nilpotent group and G0 is its derived subgroup, then (G, G0 ) has relative Property
(FH); this result is due to Houghton, in a more general formulation encompassing
polycyclic groups (see [11]). Bounded generation by distorted unipotent elements
can also be used to obtain nontrivial examples of groups with √
Property (FW), including√the above examples SL n (Z) for n ≥ 3, and SL n (Z[ d]). The case of
SL 2 (Z[ d]) is particularly interesting because it does not have Property (T).
3.4. Subgroups of PGL 2 (k) with Property (FW). If a group G acts on a tree T
by graph automorphisms, then G acts on the set E of directed edges of T (T is
non-oriented, so each edge gives rise to a pair of opposite directed edges). Let Ev
be the set of directed edges pointing towards a vertex v. Then Ev 4Ew is the set
of directed edges lying in the segment between v and w; it is finite of cardinality
2d(v, w), where d is the graph distance. The group G commensurates the subset Ev
for every v, and #(Ev 4gEv ) = 2d(v, gv). As a consequence, if G has Property
(FW), then it has Property (FA) in the sense that every action of G on a tree has
bounded orbits. This argument can be combined with Proposition 5.B.1 of [11] to
obtain the following lemma.
Lemma 3.4 (See [11]). Let G be a group with Property (FW), then all finite index
subgroups of G have Property (FW), and hence have Property (FA). Conversely, if
a finite index subgroup of G has Property (FW), then so does G.
On the other hand, Property (FA) is not stable by taking finite index subgroups.
Lemma 3.5. Let k be an algebraically closed field and Λ be a subgroup of GL 2 (k).
(1) Λ has a finite orbit on the projective line if and only if it is virtually solvable,
if and only if its Zariski closure does not contain SL 2 .
(2) Assume that all finite index subgroups of Λ have Property (FA) (e.g., Λ has
Property FW). If the action of Λ on the projective line preserves a nonempty, finite set, then Λ is finite.

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The proof of the first assertion is standard and omitted. The second assertion
follows directly from the first one.
In what follows, we denote by Z ⊂ Q the ring of algebraic integers (in some
fixed algebraic closure Q of Q).
Theorem 3.6 (Bass [2]). Let k be an algebraically closed field.
(1) If k has positive characteristic, then GL 2 (k) has no infinite subgroup with
Property (FA).
(2) Suppose that k has characteristic zero and that Γ ⊂ GL 2 (k) is a countable
subgroup with Property (FA), and is not virtually abelian. Then Γ acts irreducibly on k2 , and is conjugate to a subgroup of GL 2 (Z). If moreover
Γ ⊂ GL 2 (K) for some subfield K ⊂ k containing Q, then we can choose
the conjugating matrix to belong to GL 2 (K).
On the proof. The original statement [2, Theorem 6.5] yields this statement, except
the last fact, and assumes that Γ is contained in GL 2 (M ) with M a finitely generated
field. The latter condition is actually automatic: indeed, being a countable group
with Property (FA), Γ is finitely generated [35, §6, Th. 15], and one can choose K
to be the field generated by entries of a finite generating subset.
For the last assertion, we have Γ ∪ BΓB −1 ⊂ GL 2 (K) for some B ∈ GL 2 (k)
such that BΓB −1 ⊂ GL 2 (Z); we claim that this implies that B ∈ k∗ GL 2 (K). First,
since Γ is absolutely irreducible, this implies that BM2 (K)B −1 ⊂ M2 (K). The
conclusion follows from Lemma 3.7 below, which can be of independent interest.

Lemma 3.7. Let K ⊂ L be fields. Then the normalizer {B ∈ GL 2 (L) : BM2 (K)B −1 ⊂
M2 (K)} is reduced to L∗ GL 2 (K) = {λA : λ ∈ L∗ , A ∈ GL 2 (K)}.
Proof. Write


b 1 b2
B=
.
b3 b4
Since BAB −1 ∈ M2 (K) for the three elementary matrices A ∈ {E11 , E12 , E21 },
we deduce by a plain computation that bi bj /bk b` ∈ K for all 1 ≤ i, j, k, ` ≤ 4 such
that bk b` 6= 0. In particular, for all indices i and j such that bi and bj are nonzero,
the quotient bi /bj = bi bj /b2j belongs to K. It follows that B ∈ L∗ GL 2 (K).

Corollary 3.8. Let k be an algebraically closed field. Let C be a projective curve
over k, and let k(C) be the field of rational functions on the curve C. Let Γ be an
infinite subgroup of PGL 2 (k(C)). If Γ has Property (FA), then

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(1) the field k has characteristic 0;
(2) there is an element of PGL 2 (k(C)) that conjugates Γ to a subgroup of
PGL 2 (Z) ⊂ PGL 2 (k(C)).

4. A CATEGORAL LIMIT CONSTRUCTION
The purpose of this section is to describe a general categorical construction,
which can be used to construct various actions of groups of birational transformations, such as Manin’s construction of the Picard-Manin space (see [30, 8]), as
well as the commensurating action which is the main construction of this paper. A
closely related construction is performed by V. Jones in [24] to construct representations of Thompson’s groups, although it does not directly apply here.
4.1. Categories of projective varieties. Here, in a category C, arrows between any
two objects X and Y are assumed to form a set HomC (X, Y ). Recall that a category
is small if its objects form a set, and is essentially small if it is equivalent to a small
category, or equivalently if there is a subset of the collection of objects meeting
every isomorphism class. A subcategory C of a category D is full if all arrows of D
between objects of C are also arrows of C.
Example 4.1. Our main example will be the following. Fix an algebraically closed
field k. Let V = Vk be the category whose objects are irreducible (reduced) projective k-varieties and whose arrows are birational morphisms. Let V ] be the category
with the same objects, but whose arrows are birational maps. Similarly, one can
consider the category VN of irreducible (reduced) normal projective varieties, with
arrows given by birational morphisms, and the category VN ] with the same objects
but whose arrows are all birational maps. By construction, VN is a full subcategory
of V, which is a subcategory of V ] .
4.2. Relative thinness and well-cofiltered categories. Given a category C and an
object X ∈ Ob(C), let us define the category CX whose objects are pairs (Y, f ) with
Y ∈ Ob(C) and f ∈ HomC (Y, X), and whose arrows (Y, f ) → (Z, g) are given by
arrows u ∈ HomC (Y, Z) such that g ◦ u = f . A category is thin if there is at most
one arrow between any two objects. Let us say that a category is relatively thin if
the category CX is thin for all X ∈ Ob(X).
Example 4.2. A category in which every arrow is invertible is relatively thin, and
so are all its subcategories. This applies to the categories of Example 4.1: the
category Vk] of birational maps between irreducible projective varieties, and to its
subcategory Vk , and similarly to VN ]k and its subcategory VN k .

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

13

Recall that a category is cofiltered if it satisfies the following two properties (a)
and (b):
(a) for any pair of objects X1 , X2 , there exists an object Y with arrows X1 ←
Y → X2 ;
(b) for any pair of objects X, Y and arrows u1 , u2 : X → Y , there exists an
object W and an arrow w : W → X such that u1 ◦ w = u2 ◦ w.
Note that (b) is automatically satisfied when the category is thin. We say that a
category C is well-cofiltered if it is relatively thin and for every object X ∈ Ob(C),
the category CX is cofiltered (note that we do not require C to be cofiltered).
Example 4.3. Coming again to the categories of Example 4.1, the category Vk is
essentially small and well-cofiltered. It is relatively thin, as mentioned in Example
4.2. To show that (Vk )X is cofiltered, consider two birational morphisms f1 : X1 →
X and f2 : X2 → X, and denote by h the composition f2−1 ◦ f1 . The graph Gh
is a projective subvariety of X1 × X2 . One can compose the projection of Gh
onto X1 with f1 (resp. onto X2 with f2 ) to get a birational morphism Gh → X;
this birational morphism is an object in (Vk )X that dominates f1 : X1 → X and
f2 : X2 → X, as in property (a).
The full subcategory VN k of Vk enjoys the same properties. When k has characteristic zero, the resolution of indeterminacies implies that its full subcategory of
non-singular varieties (and birational morphisms) is also well-cofiltered.
4.3. Filtering inductive limits.
4.3.1. We shall say that a category E admits filtering inductive limits if for every
small, thin and cofiltered category D and every contravariant functor F : D →
E, the colimit of F exists (and then it also exists when “small” is replaced with
“essentially small”). For example, the category of sets and the category of groups
admit filtering inductive limits (see [36], § 1.4, for colimits).
4.3.2. Let us consider an essentially small category C, a category E admitting
filtering inductive limits, and a contravariant functor F : C → E; we denote the
functor F by X 7→ FX on objects and u 7→ Fu on arrows. Assume that C is wellcofiltered. Then, for every object X ∈ Ob(C), we can restrict the functor F to CX
and take the colimit F̃X of this restriction F : CX → E. Roughly speaking, F̃X is
the inductive limit in E of all FY for Y ∈ CX . So, for every arrow u : Y → X in C,
there is an arrow in E, φu : FY → F̃X in E; and for every arrow v : Z → Y in CX ,

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

14

the following diagram commutes
FO Z
Fv

FY

φu◦v
φu

/

!

F̃X .

The colimits F̃X satisfy a universal property. To describe it, consider an object
E ∈ Ob(E), together with arrows ψY : FY → E for all Y ∈ CX , and assume that
for every arrow v : Z → Y in CX we have the relation ψZ ◦ Fv = ψY . Then, there
exists a unique arrow ψ : F̃X → E in E such that for every (Y, u) ∈ Ob(CX ) the
following diagram commutes:
F̃X
O

ψ

Fu

/

>E

ψY

FY .
This construction provides a bijection ΦX from the inductive limit lim
←−Y ∈CX HomE (FY , E)
to Hom(F̃X , E) whose reciprocal bijection maps an element ψ ∈ Hom(F̃X , E) to
the family of arrows (ψ ◦ Fu )(Y,u)∈CX .
4.3.3. We can now define the covariant functor α associated to F . At the level
of objects, α maps X ∈ Ob(C) to the limit F̃X . Let us now describe α at the
level of arrows. If we fix (Y, u) ∈ Ob(CX ), the family of arrows (φu◦v : FZ →
F̃X )(Z,v)∈Ob(CY ) corresponds under ΦY to an arrow αu : F̃Y → F̃X . For every
(Z, v) ∈ Ob(CY ), the following diagram commutes
F̃OY
φv

αu

/

F̃
= X

(4.1)

φu◦v

FZ ,
and this characterizes the arrow αu . If (W, w) ∈ CY , the uniqueness readily proves
that αu◦w = αu ◦ αw , that is, X 7→ F̃X , u 7→ αu is a covariant functor C → E,
denoted α, and called the relative colimit functor associated to F .
4.3.4. Note that the previous diagram can be essentially rewritten in the form of
the commutative square on the left of the next equation. The commutative diagram
on the right only refers to u; it is obtained by composing the left diagram with the

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

15

map Fu and by using the equalities φv ◦ Fv = φidY and φu ◦ Fu = φidX .
αu

F̃OY

/

φv

F̃X
O

Fv

/

F̃X
O

φidY

φu

FZ o

αu

F̃OY

φidX

FY o

FY

(4.2)

Fu

FX .

Lemma 4.4. Suppose that C is well-cofiltered. Then α maps arrows of C to invertible arrows (i.e. isomorphisms) of E.
Proof. Fix (Y, u) ∈ CX . The proof consists in constructing a map, and then show
that it is the inverse map of αu .
Consider (S, s) ∈ Ob(CX ). By assumption, in C we can find a commutative
diagram as the one on the left of the following equation; hence, in E we obtain the
diagram on the right, where g = φx ◦ Fw by definition.
Y o
u



Xo

x

D


s

Fx

FOY

w

φx

FD
O

Fw

Fu

S,

/

Fs

FX

/

/

F̃
= Y

g

FS .

A priori g depends on the choice of (D, x, w); let us show that it only depends on
(u, s) and, for that purpose, let us denote g temporarily by g = gD (x, w being
implicit). First consider the case of a commutative diagram as the one on the left
in the next equation; in E, this diagram induces the diagram depicted on the right,
where everything not involving gD or gD0 is commutative.
Y `o

x
x0

q

D0

u

D

Fx

FOY

Fx0

~
w

!
Fu



v

Fq

/4

F̃
<D Y

φx 0

F a



S,

φx

FO
= D
Fw

D0

w0

Xo

/

Fs

FX

Fw0

/

gD ,gD0

FS ,

Thus, by definition gD0 = φx0 ◦ Fw0 = φx ◦ Fq ◦ Fw0 = φx ◦ Fw = gD . Now consider,
more generally two objects D0 and D00 and four arrows forming a diagram in CX :
YO o
D00

/

D0


S;

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

16

we have to show that gD0 = gD00 . Since C is well-cofiltered, CX is thin and cofiltered,
and we can complete the previous diagram into the one on the left of the following
equation. Since this diagram is in CX which is a thin category, it is commutative;
which means that if we complete it with both composite arrows D → Y and both
composite arrows D → S, the resulting arrows coincide; the resulting diagram, on
the right of the equation, is a commutative one.
DO 0

DO 0
Y a

}

!

D

>

Y ao

S

}



D

/

!
>

S



D00 ;

D00 ;

Using the previous case, we deduce gD0 = gD = gD00 . Thus, we have seen that
gD does not depend on the choice of D; we now write it as gu,s . In particular,
when S ∈ (CX )Y , we can choose D = S (and w the identity); we thus deduce that
gu,s = φx (where s = u ◦ x).
x
w
Consider (T, t) ∈ CS and choose D ∈ Ob(CX ) with Y ← D → T in CX . Then
we have the diagram in E
FOY

Fx

/

FD
O

φx

Fw

FO T

gu,s◦t

Ft

FX

Fs

/

!
/

F̃
> Y

gu,s

FS ,

where the left rectangle is commutative as well as the upper right triangle; since
by definition gu,s = φx ◦ Ft◦w = φx ◦ Fw ◦ Ft = gu,s◦t ◦ Ft , the lower right
triangle is also commutative. So the family (gs : FS → F̃Y )(S,s)∈Ob(CX ) defines
an element gu : F̃X → F̃Y . Namely, for every (S, s) ∈ Ob(CX ) the following
diagram commutes
F̃X
O
φs

FS ,
and gu is characterized by this property.

gu

/

F̃
> Y

gu,s

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

17

We now combine this with the map αu , and make use of the same notation as the
one in Equations (4.1) and (4.2). When S = Z (so s = u ◦ v, gu,s = φv ), we obtain
the commutative diagram.
F̃Y a

αu

/

gu

F̃X
O

φs

φv

/

F̃
= Y

φv

FZ ,
Since this holds for all (Z, v) ∈ (CX )Y , the universal property of F̃Y implies that
gu ◦ αu is the identity of F̃Y .
On the other hand, turning back to the notation of the beginning of the proof,
both triangles in the following diagram are commutative
αu

F̃OY
φx

/

F̃O
= X

φu◦x

FD o

Fw

φs

FS ;

since gu,s = φx ◦ Fw , this implies that the right triangle of the following diagram is
commutative, the left-hand triangle from above also being commutative
F̃X a

gu

/

F̃OY

αu

/

F̃
= X

gu,s
φs

φs

FS .
Since this holds for all (S, s) ∈ CX , by the universal property of F̃X , we obtain that
αu ◦ gu is the identity of F̃X . This ends the proof that αu is invertible.

4.4. Good right-localization and extensions. Given a category D with a subcategory C with the same objects, we say that (C, D) is a good right-localization if (i)
every arrow u : X → Y in D admits an inverse u−1 : Y → X and (ii) every arrow
in D can be decomposed as g ◦ f −1 where f and g are arrows of C.
Lemma 4.5. Let (C, D) be a good right-localization. Then C is well-cofiltered.
Proof. Clearly any category in which all arrows are invertible is relatively thin. It
follows that D and its subcategory C are relatively thin. Now consider a pair of
objects (Y, u) and (Z, v) of CX . Then v −1 ◦ u is an arrow of D (because all arrows
s
t
are invertible in D), and it can be decomposed as Y ← W → Z, with s and t arrows
of C. By definition v ◦ t = u ◦ s determines an arrow W → X; endowing W with
the resulting composite arrow to X, the arrows s and t become arrows in CX .


COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

18

Lemma 4.6. Let (C, D) be a good right-localization, and let E be another category.
Consider a (covariant) functor β from C to the category E mapping every arrow
to an invertible arrow of E. Then β has a unique extension to a functor from the
category D to the category E.
Proof. The uniqueness is clear. For the existence, consider an arrow u in D. We
wish to map u to β(g) ◦ β(f )−1 , where u = g ◦ f −1 . We have to prove that this does
not depend on the choice of (f, g). Thus write u = g1 ◦ f1−1 = g2 ◦ f2−1 . Since CX is
well-cofiltered (Lemma 4.5), we can produce a diagram as follows in C, where the
left “square” and the whole square are commutative:

f1

X`

~

YO 1
h1

g1

Z
f2



h2

Y
> 2
g2

X2
Then the right square is also commutative: indeed g1 ◦ h1 = (g1 ◦ f1−1 ) ◦ (f1 ◦ h1 ) =
(g2 ◦ f2−1 ) ◦ (f2 ◦ h2 ) = g2 ◦ h2 . Then
β(g1 ) ◦ β(f1 )−1 = β(g1 ) ◦ β(h1 ) ◦ β(h1 )−1 ◦ β(f1 )−1 = β(g1 ◦ h1 ) ◦ β(f1 ◦ h1 )−1
= β(g2 ◦ h2 ) ◦ β(f2 ◦ h2 )−1 = β(g2 ) ◦ β(h2 ) ◦ β(h2 )−1 ◦ β(f2 )−1 = β(g2 ) ◦ β(f2 )−1 ;
hence we can define without ambiguity β(u) = β(g) ◦ β(f )−1 .
We have to prove β(v) ◦ β(u) = β(v ◦ u) for any arrows u, v of D. This already
holds for u, v arrows of C. Write u = g ◦ f −1 , v = j ◦ h−1 with f , g, h, and j
arrows of C. Write h−1 ◦ g = t ◦ s−1 with s, t arrows of C. Then g ◦ s = h ◦ t,
so β(g) ◦ β(s) = β(g ◦ s) = β(h ◦ t) = β(h) ◦ β(t), which can be rewritten
β(h)−1 ◦ β(g) = β(t) ◦ β(s)−1 . In turn, we get
β(v ◦ u) = β(j ◦ h−1 ◦ g ◦ f −1 ) = β(j ◦ t ◦ s−1 ◦ f −1 ) = β(j ◦ t) ◦ β(f ◦ s)−1
= β(j) ◦ β(t) ◦ β(s)−1 ◦ β(f )−1 = β(j) ◦ β(h)−1 ◦ β(g) ◦ β(f )−1 = β(v) ◦ β(u). 
Combining Lemmas 4.4, 4.5 and 4.6, we deduce:
Proposition 4.7. Consider a good right-localization (C, D) and a category E admitting filtering inductive limits. Let F be a contravariant functor from C to E. Then
the relative colimit functor α, defined by
X 7→ F̃X = −
lim
→Y →X F (Y ),

u 7→ αu ,

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

19

has a unique extension to a covariant functor from D to E.



5. I RREDUCIBLE HYPERSURFACES
Let X be a normal projective variety. In this chapter, we make use of the categorical construction of Section 4.3 to define an action of Bir(X) on the set of all
irreducible hypersurfaces in all “models” Y → X of X. We also sketch an application to the construction of Picard-Manin spaces in Section 5.5.
5.1. Localizations in categories of projective varieties. Consider the categories
V, V ] , VN , and VN ] from examples 4.1 and 4.3.
Proposition 5.1. Let k be an algebraically closed field.
(1) The categories V and VN are well-cofiltered (as defined in §4.2).
(2) The pairs (V, V ] ) (VN , VN ] ) are good right-localizations of categories (as
defined in §4.4).
Proof. Property (1) follows from Example 4.3. Let us now prove the second property. Clearly, every arrow in V ] is invertible. Let f : X 99K X 0 be a birational
g0

g

map. The graph Gf is an irreducible variety, both projections X ← Gf → X 0 are
g

g0

birational morphisms and f = g 0 ◦ g −1 . Since f = g 0 ◦ g −1 with X ← Gf → X 0 ,
we deduce that (V, V ] ) is a good right-localization.
For VN , we only need to compose with the normalization map Y → Gf to get
the result (see [36], §9.7).

To spare the reader from going into too much category theory, let us state explicitly Proposition 4.7 in this case:
Corollary 5.2. Let E be a category admitting filtering inductive limits. Consider a
contravariant functor F from VN to the category E. Then the covariant functor α,
]
X 7→ F̃X = −
lim
→Y →X F (Y ), u 7→ αu has a unique extension to a functor from VN
to E.

5.2. The functor of irreducible hypersurfaces. Let us construct the functor to
which we will apply Corollary 5.2. For X ∈ V, define Hyp(X) as the set of irreducible and reduced hypersurfaces of X.
Proposition 5.3. Let f : Y → X be a birational morphism between two irreducible
projective varieties (an arrow in V).
(1) The number of T ∈ Hyp(Y ) such that f (T ) is not a hypersurface is finite
(these T are precisely the hypersurfaces contracted by f ).

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

20

(2) For every S ∈ Hyp(X), the number of S 0 ∈ Hyp(Y ) such that f (S 0 ) = S is
positive and finite.
(3) If X is normal, this number is equal to 1, and the unique preimage S 0 of S
is the strict transform f ◦ (S) = (f −1 )◦ (S). In particular, S 7→ S 0 = f ◦ (S)
is injective and its image has finite complement: this complement Hyp(Y ) r
f ◦ (Hyp(X)) is the set of hypersurfaces T that are contracted by f .
Proof. Let U ⊂ Y be a Zariski-dense open subset on which f induces an isomorphism onto its image. Let F be the complement of U .
(1) If f (T ) is not a hypersurface, then T ⊂ F . So T is one of the irreducible
components of F , which leaves finitely many possibilities.
(2) Since Y is projective, f is surjective, and hence f (T ) = S, where T =
−1
f (S); T is a proper subvariety. Then at least one irreducible component T 0 of T
satisfies f (T 0 ) = S, and conversely, every S 0 ∈ Hyp(Y ) such that f (S 0 ) = S has to
be an irreducible component of T , hence there are finitely many of them.
(3) We now use Theorem 2.1: since X is normal and Y is projective, the indeterminacy set Ind(f −1 ) ⊂ X has codimension ≥ 2. Hence the strict transform
(f −1 )◦ (S) of S is well-defined and is equal to f −1 (X r Ind(f −1 ). The total transform of S by f −1 may contain additional components of codimension 1, but all of
them are contracted into Ind(f −1 ), which has codimension 2 (hence are not equal
to S). This proves that S 0 = (f −1 )◦ (S). Since f (S 0 ) = S, the map f ◦ := (f −1 )◦ is
injective. Moreover, by construction, its image is made of hypersurfaces which are
not contained in f −1 (Ind(f )). Since every element T ∈ Hyp(Y ) which is not contracted coincides with f ◦ (f (T )), the image of f ◦ is in fact equal to the complement
of the set of contracted hypersurfaces.

From Proposition 5.3, the map X 7→ Hyp(X) defines a contravariant functor
from VN to the category of sets, mapping an arrow f : Y → X to the (injective)
map
f ◦ : Hyp(X) → Hyp(Y ).
If F denoted the functor Hyp as in Section 4.3, then we would have FX = Hyp(X)
˜
and Ff = f ◦ . For X in the category VN , define Hyp(X)
as the filtering inductive
limit
˜
Hyp(X)
= lim
−→Y →X Hyp(Y ).
˜
By construction, X 7→ Hyp(X)
is a covariant functor from VN to the category
of sets. By Corollary 5.2, it has a unique extension to a functor from VN ] to the
˜
category of sets. The image of an object Y ∈ VN ] is denoted Hyp(Y
), and the

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

21

image of an arrow f : Y 99K Y 0 , that is of a birational map between two normal
projective varieties Y and Y 0 , is denoted by f• . By construction, f• is a bijection
0
˜
˜
from Hyp(Y
) to Hyp(Y
).
For an arrow u in VN , i.e. a birational morphism u : Y → X between two normal projective varieties, we rewrite the commutative square on the right of Equation (4.2) as
u•
˜
˜
Hyp(Y
) ∼ / Hyp(X)
O

O

iY

?

Hyp(Y ) o

u◦

?

iX

? _ Hyp(X).

The two injections iX and iY will simply be viewed as inclusions in what follows. So the bijection u−1
• extends the injection which is given by the strict trans◦
form u . Since the image of u◦ has finite complement, the symmetric difference
Hyp(Y )4u−1
• (Hyp(X)) is finite. This latter property passes to inverses and compositions; hence, for every birational map v : X 99K X 0 between normal irreducible
projective varieties, the symmetric difference Hyp(X)4v•−1 (Hyp(X 0 )) is finite. To
give a more precise statement, let us introduce the following notation: given a birational map v : X 99K X 0 between normal irreducible projective varieties, define
exc(v) by
exc(v) = # {S ∈ Hyp(X); v contracts S} .
This is the number of contracted hypersurfaces S ∈ Hyp(X) by v.
Proposition 5.4. Let v : X 99K X 0 be a birational transformation between normal
irreducible projective varieties. Let S be an element of Hyp(X).
(1) If S ∈ (v −1 )◦ Hyp(X 0 ), then v• (S) = v◦ (S) ∈ Hyp(X 0 ).
(2) If S ∈
/ (v −1 )◦ Hyp(X 0 ), then v◦ (S) has codimension ≥ 2 (i.e. v contracts
0
˜
S), and v• (S) is an element of Hyp(X
) r Hyp(X 0 ).
(3) The symmetric difference v• (Hyp(X))4Hyp(X 0 ) contains exc(v)+exc(v −1 )
elements.
Proof. Let U be the complement of Ind(v) in X 0 . Since, by Theorem 2.1, Ind(v)
has codimension ≥ 2, no S ∈ Hyp(X) is contained in Ind(v).
Let us prove (1). When v is a birational morphism the assertion follows from
Proposition 5.3. To deal with the general case, write v = g ◦ f −1 where f : Y → X
and g : Y → X 0 are birational morphisms from a normal variety Y . Since f is a
birational morphism, f • (S) = f ◦ (S) ⊂ Hyp(Y ); since S is not contracted by v,
g• (f ◦ (S)) = g◦ (f ◦ (S)) ∈ Hyp(X 0 ). Thus, v• (S) = g• (f • (S)) coincides with the
strict transform v◦ (S) ∈ Hyp(X 0 ).

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

22

Now let us prove (2), assuming thus that S ∈
/ (v −1 )◦ Hyp(X 0 ). Let S 00 ∈ Hyp(Y )
be the hypersurface (f −1 )• (S) = (f −1 )◦ (S). Then f (S 00 ) = S. If g◦ (S 00 ) is a
hypersurface S 0 , then (v −1 )◦ (S 0 ) = S, contradicting S ∈
/ (v −1 )◦ Hyp(X 0 ). Thus,
g contracts S 00 onto a subset S 0 ⊂ X 0 of codimension ≥ 2. Since S 0 = v◦ (S),
assertion (2) is proved.
Assertion (3) follows from the previous two assertions.

Example 5.5. Let g be a birational transformation of Pnk of degree d, meaning that
g ∗ (H) ' dH where H denotes a hyperplane of Pnk , or equivalently that g is defined
by n + 1 homogeneous polynomials of the same degree d without common factor of
positive degree. The exceptional set of g has degree (n+1)(d−1); as a consequence,
excPnk (g) ≤ (n+1)(d−1). More generally, if H is a polarization of X, then excX (g)
is bounded from above by a function that depends only on the degree degH (g) :=
(g ∗ H) · H dim(X)−1 .
˜
5.3. Action of Bir(X) on Hyp(X).
Let us now restrict the functor f 7→ f• to the
elements of Bir(X). The existence of this functor and Proposition 5.4 give the
following theorem.
Theorem 5.6. Let X be a normal projective variety. The group Bir(X) acts faith˜
fully by permutations on the set Hyp(X)
via the homomorphism

˜
Bir(X) → Perm(Hyp(X))
g
7→
g•
˜
This action commensurates the subset Hyp(X) of Hyp(X):
for every g ∈ Bir(X),
|g• (Hyp(X))4Hyp(X)| = exc(g) + exc(g −1 ).
The only thing that has not been proven yet is the fact that the homomorphism
˜
f ∈ Bir(X) 7→ Perm(Hyp(X))
is injective. But the kernel of this homomorphism
is made of birational transformations f such that f◦ (W ) = W for every irreducible
hypersurface W of X. Since X is projective, one can embed X in some projective
space Pm
k ; then, every point of X(k) is the intersection of finitely many irreducible
hyperplane sections of X: since all these sections are fixed by f , every point is fixed
by f , and f is the identity.
5.4. Products of varieties. Let X, Y be irreducible, normal projective varieties.
We consider the natural embedding of Bir(X) into Bir(X × Y ), given by the birational action f · (x, y) = (f (x), y), f ∈ Bir(X). There is a natural injection jY of
Hyp(X) into Hyp(X × Y ), given by S 7→ S × Y , which naturally extends to an

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

23

˜
˜
injection of Hyp(X)
into Hyp(X
× Y ); this inclusion is Bir(X)-equivariant. The
following proposition will be applied to Corollary 6.7.
Proposition 5.7. Let a group Γ act on X by birational transformations. Then Γ
˜
˜
transfixes Hyp(X) in Hyp(X)
if and only if it transfixes Hyp(X ×Y ) in Hyp(X
×Y ).
More precisely, the subset Hyp(X × Y ) r jY (Hyp(X)) is Bir(X)-invariant.
Proof. The reverse implication is immediate, since any restriction of a transfixing
action is transfixing. The direct implication follows from the latter statement, which
we now prove. Consider S ∈ Hyp(X × Y ) r jY (Hyp(X)). This means that S is an
irreducible hypersurface of X × Y whose projection to X is surjective. Now, for
γ ∈ Bir(X), γ induces an isomorphism between open dense subset U, V of X, and
hence between U × Y and V × Y ; in particular, γ does not contract S. This shows
that γ stabilizes Hyp(X × Y ) r jY (Hyp(X)).

5.5. Manin’s construction. Instead of looking at the functor X 7→ Hyp(X) from
the category of normal projective varieties to the category of sets, one can consider
the Néron-Severi functor X 7→ NS(X) from the category of smooth projective varieties to the category of abelian groups. In characteristic zero, or for surfaces in arbitrary characteristic, the resolution of singularities shows that smooth projective varieties, together with birational morphisms, form a good right localization of the category of smooth projective varieties with birational maps between them. Thus, one
˜
can construct a functor, the relative colimit of Néron-Severi groups, X 7→ NS(X)
˜
˜
that maps birational maps X 99K Y to group isomorphisms NS(X)
→ NS(Y
). In
dimension 2, this construction is known as the Picard-Manin space (see [8, 30]).
One may also replace NS(X) by other cohomology groups if they behave contravariantly with respect to birational morphisms (see [7] for instance).
6. P SEUDO - REGULARIZATION OF BIRATIONAL TRANSFORMATIONS
˜
In this section, we make use of the action of Bir(X) on Hyp(X)
to characterize
and study groups of birational transformations that are pseudo-regularizable, in the
sense of Definition 1.1. As before, k is an algebraically closed field.
6.1. An example. Consider the birational transformation f (x, y) = (x + 1, xy) of
P1k × P1k . The vertical curves Ci = {x = −i}, i ∈ Z, are exceptional curves for the
cyclic group Γ = hf i: each of these curves is contracted by an element of Γ onto a
point, namely f◦i+1 (Ci ) = (1, 0).
Let ϕ : Y 99K P1k × P1k be a birational map, and let U be a non-empty open subset
of Y . Consider the subgroup ΓY := ϕ−1 ◦ Γ ◦ ϕ of Bir(Y ). If i is large enough,

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

24

ϕ◦−1 (Ci ) is an irreducible curve Ci0 ⊂ Y , and these curves Ci0 are pairwise distinct,
so that most of them intersect U. For positive integers m, f i+m maps Ci onto (m, 0),
and (m, 0) is not an indeterminacy point of ϕ−1 if m is large. Thus, ϕ−1 ◦ f m ◦ ϕ
contracts Ci0 , and ϕ−1 ◦ f m ◦ ϕ is not a pseudo-automorphism of U. This argument
proves the following lemma.
Lemma 6.1. Let X be the surface P1k ×P1k . Let f : X 99K X be defined by f (x, y) =
(x + 1, xy), and let Γ be the subgroup generated by f ` , for some ` ≥ 1. Then the
cyclic group Γ is not pseudo-regularizable.
This shows that Theorem A requires an assumption on the group Γ. More generally, consider a subgroup Γ ⊂ Bir(X) such that Γ
(a) contracts a family of hypersurfaces Wi ⊂ X whose union is Zariski dense
(b) the union of the family of strict transforms f◦ (Wi ), for f ∈ Γ contracting
Wi , form a subset of X whose Zariski closure has codimension at most 1.
Then, Γ cannot be pseudo-regularized.
6.2. Characterization of pseudo-Isomorphisms. Recall that f• denotes the bi0
˜
˜
jection Hyp(X)
→ Hyp(X
) which is induced by a birational map f : X 99K X 0 .
Also, for any nonempty open subset U ⊂ X, we define Hyp(U ) = {H ∈ Hyp(X) :
H ∩ U 6= ∅}; it has finite complement in Hyp(X).
Proposition 6.2. Let f : X 99K X 0 be a birational map between normal projective
varieties. Let U ⊂ X and U 0 ⊂ X 0 be two dense open subsets. Then, f induces a
pseudo-isomorphism U → U 0 if and only if f• (Hyp(U )) = Hyp(U 0 ).
Proof. If f restricts to a pseudo-isomorphism U → U 0 , then f maps every hypersurface of U to a hypersurface of U 0 by strict transform. And (f −1 )◦ is an inverse
for f◦ : Hyp(U ) → Hyp(U 0 ). Thus, f• (Hyp(U )) = f◦ (Hyp(U ) = Hyp(U 0 ).
Let us now assume that f• (Hyp(U )) = Hyp(U 0 ). Since X and X 0 are normal,
Ind(f ) and Ind(f −1 ) have codimension ≥ 2 (Theorem 2.1).
Let fU,U 0 be the birational map from U to U 0 which is induced by f . The indeterminacy set of fU,U 0 is contained in the union of the set Ind(f ) ∩ U and the set
of points x ∈ U r Ind(f ) which are mapped by f in the complement of U 0 ; this
second part of Ind(fU,U 0 ) has codimension 2, because otherwise there would be an
irreducible hypersurface W in U which would be mapped in X 0 r U 0 , contradicting
the equality f• (Hyp(U )) = Hyp(U 0 ). Thus, the indeterminacy set of fU,U 0 has codimension ≥ 2. Changing f in its inverse f −1 , we see that the indeterminacy set of
fU−10 ,U : U 0 99K U 0 has codimension ≥ 2 too.

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

25

If fU,U 0 contracted an irreducible hypersurface W ⊂ U onto a subset of U 0 of
codimension ≥ 2, then f• (W ) would not be contained in Hyp(U 0 ) (it would corre0
˜
spond to an element of Hyp(X
)rHyp(X 0 ) by Proposition 5.4). Thus, fU,U 0 satisfies
the first property of Proposition 2.4 and, therefore, is a pseudo-isomorphism.



6.3. Characterization of pseudo-regularization. Let X be a (irreducible, reduced)
normal projective variety. Let Γ be a subgroup of Bir(X). Assume that the action
˜
˜
of Γ on Hyp(X)
fixes (globally) a subset A ⊂ Hyp(X)
such that
|A4Hyp(X)| < +∞.
In other words, A is obtained from Hyp(X) by removing finitely many hypersur˜
faces Wi ∈ Hyp(X) and adding finitely many hypersurfaces Wj0 ∈ Hyp(X)
r
0
Hyp(X). Each Wj comes from an irreducible hypersurface in some model πj : Xj →
X, and there is a model π : Y → X that covers all of them (i.e. π ◦ πj−1 is a morphism from Y to Xj for every j). Then, π ◦ (A) is a subset of Hyp(Y ). Changing X
into Y , A into π ◦ (A), and Γ into π −1 ◦ Γ ◦ π, we may assume that
(1) A = Hyp(X) r {E1 , . . . , E` } where the Ei are ` distinct irreducible hypersurfaces of X,
˜
(2) the action of Γ on Hyp(X)
fixes the set A.
In what follows, we denote by U the non-empty Zariski open subset X r ∪i Ei and
by ∂X the boundary X r U = E1 ∪ · · · ∪ E` ; ∂X is considered as the boundary of
the compactification X of U.
Lemma 6.3. The group Γ acts by pseudo-automorphisms on the open subset U. If
U is smooth and there is an ample divisor D whose support coincides with ∂X, then
Γ acts by automorphisms on U.
In this statement, we say that the support of a divisor D coincides with ∂X if
P
D = i ai Ei with ai > 0 for every 1 ≤ i ≤ `.
Proof. Since A = Hyp(U) is Γ-invariant, Proposition 6.2 shows that Γ acts by
pseudo-automorphisms on U.
Since D is an ample divisor, some positive multiple mD is very ample, and the
complete linear system |mD| provides an embedding of X in a projective space.
The divisor mD corresponds to a hyperplane section of X in this embedding, and
the open subset U is an affine variety because the support of D is equal to ∂X.
Proposition 2.8 concludes the proof of the lemma.

˜
By Theorem 5.6, every subgroup of Bir(X) acts on Hyp(X)
and commensu˜
rates Hyp(X). If Γ transfixes Hyp(X), there is an invariant subset A of Hyp(X)

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

26

for which A4Hyp(X) is finite. Thus, one gets the following characterization of
pseudo-regularizability (the converse being immediate).
Theorem 6.4. Let X be a normal projective variety over an algebraically closed
field k. Let Γ be a subgroup of Bir(X). Then Γ transfixes the subset Hyp(X) of
˜
Hyp(X)
if and only if Γ is pseudo-regularizable.
Of course, this theorem applies directly when Γ ⊂ Bir(X) has property (FW)
because Theorem 5.6 shows that Γ commensurates Hyp(X).
Remark 6.5. Assuming char(k) = 0, we may work in the category of smooth
varieties (see Example 4.3 and § 5.5). As explained in Remark 1.2 and Lemma 6.3,
there are two extreme cases, corresponding to an empty or an ample boundary B =
∪i Ei .
If U = Y , Γ acts by pseudo-automorphisms on the projective variety Y . As
explained in Theorem 2.7, Γ is an extension of a subgroup of GL (NS(Y )) by an
algebraic group (which is almost contained in Aut(Y )0 ).
If U is affine, Γ acts by automorphisms on U. The group Aut(U) may be huge
(for instance if U is the affine space), but there are techniques to study groups of
automorphisms that are not available for birational transformations. For instance
Γ is residually finite and virtually torsion free if Γ is a group of automorphisms
generated by finitely many elements (see [3]).
6.4. Distorted elements. Theorem 6.4 may be applied when Γ has Property (FW),
or for pairs (Λ, Γ) with relative Property (FW). Here is one application:
Corollary 6.6. Let X be an irreducible projective variety. Let Γ be a distorted
cyclic subgroup of Bir(X). Then Γ is pseudo-regularizable.
The contraposition is useful to show that some elements of Bir(X) are undistorted. Let us state it in a strong “stable” way.
Corollary 6.7. Let X be a normal irreducible projective variety and let f be an
element of Bir(X) such that the cyclic group hf i does not transfix Hyp(X) (i.e., f is
not pseudo-regularizable). Then the cyclic subgroup hf i is undistorted in Bir(X),
and more generally for every irreducible projective variety, the cyclic subgroup
hf × IdY i is undistorted in Bir(X × Y ).
The latter consequence indeed follows from Proposition 5.7. This can be applied
to various examples, such as those in Example 7.9.

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

27

7. I LLUSTRATING RESULTS
7.1. Surfaces whose birational group is transfixing. If X is a projective curve,
Bir(X) always transfixes Hyp(X), since it acts by automorphisms on a smooth
model of X. We now consider the same problem for surfaces, starting with the
following result, which holds in arbitrary dimension.
Proposition 7.1. Let X be a normal irreducible variety of positive dimension over
an algebraically closed field k. Then Bir(X × P1 ) does not transfix Hyp(X × P1 ).
Proof. We can suppose that X is affine and work in the model X × A1 . For
ϕ a nonzero regular function on X, define a regular self-map f of X × A1 by
f (x, t) = (x, ϕ(x)t). Denoting by Z(ϕ) the zero set of ϕ, we remark that f induces
an automorphism of the open subset (X r Z(ϕ)) × A1 . In particular, it induces a
permutation of Hyp((X r Z(ϕ)) × A1 ). Moreover, since f contracts the complement Z(ϕ) × A1 to the subset Z(ϕ) × {0}, which has codimension ≥ 2, its action
˜
on Hyp(X
× A1 ) maps the set of codimension 1 components of Z(ϕ) × A1 outside
M = Hyp(X × A1 ). Therefore M r f −1 (M ) is the set of irreducible components of
Z(ϕ) × A1 . Its cardinal is equal to the number of irreducible components of Z(ϕ).
When ϕ varies, this number is unbounded; hence, Bir(X × A1 ) does not transfix
Hyp(X × A1 ).

Varieties that are birational to the product of a variety and the projective line are
said to be ruled. Proposition 7.1 states that for any ruled irreducible projective
variety Y of dimension ≥ 2, Bir(Y ) does not transfix Hyp(Y ). The converse holds
for surfaces, by the following theorem.
Theorem 7.2. Let k be an algebraically closed field. Let X be an irreducible
normal projective surface over k. The following are equivalent:
(1)
(2)
(3)
(4)

Bir(X) does not transfix Hyp(X);
the Kodaira dimension of X is −∞;
X is ruled;
there is no irreducible projective surface Y that is birationally equivalent to
X, and such that Bir(Y ) = Aut(Y ).

Proof. The equivalence between (2) and (3) is classical (see [1]). The group Aut(Y )
˜
fixes Hyp(Y ) ⊂ Hyp(Y
), hence (1) implies (4). If the Kodaira dimension of X is
≥ 0, then X has a unique minimal model X0 , and Bir(X0 ) = Aut(X0 ). Thus, (4)
implies (2). Finally, Proposition 7.1 shows that (3) implies (1).


COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

28

Theorem 7.3. Let X be an irreducible projective surface over an algebraically
closed field k. The following are equivalent:
(1) some finitely generated subgroup of Bir(X) does not transfix Hyp(X);
(2) some cyclic subgroup of Bir(X) does not transfix Hyp(X);
(3) • k has characteristic 0, and X is birationally equivalent to the product
of the projective line with a curve of genus 0 or 1, or
• k has positive characteristic, and X is a rational surface.
Example 7.4. Let k be an algebraically closed field that is not algebraic over a finite
field. Let t be an element of infinite order in the multiplicative group k∗ . Then the
birational transformation g of P2k given, in affine coordinates, by (x, y) 7→ (tx +
1, xy) does not transfix Hyp(P2k ). Indeed, it is easy to show that the hypersurface
C = {x = 0} satisfies, for n ∈ Z, f n (C) ∈ Hyp(P2k ) if and only if n ≤ 0.
Example 7.5. The example of non-transfixing element in Example 7.4 works under
a small restriction on k. Here is an example over an arbitrary algebraically closed
field k. Let L and L0 be two lines in P2k intersecting transversally at a point q.
Let f be a birational transformation of P2k that contracts L0 onto q and fixes the
line L. For instance, in affine coordinates, the monomial map (x, y) 7→ (x, xy)
contracts the y-axis onto the origin, and fixes the x-axis. Assume that there is an
open neighborhood U of q such that f does not contract any curve in U except the
line L0 . Let C be an irreducible curve that intersects L and L0 transversally at q.
Then, for every n ≥ 1, the strict transform f◦n (C) is an irreducible curve, and the
order of tangency of this curve with L goes to infinity with n. Thus, the degree of
f◦n (C) goes to infinity too and the f◦n (C) form an infinite sequence in Hyp(P2k ).
Now, assume that C is contracted by f −1 onto a point p, p ∈
/ Ind(f ), and p is
−1
−m
fixed by f . Then, for every m ≥ 1, f• (C) is not in Hyp(P2k ). This shows
that the orbit of C under the action of f• intersects Hyp(P2k ) and its complement
2
2
n
−m
˜
Hyp(P
k ) r Hyp(Pk ) on the infinite sets {f◦ (C) ; n ≥ 1} and {f• (C) ; m ≥ 1}.
In particular, f does not transfix Hyp(P2k ).
Since such maps exist over every algebraically closed field k, this example shows
that property (2) of Theorem 7.3 is satisfied for every rational surface X.
Proof. Trivially (2) implies (1).
Suppose that (3) holds and let us prove (2). The case X = P1 × P1 is already
covered by Lemma 6.1 in characteristic zero, and by the previous example in positive characteristic. The case X = C × P1 in characteristic zero, where C is an
elliptic curve, is similar. To see it, fix a point t0 ∈ C and a rational function ϕ on C

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

29

that vanishes at t0 . Then, since k has characteristic zero, one can find a translation
s of C of infinite order such that the orbit {sn (t0 ) : n ∈ Z} does not contain any
other zero or pole of ϕ (here we use that the characteristic of k is 0). Consider the
birational transformation f ∈ Bir(X) given by f (t, x) = (s(t), ϕ(t)x). Let H be
the hypersurface {t0 } × C. Then for n ∈ Z, we have (f• )n H ∈ Hyp(X) if and only
if n ≤ 0. Hence the action of the cyclic group hf i does not transfix Hyp(X).
Let us now prove that (1) implies (3). Applying Theorem 7.2, and changing
X to a birationally equivalent surface if necessary, we assume that X = C × P1
for some (smooth irreducible) curve C. We may now assume that the genus of C
is ≥ 2, or ≥ 1 in positive characteristic, and we have to show that every finitely
generated group Γ of Bir(X) transfixes Hyp(X). Since the genus of C is ≥ 1, the
group Bir(X) preserves the fibration X → C; this gives a surjective homomorphism
Bir(X) → Aut(C). Now let us fully use the assumption on C: if its genus is ≥ 2,
then Aut(C) is finite; if its genus is 1 and k has positive characteristic, then Aut(C)
is locally finite1, and in particular the projection of Γ on Aut(C) has a finite image.
Thus the kernel of this homomorphism intersects Γ in a finite index subgroup Γ0 .
It now suffices to show that Γ0 transfixes Hyp(X). Every f ∈ Γ0 has the form
f (t, x) = (t, ϕt (x)) for some rational map t 7→ ϕt from C to PGL 2 ; define Uf ⊂ C
as the open and dense subset on which ϕγ is regular: by definition, f restricts to an
automorphism of Uf × P1 . Let S be a finite generating subset of Γ0 , and let US be
the intersection of the open subsets Ug , for g ∈ S. Then Γ0 acts by automorphisms
on US × P1 and its action on Hyp(X) fixes the subset Hyp(US ). Hence Γ transfixes
Hyp(X).

It would be interesting to obtain characterizations of the same properties in dimension 3 (see Question 11.2).
7.2. Transfixing Jonquières twists. Let X be an irreducible normal projective surface and π a morphism onto a smooth projective curve C with rational connected
fibers. Let Birπ (X) be the subgroup of Bir(X) permuting the fibers of π. Since C
is a smooth projective curve, the group Bir(C) coincides with Aut(C) and we get a
canonical homomorphism rC : Birπ (X) → Aut(C).
The main examples to keep in mind are provided by P1 ×P1 , Hirzebruch surfaces,
and C × P1 for some genus 1 curve C, π being the first projection.
Let Hypπ (X) denote the set of irreducible curves which are contained in fibers
˜ π (X) = Hypπ (X) t (Hyp(X)
˜
˜
of π, and define Hyp
r Hyp(X)), so that Hyp(X)
=
1

Every finitely generated subgroup is finite.

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

30

˜ π (X) t (Hyp(X) r Hypπ (X)). An irreducible curve H ⊂ X is an element of
Hyp
Hyp(X) r Hypπ (X) if and only if its projection π(H) coincides with C; this curves
are said to be transverse to π.
˜
˜ π (X) t (Hyp(X) r Hypπ (X))
Proposition 7.6. The decomposition Hyp(X)
= Hyp
is Birπ (X)-invariant.
Proof. Let H ⊂ X be an irreducible curve which is transverse to π. Since Birπ (X)
acts by automorphisms on C, H can not be contracted by any element of Birπ (X);
more precisely, for every g ∈ Birπ (X), g• (H) is an element of Hyp(X) which is
transverse to π. Thus the set of transverse curves is Birπ (X)-invariant.

This proposition and the proof of Theorem 7.3 lead to the following corollary.
Corollary 7.7. Let G be a subgroup of Birπ (X). If π maps the set of indeterminacy
points of the elements of G into a finite subset of C, then G transfixes Hyp(X).
In the case of cyclic subgroups, we establish a converse under the mild assumption of algebraic stability. Recall that a birational transformation f of a smooth
projective surface is algebraically stable if the forward orbit of Ind(f −1 ) does not
intersect Ind(f ). By [14], given any birational transformation f of a surface X,
there is a birational morphism u : Y → X, with Y a smooth projective surface,
such that fY := u−1 ◦ f ◦ u is algebraically stable. If π : X → C is a fibration,
as above, and f is in Birπ (X), then fY preserves the fibration π ◦ u. Thus, we
may always assume that X is smooth and f is algebraically stable after a birational
conjugacy.
Proposition 7.8. Let X be a smooth projective surface, and π : X → C a rational
fibration. If f ∈ Birπ (X) is algebraically stable, then f transfixes Hyp(X) if, and
only if the orbit of π(Ind(f )) under the action of rC (f ) is finite.

For X = P1 × P1 , the reader can check (e.g., conjugating a suitable automorphism) that the proposition fails without the algebraic stability assumption.
Proof. Denote by A ⊂ Aut(C) the subgroup generated by rC (f ). Consider a fiber
F ' P1 which is contracted to a point q by f . Then, there is a unique indeterminacy
point p of f on F . If the orbit of π(q) under the action of A is infinite, the orbit of
q under the action of f is infinite too. Set qn = f n−1 (q) for n ≥ 1 (so that q1 = q);
this sequence of points is well defined because f is algebraically stable: for every
n ≥ 1, f is a local isomorphism from a neighborhood of qn to a neighborhood
˜
of qn+1 . Then, the image of F in Hyp(X)
under the action of f n is an element

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

31

˜
of Hyp(X)
r Hyp(X): it is obtained by a finite number of blow-ups above qn .
Since the points qn form an infinite set, the images of F form an infinite subset of
˜
Hyp(X)
r Hyp(X). Together with the previous corollary, this argument proves the
proposition.

Example 7.9. Consider X = P1 × P1 , with π(x, y) = x (using affine coordinates).
Start with fa (x, y) = (ax, xy), for some non-zero parameter a ∈ k. The action of
rC (fa ) on C = P1 fixes the images 0 and ∞ of the indeterminacy points of fa . Thus,
˜
fa transfixes Hyp(X)
by Corollary 7.7. Now, consider ga (x, y) = (ax, (x + 1)y).
Then, the orbit of −1 under multiplication by a is finite if and only if a is a root
of unity; thus, if a is not a root of unity, ga does not transfix Hyp(X). Section 6.1
provides more examples of that kind.
8. B IRATIONAL TRANSFORMATIONS OF SURFACES I
From now on, we work in dimension 2: X, Y , and Z will be smooth projective
surfaces over the algebraically closed field k. (In dimension 2, the resolution of
singularities is available in all characteristics, so that we can always assume the
varieties to be smooth.)
8.1. Regularization. In this section, we refine Theorem 6.4, in order to apply results of Danilov and Gizatullin. Recall that a curve C in a smooth surface Y has
normal crossings if each of its singularities is a simple node with two transverse
tangents. In the complex case, this means that C is locally analytically equivalent
to {xy = 0} (two branches intersecting transversally) in an analytic neighborhood
of each of its singularities.
Theorem 8.1. Let X be a smooth projective surface, defined over an algebraically
closed field k. Let Γ be a subgroup of Bir(X) that transfixes the subset Hyp(X) of
˜
Hyp(X).
There exists a smooth projective surface Z, a birational map ϕ : Z 99K X
and a dense open subset U ⊂ Z such that, writing the boundary ∂Z := Z r U as a
finite union of irreducible components Ei ⊂ Z, 1 ≤ i ≤ `, the following properties
hold:
(1) The boundary ∂Z is a curve with normal crossings.
(2) The subgroup ΓZ := ϕ−1 ◦ Γ ◦ ϕ ⊂ Bir(Z) acts by automorphisms on the
open subset U.
(3) For all i ∈ {1, . . . , `} and g ∈ ΓZ , the strict transform of Ei under the
action of g on Z is contained in ∂Z: either g◦ (Ei ) is a point of ∂Z or
g◦ (Ei ) is an irreducible component Ej of ∂Z.

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

32

(4) For all i ∈ {1, . . . , `}, there exists an element g ∈ ΓZ that contracts Ei to a
point g◦ (Ei ) ∈ ∂Z. In particular, Ei is a rational curve.
(5) The pair (Z, U) is minimal for the previous properties, in the following
sense: if one contracts a smooth curve of self-intersection −1 in ∂Z, then
the boundary stops to be a normal crossing divisor.
Before starting the proof, note that the boundary ∂Z may a priori contain an
irreducible rational curve E with a node.
Proof. We apply Theorem 6.4 (which works in positive characteristic too, because
X is a surface), and get a birational morphism ϕ0 : Y0 → X and an open subset U0
of Y0 that satisfy properties (1) and (3), except that we only know that the action of
Γ0 := ϕ−1
0 ◦ Γ ◦ ϕ0 on U0 is by pseudo-automorphisms (not yet by automorphisms).
We shall progressively modify the triple (Y0 , U0 , ϕ0 ) to obtain a surface Z with
properties (1) to (5).
Step 1.– First, we blow-up the singularities of the curve ∂Y0 = Y0 r U0 to get a
boundary that is a normal crossing divisor. This replaces the surface Y0 by a new
one, still denoted Y0 . This modification adds new components to the boundary ∂Y0
but does not change the fact that Γ0 acts by pseudo-automorphisms on U0 . Let `0 be
the number of irreducible components of Y0 r U0 .
Step 2.– Consider a point q in U0 , and assume that there is a curve Ei of ∂Y0
that is contracted to q by an element g ∈ Γ0 ; fix such a g, and denote by D the
union of the curves Ej such that g◦ (Ej ) = q. By construction, g is a pseudoautomorphism of U0 . The curve D does not intersect the indeterminacy set of g,
since otherwise there would be a curve C containing q that is contracted by g −1 .
And D is a connected component of ∂Y0 , because otherwise g maps one of the Ej
to a curve that intersects U0 . Thus, there are small neighborhoods W of D and
W 0 of q such that W ∩ ∂Y0 = D and g realizes an isomorphism from W r D to
W 0 r {q}, contracting D onto the smooth point q ∈ Y0 . As a consequence, there is
a birational morphism π1 : Y0 → Y1 such that
(1) Y1 is smooth
(2) π1 contracts D onto a point q1 ∈ Y1
(3) π1 is an isomorphism from Y0 r D to Y1 r {q1 }.
In particular, π1 (U0 ) is an open subset of Y1 and U1 = π1 (U0 ) ∪ {q1 } is an open
neighborhood of q1 in Y1 .
Then, Γ1 := π1 ◦ Γ0 ◦ π1−1 acts birationally on Y1 , and by pseudo-automorphisms
on U1 . The boundary ∂Y1 = Y1 r U1 contains `1 irreducible components, with

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

33

`1 < `0 (the difference is the number of components of D), and is a normal crossing
divisor because D is a connected component of ∂Y0 .
Repeating this process, we construct a sequence of surfaces πk : Yk−1 → Yk and
open subsets πk (Uk−1 ) ⊂ Uk ⊂ Yk such that the number of irreducible components
of ∂Yk = Yk r Uk decreases. After a finite number of steps (at most `0 ), we may
assume that Γk ⊂ Bir(Yk ) does not contract any boundary curve onto a point of the
open subset Uk . On such a model, Γk acts by automorphisms on Uk .
We fix such a model, which we denote by the letters Y , U, ∂Y , ϕ. The new birational map ϕ : Y 99K X is the composition of ϕ0 with the inverse of the morphism
Y0 → Yk . On such a model, properties (1) and (2) are satisfied. Moreover, (3)
follows from (2). We now modify Y further to get property (4).
Step 3.– Assume that the curve Ei ⊂ Y r U is not contracted by Γ. Let F be
the orbit of Ei : F = ∪g∈Γ g◦ (Ei ); this curve is contained in the boundary ∂Y of the
open subset U. Changing U into
U 0 = U ∪ (F r ∂Y r F ),
the group Γ also acts by pseudo-automorphisms on U 0 . This operation decreases
the number ` of irreducible components of the boundary. Thus, combining steps 2
and 3 finitely many times, we reach a model that satisfies Properties (1) to (4). We
continue to denote it by Y .
Step 4.– If the boundary ∂Y contains a smooth (rational) curve Ei of self-intersection −1, it can be blown down to a smooth point q by a birational morphism
π : Y → Y 0 ; the open subset U is not affected, but the boundary ∂Y 0 has one
component less. If Ei was a connected component of ∂Y , then U 0 = π(U) ∪ {q}
is a neighborhood of q and one replaces U by U 0 , as in step 2. Now, two cases may
happen. If the boundary ∂Y 0 ceases to be a normal crossing divisor, we come back
to Y and do not apply this surgery. If ∂Y 0 has normal crossings, we replace Y by
this new model. In a finite number of steps, looking successively at all (−1)-curves
and iterating the process, we reach a new surface Z on which all five properties are
satisfied.

Remark 8.2. One may also remove property (5) and replace property (1) by
(1’) The Ei are rational curves, and none of them is a smooth rational curve with
self-intersection −1.
But doing so, we may lose the normal crossing property. To get property (1’), apply
the theorem and argue as in step 4.

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

34

8.2. Constraints on the boundary. We now work on the new surface Z given by
Theorem 8.1. Thus, Z is the surface, Γ the subgroup of Bir(Z), U the open subset
on which Γ acts by automorphisms, and ∂Z the boundary of U.
Proposition 8.3 (Gizatullin, [17] § 4). There are four possibilities for the geometry
of the boundary ∂Z = Z r U.
(1)
(2)
(3)
(4)

∂Z is empty.
∂Z is a cycle of rational curves.
∂Z is a chain of rational curves.
∂Z is not connected; it is the disjoint union of finitely many smooth rational
curves of self-intersection 0.

Moreover, in cases (2) and (3), the open subset U is the blow-up of an affine surface.
Thus, there are four possibilities for ∂Z, which we study successively. We shall
start with (1) and (4) in sections 8.3 and 8.4. Then case (3) is dealt with in Section 8.5. Case (2) is slightly more involved: it is treated in Section 9.
Before that, let us explain how Proposition 8.3 follows from Section 5 of [17].
First, we describe the precise meaning of the statement, and then we explain how
the original results of [17] apply to our situation.
The boundary and its dual graph .– Consider the dual graph GZ of the boundary ∂Z. The vertices of GZ are in one to one correspondence with the irreducible
components Ei of ∂Z. The edges correspond to singularities of ∂Z: each singular point q gives rise to an edge connecting the components Ei that determine the
two local branches of ∂Z at q. When the two branches correspond to the same
irreducible component, one gets a loop of the graph GZ .
We say that ∂Z is a chain of rational curves if the dual graph is of type A` : ` is
the number of components, and the graph is linear, with ` vertices. Chains are also
called zigzags by Gizatullin and Danilov.
We say that ∂Z is a cycle if the dual graph is isomorphic to a regular polygon
with ` vertices. There are two special cases: when ∂Z is reduced to one component,
this curve is a rational curve with one singular point and the dual graph is a loop
(one vertex, one edge); when ∂Z is made of two components, these components
intersect in two distinct points, and the dual graph is made of two vertices with two
edges between them. For ` = 3, 4, . . ., the graph is a triangle, a square, etc.
Gizatullin’s original statement.– To describe Gizatullin’s article, let us introduce some useful vocabulary. Let S be a projective surface, and C ⊂ S be a curve;
C is a union of irreducible components, which may have singularities. Assume that

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

35

S is smooth in a neighborhood of C. Let S0 be the complement of C in S, and
let ι : S0 → S be the natural embedding of S0 in S. Then, S is a completion of
S0 : this completion is marked by the embedding ι : S0 → S, and its boundary is
the curve C. Following [17] and [18, 19], we only consider completions of S0 by
curves (i.e. S rι(S0 ) is of pure dimension 1), and we always assume S to be smooth
in a neighborhood of the boundary. Such a completion is
(i) simple if the boundary C has normal crossings;
(ii) minimal if it is simple and minimal for this property: if Ci ⊂ C is an
exceptional divisor of the first kind then, contracting Ci , the image of C is
not a normal crossing divisor anymore. Equivalently, Ci intersects at least
three other components of C. Equivalently, if ι0 : S0 → S 0 is another simple
completion, and π : S → S 0 is a birational morphism such that π ◦ ι = ι0 ,
then π is an isomorphism.
If S is a completion of S0 , one can blow-up boundary points to obtain a simple
completion, and then blow-down some of the boundary components Ci to reach a
minimal completion.
Now, consider the group of automorphisms of the open surface S0 . This group
Aut(S0 ) acts by birational transformations on S. An irreducible component Ei of
the boundary C is contracted if there is an element g of Aut(S0 ) that contracts Ei :
g◦ (Ei ) is a point of C. Let E be the union of the contracted components. In [17],
Gizatullin proves that E satisfies one of the four properties stated in Proposition 8.3;
moreover, in cases (2) and (3), E contains an irreducible component Ei with Ei2 > 0
(see Corollary 4, Section 5 of [17]).
Thus, Proposition 8.3 follows from the properties of the pair (Z, U, Γ): the open
subset U plays the role of S0 , and Z is the completion S; the boundary ∂Z is the
curve C: it is a normal crossing divisor, and it is minimal by construction. Since
every component of ∂Z is contracted by at least one element of Γ ⊂ Aut(U), ∂Z
coincides with Gizatullin’s curve E. The only thing we have to prove is the last
sentence of Proposition 8.3, concerning the structure of the open subset U.
First, let us show that E = ∂Z supports an effective divisor D such that D2 > 0
and D·F ≥ 0 for every irreducible curve. To do so, fix an irreducible component E0
of ∂Z with positive self-intersection. Assume that ∂Z is a cycle, and list cyclically
the other irreducible components: E1 , E2 , ..., up to Em , with E1 and Em intersecting
E0 . First, one defines a1 = 1. Then, one chooses a2 > 0 such that a1 E1 + a2 E2
intersects positively E1 , then a3 > 0 such that a1 E1 + a2 E2 + a3 E3 intersects
P
positively E1 and E2 , ..., up to m
i=1 ai Ei that intersects all components Ei , 1 ≤

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

36

i ≤ m − 1 positively. Since E02 > 0 and E0 intersects Em , one can find a coefficient
a0 for which the divisor
m
X
D=
ai Ei
i=0
2

satisfies D > 0 and D · Ei > 0 for all Ei , 0 ≤ i ≤ m. This implies that D
intersects every irreducible curve F non-negatively. Thus, D is big and nef (see
[26], Section 2.2). A similar proof applies when ∂Z is a zigzag.
Let W be the subspace of NS(X) spanned by classes of curves F with D · F = 0.
Since D2 > 0, Hodge index theorem implies that the intersection form is negative definite on W . Thus, Mumford-Grauert contraction theorem provides a birational morphism τ : Z → Z 0 that contracts simultaneously all curves F with class
[F ] ∈ W and is an isomorphism on Z rF ; in particular, τ is an isomorphism from a
neighborhood V of ∂Z onto its image τ (V) ⊂ Z 0 . The modification τ may contract
curves that are contained in U, and may create singularities for the new open subset
U 0 = τ (U), but does not modify Z near the boundary ∂Z. Now, on Z 0 , the divisor D0 = τ∗ (D) intersects every effecitve curve positively and satisfies (D0 )2 > 0.
Nakai-Moishezon criterion shows that D0 is ample (see [26], Section 1.2.B); consequently, there is an embedding of Z 0 into a projective space and a hyperplane section
H of Z 0 for which Z 0 r H coincides with U 0 . This proves that U is a blow-up of the
affine (singular) surface U 0 .
8.3. Projective surfaces and automorphisms. In this section, we (almost always)
assume that Γ acts by regular automorphisms on a projective surface X. This
corresponds to case (1) in Proposition 8.3. Our goal is the special case of Theorem B which is stated below as Theorem 8.8. We shall assume that Γ has property
(FW) in some of the statements (this was not a hypothesis in Theorem thm:FWregularization-surfaces).
We may, and shall, assume that X is smooth. We refer to [1, 4, 20] for the
classification of surfaces and the main notions attached to them.
8.3.1. Action on the Néron-Severi group. The intersection form is a non-degenerate
quadratic form qX on the Néron-Severi group NS(X), and Hodge index theorem
asserts that its signature is (1, ρ(X) − 1), where ρ(X) denotes the Picard number,
i.e. the rank of the lattice NS(X) ' Zρ .
The action of Aut(X) on the Néron-Severi group NS(X) provides a linear representation preserving the intersection form qX . This gives a morphism
Aut(X) → O (NS(X); qX ).

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

37

Fix an ample class a in NS(X) and consider the hyperboloid
HX = {u ∈ NS(X) ⊗Z R; qX (u, u) = 1 and qX (u, a) > 0}.
This set is one of the two connected components of {u; qX (u, u) = 1}. With the riemannian metric induced by (−qX ), it is a copy of the hyperbolic space of dimension
ρ(X) − 1; the group Aut(X) acts by isometries on this space (see [9]).
Proposition 8.4. Let X be a smooth projective surface. Let Γ be a subgroup of
Aut(X). If Γ has Property (FW), its action on NS(X) fixes a very ample class, the
image of Γ in O (NS(X); qX ) is finite, and a finite index subgroup of Γ is contained
in Aut(X)0 .
Proof. The image Γ∗ of Γ is contained in the arithmetic group O (NS(X); qX ).
The Néron-Severi group NS(X) is a lattice Zρ and qX is defined over Z. Thus,
O (NS(X);
qX ) is a standard arithmetic group in the sense of [6], § 1.1. The main results
of [6] imply that the action of Γ∗ on the hyperbolic space HX has a fixed point. Let
u be such a fixed point. Since qX is negative definite on the orthogonal complement
u⊥ of u in NS(X), and Γ∗ is a discrete group acting by isometries on it, we deduce
P
that Γ∗ is finite. If a is a very ample class, the sum γ∈Γ∗ γ ∗ (a) is an invariant, very
ample class.
The kernel K ⊂ Aut(X) of the action on NS(X) contains Aut(X)0 as a finite
index subgroup. Thus, if Γ has Property (FW), it contains a finite index subgroup
that is contained in Aut(X)0 .

8.3.2. Non-rational surfaces. In this paragraph, we assume that the surface X is
not rational. The following proposition classifies subgroups of Bir(X) with Property (FW); in particular, such a group is finite if the Kodaira dimension of X is
non-negative (resp. if the characteristic of k is positive). Recall that we denote by
Z ⊂ Q the ring of algebraic integers.
Proposition 8.5. Let X be a smooth, projective, and non-rational surface, over the
algebraically closed field k. Let Γ be an infinite subgroup of Bir(X) with Property (FW). Then k has characteristic 0, and there is a birational map ϕ : X 99K
C × P1k that conjugates Γ to a subgroup of Aut(C × P1k ). Moreover, there is a finite
index subgroup Γ0 of Γ such that ϕ ◦ Γ0 ◦ ϕ−1 , is a subgroup of PGL 2 (Z), acting
on C × P1k by linear projective transformations on the second factor.
Proof. Assume, first, that the Kodaira dimension of X is non-negative. Let π : X →
X0 be the projection of X on its (unique) minimal model (see [20], Thm. V.5.8).

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

38

The group Bir(X0 ) coincides with Aut(X0 ); thus, after conjugacy by π, Γ becomes
a subgroup of Aut(X0 ), and Proposition 8.4 provides a finite index subgroup Γ0 ≤ Γ
that is contained in Aut(X0 )0 . Note that Γ0 inherits Property (FW) from Γ.
If the Kodaira dimension of X is equal to 2, the group Aut(X0 )0 is trivial; hence
Γ0 = {IdX0 } and Γ is finite. If the Kodaira dimension is equal to 1, Aut(X0 )0
is either trivial, or isomorphic to an elliptic curve, acting by translations on the
fibers of the Kodaira-Iitaka fibration of X0 (this occurs, for instance, when X0 is the
product of an elliptic curve with a curve of higher genus). If the Kodaira dimension
is 0, then Aut(X0 )0 is also an abelian group (either trivial, or isomorphic to an
abelian surface). Since abelian groups with Property (FW) are finite, the group Γ0
is finite, and so is Γ.
We may now assume that the Kodaira dimension kod(X) is negative. Since X
is not rational, then X is birationally equivalent to a product S = C × P1k , where
C is a curve of genus g(C) ≥ 1. Denote by k(C) the field of rational functions on
the curveC. We fix a local coordinate x on C and denote the elements of k(C) as
functions a(x) of x. The semi-direct product Aut(C) n PGL 2 (k(C)) acts on S by
birational transformations of the form


a(x)y + b(x)
1
,
(x, y) ∈ C × Pk 7→ f (x),
c(x)y + d(x)
and Bir(S) coincides with this group Aut(C) n PGL 2 (k(C)); indeed, the first projection π : S → C is equivariant under the action of Bir(S) because every rational
map P1k → C is constant.
Since g(C) ≥ 1, Aut(C) is virtually abelian. Property (FW) implies that there
is a finite index, normal subgroup Γ0 ≤ Γ that is contained in PGL 2 (k(C)). By
Corollary 3.8, every subgroup of PGL 2 (k(C)) with Property (FW) is conjugate to a
subgroup of PGL 2 (Z) or a finite group if the characteristic of the field k is positive.
We may assume now that the characteristic of k is 0 and that Γ0 ⊂ PGL 2 (Z) is
infinite. Consider an element g of Γ; it acts as a birational transformation on the
surface S = C × P1k , and it normalizes Γ0 :
g ◦ Γ0 = Γ0 ◦ g.
Since Γ0 acts by automorphisms on S, the finite set Ind(g) is Γ0 -invariant. But a
subgroup of PGL 2 (k) with Property (FW) preserving a non-empty, finite subset of
P1 (k) is a finite group. Thus, Ind(g) must be empty. This shows that Γ is contained
in Aut(S).


COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

39

8.3.3. Rational surfaces. We now assume that X is a smooth rational surface, that
Γ ≤ Bir(X) is an infinite subgroup with Property (FW), and that Γ contains a
finite index, normal subgroup Γ0 that is contained in Aut(X)0 . Recall that a smooth
surface Y is minimal if it does not contain any smooth rational curve of the first
kind, i.e. with self-intersection −1. Every exceptional curve of the first kind E ⊂
X is determined by its class in NS(X) and is therefore invariant under the action
of Aut(X)0 . Contracting such (−1)-curves one by one, we obtain the following
lemma.
Lemma 8.6. There is a birational morphism π : X → Y onto a minimal rational surface Y that is equivariant under the action of Γ0 ; Y does not contain any
exceptional curve of the first kind and Γ0 becomes a subgroup of Aut(Y )0 .
Let us recall the classification of minimal rational surfaces and describe their
groups of automorphisms. First, we have the projective plane P2k , with Aut(P2k ) =
PGL 3 (k) acting by linear projective transformations. Then comes the quadric P1k ×
P1k , with Aut(P1k × P1k )0 = PGL 2 (k) × PGL 2 (k) acting by linear projective transformations on each factor; the group of automorphisms of the quadric is the semidirect product of PGL 2 (k) × PGL 2 (k) with the group of order 2 generated by the
permutation of the two factors, η(x, y) = (y, x). Then, for each integer m ≥ 1, the
Hirzebruch surface Fm is the projectivization of the rank 2 bundle O ⊕ O(m) over
P1k ; it may be characterized as the unique ruled surface Z → P1k with a section C
of self-intersection −m. Its group of automorphisms is connected and preserves the
ruling. This provides a homomorphism Aut(Fm ) → PGL 2 (k) that describes the action on the base of the ruling, and it turns out that this homomorphism is surjective.
If we choose coordinates for which the section C intersects each fiber at infinity, the
kernel Jm of this homomorphism acts by transformations of type
(x, y) 7→ (x, αy + β(x))
where β(x) is a polynomial function of degree ≤ m. In particular, Jm is solvable.
In other words, Aut(Fm ) is isomorphic to the group
(GL 2 (k)/µm ) n Wm
where Wm is the linear representation of GL 2 (k) on homogeneous polynomials
of degree m in two variables, and µm is the kernel of this representation: it is
the subgroup of GL 2 (k) given by scalar multiplications by roots of unity of order
dividing m.

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

40

Lemma 8.7. Given the above conjugacy π : X → Y , the subgroup π ◦ Γ ◦ π −1 of
Bir(Y ) is contained in Aut(Y ).
Proof. Assume that the surface Y is the quadric P1k × P1k . Then, according to Theorem 3.6, Γ0 is conjugate to a subgroup of PGL 2 (Z) × PGL 2 (Z). If g is an element
of Γ, its indeterminacy locus is a finite subset Ind(g) of P1k × P1k that is invariant under the action of Γ0 , because g normalizes Γ0 . Since Γ0 is infinite and has Property
(FW), this set Ind(g) is empty (Lemma 3.5). Thus, Γ is contained in Aut(P1k × P1k ).
The same argument applies for Hirzebruch surfaces. Indeed, Γ0 is an infinite
subgroup of Aut(Fm ) with Property (FW). Thus, up to conjugacy, its projection
in PGL 2 (k) is contained in PGL 2 (Z). If it were finite, a finite index subgroup
of Γ0 would be contained in the solvable group Jm , and would therefore be finite
too by Property (FW); this would contradict |Γ0 | = ∞. Thus, the projection of
Γ0 in PGL (Z) is infinite. If g is an element of Γ, Ind(g) is a finite, Γ0 -invariant
subset, and by looking at the projection of this set in P1k one deduces that it is empty
(Lemma 3.5). This proves that Γ is contained in Aut(Fm ).
Let us now assume that Y is the projective plane. Fix an element g of Γ, and
assume that g is not an automorphism of Y = P2 ; the indeterminacy and exceptional
sets of g are Γ0 invariant. Consider an irreducible curve C in the exceptional set
of g, together with an indeterminacy point q of g on C. Changing Γ0 in a finite
index subgroup, we may assume that Γ0 fixes C and q; in particular, Γ0 fixes q, and
permutes the tangent lines of C through q. But the algebraic subgroup of PGL 3 (k)
preserving a point q and a line through q does not contain any infinite group with
Property (FW) (Lemma 3.5). Thus, again, Γ is contained in Aut(P2k ).

8.3.4. Conclusion, in Case (1). Putting everything together, we obtain the following particular case of Theorem B.
Theorem 8.8. Let X be a smooth projective surface over an algebraically closed
field k. Let Γ be an infinite subgroup of Bir(X) with Property (FW). If a finite index
subgroup of Γ is contained in Aut(X), there is a birational morphism ϕ : X → Y
that conjugates Γ to a subgroup ΓY of Aut(Y ), with Y in the following list:
(1) Y is the product of a curve C by P1k , the field k has characteristic 0, and a
finite index subgroup Γ0Y of ΓY is contained in PGL 2 (Z), acting by linear
projective transformations on the second factor;
(2) Y is P1k × P1k , the field k has characteristic 0, and ΓY is contained in
PGL 2 (Z) × PGL 2 (Z);
(3) Y is a Hirzebruch surface Fm and k has characteristic 0;

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

41

(4) Y is the projective plane P2k .
In particular, Y = P2k if the characteristic of k is positive.
Remark 8.9. Denote by ϕ : X → Y the birational morphism given by the theorem.
Changing Γ in a finite index subgroup, we may assume that it acts by automorphisms on both X and Y .
If Y = C × P1 , then ϕ is in fact an isomorphism. To prove this fact, denote by
ψ the inverse of ϕ. The indeterminacy set Ind(ψ) is ΓY invariant because both Γ
and ΓY act by automorphisms. From Lemma 3.5, applied to Γ0Y ⊂ PGL 2 (k), we
deduce that Ind(ψ) is empty and ψ is an isomorphism. The same argument implies
that the conjugacy is an isomorphism if Y = P1k × P1k or a Hirzebruch surface Fm ,
m ≥ 1.
Now, if Y is P2k , ϕ is not always an isomorphism. For instance, SL 2 (C) acts on
P2k with a fixed point, and one may blow up this point to get a new surface with an
action of groups with Property (FW). But this is the only possible example, i.e. X
is either P2k , or a single blow-up of P2k (because Γ ⊂ PGL 3 (C) can not preserve
more than one base point for ϕ−1 without loosing Property (FW)).
8.4. Invariant fibrations. We now assume that Γ has Property (FW) and acts by
automorphisms on U ⊂ X, and that the boundary ∂X = X rU is the union of ` ≥ 2
pairwise disjoint rational curves Ei ; each of them has self-intersection Ei2 = 0 and
is contracted by at least one element of Γ. This corresponds to the fourth possibility
in Gizatullin’s Proposition 8.3. Since Ei · Ej = 0, the Hodge index theorem implies
that the classes ei = [Ei ] span a unique line in NS(X), and that [Ei ] intersects
non-negatively every curve.
From Section 8.3.2, we may, and do assume that X is a rational surface. In
particular, the Euler characteristic of the structural sheaf is equal to 1: χ(OX ) = 1,
and Riemann-Roch formula gives
E12 − KX · E1
+ 1.
2
The genus formula implies KX ·E1 = −2, and Serre duality shows that h2 (X, E1 ) =
h0 (X, KX −E1 ) = 0 because otherwise −2 = (KX −E1 )·E1 would be non-negative
(because E1 intersects non-negatively every curve). From this, we obtain
h0 (X, E1 ) − h1 (X, E1 ) + h2 (X, E1 ) =

h0 (X, E1 ) = h1 (X, E1 ) + 2 ≥ 2.
Since E12 = 0, we conclude that the space H 0 (X, E1 ) has dimension 2 and determines a fibration π : X → P1k ; the curve E1 , as well as the Ei for i ≥ 2, are fibers
of π.

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

42

If f is an automorphism of U and F ⊂ U is a fiber of π, then f (F ) is a (complete)
rational curve. Its projection π(f (F )) is contained in the affine curve P1k r ∪i π(Ei )
and must therefore be reduced to a point. Thus, f (F ) is a fiber of π and f preserves
the fibration. This proves the following lemma.
Lemma 8.10. There is a fibration π : X → P1k such that
(1) every component Ei of ∂X is a fiber of π, and U = π −1 (V) for an open
subset V ⊂ P1k ;
(2) the generic component of π is a smooth rational curve;
(3) Γ permutes the fibers of π: there is a morphism ρ : Γ → PGL 2 (k) such that
π ◦ f = ρ(f ) ◦ π for every f ∈ Γ.
The open subset V ( P1k is invariant under the action of ρ(Γ); hence ρ(Γ) is
finite by Property (FW) and Lemma 3.5. Let Γ0 be the kernel of this morphism. Let
ϕ : X 99K P1k × P1k be a birational map that conjugates the fibration π to the first
projection τ : P1k × P1k → P1k . Then, Γ0 is conjugate to a subgroup of PGL 2 (k(x))
acting on P1k × P1k by linear projective transformations of the fibers of τ . From
Corollary 3.8, a new conjugacy by an element of PGL 2 (k(x)) changes Γ0 in an
infinite subgroup of PGL 2 (Z). Then, as in Sections 8.3.2 and 8.3.3 we conclude
that Γ becomes a subgroup of PGL 2 (Z) × PGL 2 (Z), with a finite projection on the
first factor.
Proposition 8.11. Let Γ be an infinite group with Property (FW), with Γ ⊂ Aut(U),
and U ⊂ Z as in case (4) of Proposition 8.3. There exists a birational map
ψ : Z 99K P1k × P1k that conjugates Γ to a subgroup of PGL 2 (Z) × PGL 2 (Z), with
a finite projection on the first factor.
8.5. Completions by zigzags. Two cases remain to be studied: ∂Z can be a chain
of rational curves (a zigzag in Gizatullin’s terminology) or a cycle of rational curves
(a loop in Gizatullin’s terminology). Cycles are considered in Section 9. In this
section, we rely on difficult results of Danilov and Gizatullin to treat the case of
chains of rational curves (i.e. case (3) in Proposition 8.3). Thus, in this section
(i) ∂X is a chain of smooth rational curves Ei
(ii) U = X r ∂X is an affine surface (singularities are allowed)
(iii) every irreducible component Ei is contracted to a point of ∂X by at least
one element of Γ ⊂ Aut(U) ⊂ Bir(X).
In [18, 19], Danilov and Gizatullin introduce a set of “standard completions” of
the affine surface U. As in Section 8.2, a completion (or more precisely a “marked

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

43

completion”) is an embedding ι : U → Y into a complete surface such that ∂Y =
Y r ι(U) is a curve (this boundary curve may be reducible). Danilov and Gizatullin
only consider completions for which ∂Y is a chain of smooth rational curves and
Y is smooth in a neighborhood of ∂Y ; the surface X provides such a completion.
Two completions ι : U → Y and ι0 : U → Y 0 are isomorphic if the birational map
ι0 ◦ι−1 : Y → Y 0 is an isomorphism; in particular, the boundary curves are identified
by this isomorphism. The group Aut(U) acts by pre-composition on the set of
isomorphism classes of (marked) completions.
Among all possible completions, Danilov and Gizatullin distinguish a class of
“standard (marked) completions”, for which we refer to [18] for a definition. There
are elementary links (corresponding to certain birational mappings Y 99K Y 0 ) between standard completions, and one can construct a graph ∆U whose vertices are
standard completions; there is an edge between two completions if one can pass
from one to the other by an elementary link.
Example 8.12. A completion is m-standard, for some m ∈ Z, if the boundary curve
∂Y is a chain of n + 1 consecutive rational curves E0 , E1 , . . ., En (n ≥ 1) such that
E02 = 0, E12 = −m, and Ei2 = −2 if i ≥ 2.
Blowing-up the intersection point q = E0 ∩ E1 , one creates a new chain starting
by E00 with (E00 )2 = −1; blowing down E00 , one creates a new (m + 1)-standard
completion. This is one of the elementary links.
Standard completions are defined by constraints on the self-intersections of the
components Ei . Thus, the action of Aut(U) on completions permutes the standard completions; this action determines a morphism from Aut(U) to the group of
isometries (or automorphisms) of the graph ∆U (see [18]):
Aut(U) → Iso (∆U ).
Theorem 8.13 (Danilov and Gizatullin, [18, 19]). The graph ∆U of all isomorphism
classes of standard completions of U is a tree. The group Aut(U) acts by isometries
of this tree. The stabilizer of a vertex ι : U → Y is the subgroup G(ι) of automorphisms of the complete surface Y that fix the curve ∂Y . This group is an algebraic
subgroup of Aut(Y ).
The last property means that G(ι) is an algebraic group that acts algebraically on
Y . It coincides with the subgroup of Aut(Y ) fixing the boundary ∂Y ; the fact that
it is algebraic follows from the existence of a G(ι)-invariant, big and nef divisor
which is supported on ∂Y (see the last sentence of Proposition 8.3).

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

44

The crucial assertion in this theorem is that ∆U is a simplicial tree (typically,
infinitely many edges emanate from each vertex). There are sufficiently many links
to assure connectedness, but not too many in order to prevent the existence of cycles
in the graph ∆U .
Corollary 8.14. If Γ is a subgroup of Aut(U) that has the fixed point property on
trees, then Γ is contained in G(ι) ⊂ Aut(Y ) for some completion ι : U → Y .
If Γ has Property (FW), it has Property (FA) (see Section 3.4). Thus, if it acts
by automorphisms on U, Γ is conjugate to the subgroup G(ι) of Aut(Y ), for some
zigzag-completion ι : U → Y . Theorem 8.8 of Section 8.3.3 implies that the action
of Γ on the initial surface X is conjugate to a regular action on P2k , P1k ×P1k or Fm , for
some Hirzebruch surface Fm . This action preserves a curve, namely the image of the
zigzag into the surface Y . The following examples list all possibilities, and conclude
the proof of Theorem B in the case of zigzags (i.e. case (3) in Proposition 8.3).
Example 8.15. Consider the projective plane P2k , together with an infinite subgroup
Γ ⊂ Aut(P2k ) that preserves a curve C and has Property (FW). Then, C must be a
smooth rational curve: either a line, or a smooth conic. If C is the line “at infinity”,
then Γ acts by affine transformations on the affine plane P2k r C. If the curve is the
conic x2 + y 2 + z 2 = 0, Γ becomes a subgroup of PO3 (k).
Example 8.16. When Γ is a subgroup of Aut(P1k × P1k ) that preserves a curve C
and has Property (FW), then C must be a smooth curve because Γ has no finite
orbit (Lemma 3.5). Similarly, the two projections C → P1k being equivariant with
respect to the morphisms Γ → PGL 2 (k), they have no ramification points. Thus,
C is a smooth rational curve, and its projections onto each factor are isomorphisms.
In particular, the action of Γ on C and on each factor are conjugate. From these
conjugacies, one deduces that the action of Γ on P1k × P1k is conjugate to a diagonal
embedding
γ ∈ Γ 7→ (ρ(γ), ρ(γ)) ∈ PGL 2 (k) × PGL 2 (k)
preserving the diagonal.
Example 8.17. Similarly, the group SL 2 (k) acts on the Hirzebruch surface Fm ,
preserving the zero section of the fibration π : Fm → P1k . This gives examples of
groups with Property (FW) acting on Fm and preserving a big and nef curve C.
Starting with one of the above examples, one can blow-up points on the invariant
curve C, and then contract C, to get examples of zigzag completions Y on which Γ
acts and contracts the boundary ∂Y .

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

45

9. B IRATIONAL TRANSFORMATIONS OF SURFACES II
In this section, U is a (normal, singular) affine surface with a completion X by a
cycle of ` rational curves. Every irreducible component Ei of the boundary ∂X =
X r U is contracted by at least one automorphism of U. Our goal is to classify
subgroups Γ of Aut(U) ⊂ Bir(X) that are infinite and have Property (FW): in fact,
we shall show that no such group exists. This ends the proof of Theorem B since
all the other possibilities of Proposition 8.3 have been dealt with in the previous
section.
Example 9.1. Let (A1k )∗ denote the complement of the origin in the affine line A1k ;
it is isomorphic to the multiplicative group Gm over k. The surface (A1k )∗ × (A1k )∗
is an open subset in P2k whose boundary is the triangle of coordinate lines {[x :
y : z]; xyz = 0}. Thus, the boundary is a cycle of length ` = 3. The group of
automorphisms of (A1k )∗ × (A1k )∗ is the semi-direct product
GL 2 (Z) n (Gm (k) × Gm (k));
it does not contain any infinite group with Property (FW).
9.1. Resolution of indeterminacies. Let us order cyclically the irreducible components Ei of ∂X, so that Ei ∩ Ej 6= ∅ if and only if i − j = ±1(mod `). Blowing
up finitely many singularities of ∂X, we may assume that ` = 2m for some integer
m ≥ 1; in particular, every curve Ei is smooth. (With such a modification, one
may a priori create irreducible components of ∂X that are not contracted by the
group Γ.)
Lemma 9.2. Let f be an automorphism of U and let fX be the birational extension
of f to the surface X. Then
(1) Every indeterminacy point of fX is a singular point of ∂X, i.e. one of the
intersection points Ei ∩ Ei+1 .
(2) Indeterminacies of fX are resolved by inserting chains of rational curves.
Property (2) means that there exists a resolution of the indeterminacies of fX ,
given by two birational morphisms  : Y → X and π : Y → X with f ◦  = π, such
that π −1 (∂X) = −1 (X) is a cycle of rational curves. Some of the singularities of
∂X have been blown-up into chains of rational curves to construct Y .
Proof. Consider a minimal resolution of the indeterminacies of fX . It is given by
a finite sequence of blow-ups of the base points of fX , producing a surface Y and
two birational morphisms  : Y → X and π : Y → X such that fX = π ◦ −1 .

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

46

π



f

F IGURE 1. A blow-up sequence creating two (red) branches. No branch
of this type appears for minimal resolution.
Since the indeterminacy points of fX are contained in ∂X, all necessary blow-ups
are centered on ∂X.
The total transform F = ∗ (∂X) is a union of rational curves: it is made of a
cycle, together with branches emanating from it. One of the assertions (1) and (2)
fails if and only if F is not a cycle; in that case, there is at least one branch.
Each branch is a tree of smooth rational curves, which may be blown-down onto
a smooth point; indeed, these branches come from smooth points of the main cycle
that have been blown-up finitely many times. Thus, there is a birational morphism
η : Y → Y0 onto a smooth surface Y0 that contracts the branches (and nothing
more).
The morphism π maps F onto the cycle ∂X, so that all branches of F are contracted by π. Thus, both  and π induce (regular) birational morphisms 0 : Y0 → X
and π0 : Y0 → X. This contradicts the minimality of the resolution.

Let us introduce a family of surfaces
πk : Xk → X.
First, X1 = X and π1 is the identity map. Then, X2 is obtained by blowing-up the `
singularities of ∂X1 ; X2 is a compactification of U by a cycle ∂X2 of 2` = 2m+1
smooth rational curves. Then, X3 is obtained by blowing up the singularities of
∂X2 , and so on. In particular, ∂Xk is a cycle of 2k−1 ` = 2m+k−1 curves.

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

47

Denote by Dk the dual graph of ∂Xk : vertices of Dk correspond to irreducible
components Ei of ∂Xk and edges to intersection points Ei ∩ Ej . A simple blow-up
(of a singular point) modifies both ∂Xk and Dk locally as follows

F IGURE 2. Blowing-up one point.
˜
The group Aut(U) acts on Hyp(X)
and Lemma 9.2 shows that its action stabilizes
˜
the subset B of Hyp(X) defined as
n
o
˜
B = C ∈ Hyp(X)
: ∃k ≥ 1, C is an irreducible component of ∂Xk .
In what follows, we shall parametrize B in two distinct ways by rational numbers.
9.2. Farey and dyadic parametrizations. Consider an edge of the graph D1 , and
identify this edge with the unit interval [0, 1]. Its endpoints correspond to two adjacent components Ei and Ei+1 of ∂X1 , and the edge corresponds to their intersection q. Blowing-up q creates a new vertex (see Figure 2). The edge is replaced
by two adjacent edges of D2 with a common vertex corresponding to the exceptional divisor and the other vertices corresponding to (the strict transforms of) Ei
and Ei+1 ; we may identify this part of D2 with the segment [0, 1], the three vertices
with {0, 1/2, 1}, and the two edges with [0, 1/2] and [1/2, 1].
Subsequent blow-ups may be organized in two different ways by using either a
dyadic or a Farey algorithm (see Figure 3).
In the dyadic algorithm, the vertices are labelled by dyadic numbers m/2k . The
vertices of Dk+1 coming from an initial edge [0, 1] of D1 are the points {n/2k ; 0 ≤
n ≤ 2k } of the segment [0, 1]. We denote by Dyad(k) the set of dyadic numbers
n/2k ∈ [0, 1]; thus, Dyad(k) ⊂ Dyad(k + 1). We shall say that an interval [a, b] is
a standard dyadic interval if a and b are two consecutive numbers in Dyad(k) for
some k.
In the Farey algorithm, the vertices correspond to rational numbers p/q. Adjacent
vertices of Dk coming from the initial segment [0, 1] correspond to pairs of rational
numbers (p/q, r/s) with ps − qr = ±1; two adjacent vertices of Dk give birth to
a new, middle vertex in Dk+1 : this middle vertex is (p + r)/(q + s) (in the dyadic

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

48

algorithm, the middle vertex is the “usual” euclidean middle). We shall say that an
interval [a, b] is a standard Farey interval if a = p/q and b = r/s with ps − qr =
−1. We denote by Far(k) the finite set of rational numbers p/q ∈ [0, 1] that is given
by the k-th step of Farey algorithm; thus, Far(1) = {0, 1} and Far(k) is a set of
2k+1 rational numbers p/q with 0 ≤ p ≤ q. (One can check that 1 ≤ q ≤ Fib(k),
with Fib(k) the k-th Fibonacci number.)

0/1

0/1

1/1

0/1

1/1

0/1

1/3

1/1

0/1

1/3

1/1

0/1

1/2

0/1

0/1
1/4

1/3

1/2

1/3

1/2

2/5

3/5

3/4

1/1

1/2

1/4

1/2

1/4
1/8

1/1

3/4

1/2
3/8

1/1

3/4
5/8

1/1
7/8

F IGURE 3. On the left, the Farey algorithm. On the right, the dyadic one.
By construction, the graph D1 has ` = 2m edges. The edges of D1 are in one
to one correspondance with the singularities qj of ∂X1 . Each edge determines a
subset Bj of B; the elements of Bj are the curves C ⊂ ∂Xk (k ≥ 1) such that πk (C)
contains the singularity qj determined by the edge. Using the dyadic algorithm
(resp. Farey algorithm), the elements of Bj are in one-to-one correspondence with
dyadic (resp. rational) numbers in [0, 1]. Gluing these segments cyclically together
one gets a circle S1 , together with a nested sequence of subdivisions in `, 2`, . . .,
2k−1 `, . . . intervals; each interval is a standard dyadic interval (resp. standard Farey
interval) of one of the initial edges .
Since there are ` = 2m initial edges, we may identify the graph D1 with the circle
S1 = R/Z = [0, 1]/0'1 and the initial vertices with the dyadic numbers in Dyad(m)
modulo 1 (resp. with the elements of Far(m) modulo 1). Doing this, the vertices of
Dk are in one to one correspondence with the dyadic numbers in Dyad(k + m − 1)
(resp. in Far(k + m − 1)).
Remark 9.3. (a).– By construction, the interval [p/q, r/s] ⊂ [0, 1] is a standard
Farey interval if and only if ps − qr = −1, iff it is delimited by two adjacent
elements of Far(m) for some m.

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

49

(b).– If h : [x, y] → [x0 , y 0 ] is a homeomorphism between two standard Farey intervals mapping rational numbers to rational numbers and standard Farey intervals to
standard Farey intervals, then h is the restriction to [x, y] of a unique linear projective transformation with integer coefficients:


at + b
a b
, for some element
of PGL 2 (Z).
h(t) =
c d
ct + d
(c).– Similarly, if h is a homeomorphism mapping standard dyadic intervals to intervals of the same type, then h is the restriction of an affine dyadic map
u
h(t) = 2m t + n , with m, n ∈ Z.
2
In what follows, we denote by GFar the group of self-homeomorphisms of S1 =
R/Z that are piecewise PGL 2 (Z) mapping with respect to a finite decomposition of
the circle in standard Farey intervals [p/q, r/s]. In other words, if f is an element of
GFar , there are two partitions of the circle into consecutive intervals Ii and Ji such
that the Ii are intervals with rational endpoints, h maps Ii to Ji , and the restriction
f : Ii → Ji is the restriction of an element of PGL 2 (Z) (see [34], §1.5.1).
Theorem 9.4. Let U be an affine surface with a compactification U ⊂ X such that
∂X := X r U is a cycle of smooth rational curves. In the Farey parametrization of
˜
the set B ⊂ Hyp(X)
of boundary curves, the group Aut(U) acts on B as a subgroup
of GFar .
Remark 9.5. There is a unique orientation preserving self-homeomorphism of the
circle that maps Dyad(k) to Far(k) for every k. This self-homeomorphism conjugates GFar to the group GDya of self-homeomorphisms of the circle that are piecewise affine with respect to a dyadic decomposition of the circle, with slopes in ±2Z ,
and with translation parts in Z[1/2]. Using the parametrization of B by dyadic numbers, the image of Aut(U) becomes a subgroup of GDya .
Remark 9.6. The reason why we keep in parallel the dyadic and Farey viewpoints
is the following: the Farey viewpoint is more natural for algebraic geometers (this is
related to toric –i.e. monomial– maps and appears clearly in [21]), while the dyadic
viewpoint is more natural to geometric group theorists, because this is the classical
setting used in the study of Thompson groups (see [34], §1.5).
Proof. Lemma 9.2 is the main ingredient. Consider the action of the group Aut(U)
on the set B. Let f be an element of Aut(U) ⊂ Bir(X). Consider an irreducible
curve E ∈ B, and denote by F its image: F = f• (E) is an element of B by

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

50

Lemma 9.2. There are integers k and l such that E ⊂ ∂Xk and F ⊂ ∂Xl . Replacing
Xk by a higher blow-up Xm → X, we may assume that flm := πl−1 ◦ f ◦ πm is
regular on a neighborhood of the curve E (Lemma 9.2). Let qk be one of the two
singularities of ∂Xm that are contained in E, and let E 0 be the second irreducible
component of ∂Xm containing q. If E 0 is blown down by flm , its image is one of
the two singularities of ∂Xl contained in F (by Lemma 9.2). Consider the smallest
integer n ≥ l such that ∂Xn contains the strict transform F 0 = f• (E 0 ); in Xn , the
curve F 0 is adjacent to the strict transform of F (still denoted F ), and f is a local
isomorphism from a neighborhood of q in Xm to a neighborhood of q 0 := F ∩ F 0 in
Xn .
Now, if one blows-up q, the exceptional divisor D is mapped by f• to the exceptional divisor D0 obtained by a simple blow-up of q: f lifts to a local isomorphism
from a neighborhood of D to a neighborhood of D0 , the action from D to D0 being
given by the differential dfq . The curve D contains two singularities of ∂Xm+1 ,
which can be blown-up too: again, f lifts to a local isomorphism if one blow-ups
the singularities of ∂Xn+1 ∩ D0 . We can repeat this process indefinitely. Let us
now phrase this remark differently. The point q determines an edge of Dm , hence
a standard Farey interval I(q). The point q 0 determines an edge of Dn , hence another standard Farey interval I(q 0 ). Then, the points of B that are parametrized by
rational numbers in I(q) are mapped by f• to rational numbers in I(q 0 ) and this
map respects the Farey order: if we identify I(q) and I(q 0 ) to [0, 1], f• is the restriction of a monotone map that sends Far(k) to Far(k) for every k. Thus, on I(q), f•
is the restriction of a linear projective transformation with integer coefficients (see
Remark 9.3-(b)). This shows that f• is an element of GFar .

9.3. Conclusion. Consider the group G∗Dya of self-homeomorphisms of the circle
S1 = R/Z that are piecewise affine with respect to a finite partition of R/Z into
dyadic intervals [xi , xi+1 [ with xi in Z[1/2]/Z for every i, and satisfy
h(t) = 2mi t + ai
with mi ∈ Z and ai ∈ Z[1/2] for every i. This group is known as the Thompson
group of the circle, and is isomorphic to the group G∗Far of orientation-preserving
self-homeomorphisms in GFar (defined in §9.2).
Theorem 9.7 (Farley, Hughes [15, 22]). Every subgroup of G∗Dya (and hence of
GFar ) with Property (FW) is a finite cyclic group.

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

51

Indeed fixing a gap in an earlier construction of Farley [15]2, Hughes proved [22]
that GFar has Property PW, in the sense that it admits a commensurating action
whose associated length function is a proper map (see also Navas’ book [34]). This
implies the conclusion, because every finite group of orientation-preserving selfhomeomorphisms of the circle is cyclic.
Thus, if Γ is a subgroup of Aut(U) with Property (FW), it contains a finite index
˜
subgroup Γ0 that acts trivially on the set B ⊂ Hyp(X).
This means that Γ0 extends
as a group of automorphisms of X fixing the boundary ∂X. Since ∂X supports
a big and nef divisor, Γ0 contains a finite index subgroup Γ1 that is contained in
Aut(X)0 .
Note that Γ1 has Property (FW) because it is a finite index subgroup of Γ. It
preserves every irreducible component of the boundary curve ∂X, as well as its singularities. As such, it must act trivially on ∂X. When we apply Theorem 8.8 to Γ1 ,
the conjugacy ϕ : X → Y can not contract ∂X, because the boundary supports an
ample divisor. Thus, Γ1 is conjugate to a subgroup of Aut(Y ) that fixes a curve
pointwise. This is not possible if Γ1 is infinite (see Theorem 8.8 and the remarks
following it).
We conclude that Γ is finite in case (2) of Proposition 8.3.
√
10. B IRATIONAL ACTIONS OF SL 2 (Z[ d])
We develop here Example 1.4. If k is an algebraically closed field of characteristic√0, therefore containing Q, we denote by σ1 and σ2 the distinct embeddings
√
of Q( d) into k. Let j1 and j2 be the resulting embeddings of SL 2 (Z[ d]) into
SL 2 (k), and j = j1 × j2 the resulting embedding into
G = SL 2 (k) × SL 2 (k).
√
Theorem 10.1. Let Γ be a finite index subgroup of SL 2 (Z[ d]). Let X be an
irreducible projective surface over an algebraically closed field k. Let α : Γ →
Bir(X) be a homomorphism with infinite image. Then k has characteristic zero,
and there exist a finite index subgroup Γ0 of Γ and a birational map ϕ : Y 99K X
such that
(1) Y is the projective plane P2 , a Hirzebruch surface Fm , or C × P1 for some
curve C;
(2) ϕ−1 α(Γ)ϕ ⊂ Aut(Y );
2

The gap in Farley’s argument lies in Prop. 2.3 and Thm. 2.4 of [15].

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

52

(3) there is a unique algebraic homomorphism β : G → Aut(Y ) such that
ϕ−1 α(γ)ϕ = β(j(γ))
for every γ ∈ Γ0 .
Theorem B ensures that the characteristic of k is 0 and that (1) and (2) are satisfied. If Y is P2 or a Hirzebruch surface Fm , then Aut(Y ) is a linear algebraic
group. If Y is a product C × P1 , a finite index subgroup of Γ preserves the projection onto P1 , so that it acts via an embedding into the linear algebraic group
Aut(P1 ) = PGL 2 (k).
When k has positive characteristic, Y is the projective plane, and the Γ-action is
given by a homomorphism Γ → PGL 3 (k). Then we use the fact that for any n,
every homomorphism f : Γ → GL n (k) has finite image. Indeed, it is well-known
that GL n (k) has no infinite order distorted elements: elements of infinite order have
some transcendental eigenvalue and the conclusion easily follows. Since Γ has an
exponentially distorted cyclic subgroup, f has infinite kernel, and infinite normal
subgroups of Γ have finite index (Margulis normal subgroup theorem).
On the other hand, in characteristic zero we conclude the proof of Theorem 10.1
with the following lemma.
√
d). Consider the embedding j
Lemma 10.2.
Let
k
be
any
field
extension
of
Q(
√
of SL 2 (Z[ d]) into G = SL 2 (k) × SL 2 (k) given by the standard embedding into
the left-hand SL 2 and its Galois conjugate in the right-hand√SL 2 . Then for every
linear algebraic group H and homomorphism f : SL 2 (Z[ d]) → H(k), there
exists a unique homomorphism f¯ : G → H of k-algebraic groups such that the
homomorphisms f and f˜ ◦ j coincide on some finite index subgroup of Γ.
Proof. The uniqueness is a consequence of Zariski density of the image of j. Let
us prove the existence. Zariski density allows to reduce to the case when H = SL n .
First, the case k = R is given by Margulis’ superrigidity, along with the fact that
every continuous real representation of SL n (R) is algebraic. The case of fields
containing R immediately follows,
√ and in turn it follows for subfields of overfields

of R (as soon as they contain Q( d)).

11. O PEN PROBLEMS
11.1. Regularization and Calabi-Yau varieties.

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

53

Question 11.1. Let Γ be a group with Property (FW). Is every birational action
of Γ regularizable ? Here regularizable is defined in the same way as pseudoregularizable, but assuming that the action on U is by automorphisms (instead of
pseudo-automorphisms).
A particular case is given by Calabi-Yau varieties, in the strict sense of a simply connected complex projective manifold X with trivial canonical bundle and
hk,0 (X) = 0 for 0 < k < dim(X). For such a variety the group Bir(X) coincides with Psaut(X). One can then ask (1) whether every infinite subgroup Γ of
Psaut(X) with property (FW) is regularizable on some birational model Y of X
(without restricting the action to a dense open subset), and (2) what are the possibilities for such a group Γ.
11.2. Transfixing birational groups.
Question 11.2. For which irreducible projective varieties X
(1) Bir(X) does not transfix Hyp(X)?
(2) some finitely generated subgroup of Bir(X) does not transfix Hyp(X)?
(3) some cyclic subgroup of Bir(X) does not transfix Hyp(X).
We have the implications: X is ruled ⇒ (3) ⇒ (2) ⇒ (1). In dimension 2, we
have: ruled ⇔ (1) /⇒(2) ⇔ (3) (see §7.1). It would be interesting to find counterexamples to these equivalences in higher dimension, and settle each of the problems
raised in Question 11.2 in dimension 3.
11.3. The affine space. The group of affine transformations of A3C contains SL 3 (C),
and this group contains many subgroups with Property (FW). In the case of surfaces,
Theorem B shows that groups of birational transformations with Property (FW)
are contained in algebraic groups, up to conjugacy. The following question asks
whether this type of theorem may hold for Aut(A3C ).
Question 11.3. Does there exist an infinite subgroup of Aut(A3C ) with Property (FW)
that is not conjugate to a group of affine transformations of A3C ?
11.4. Length functions. Recall that a length function ` on a group G is quasigeodesic if there exists M > 0 such that for every n ≥ 1 and every g ∈ G with
−1
`(g) ≤ n, there exist 1 = g0 , g1 , . . . , gn = g in G such that `(gi−1
gi ) ≤ M for all i.
−1
Equivalently G, endowed with the distance (g, h) 7→ `(g h), is quasi-isometric to
a connected graph.

COMMENSURATING ACTIONS OF BIRATIONAL GROUPS

54

Question 11.4. Given an irreducible variety X, is the length function
g ∈ Bir(X) 7→ |Hyp(X)4gHyp(X)|
quasi-geodesic? In particular, what about X = P2 and the Cremona group Bir(P2 )?
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U NIV R ENNES , CNRS, IRMAR - UMR 6625, F-35000 R ENNES
CNRS AND U NIV LYON , U NIV C LAUDE B ERNARD LYON 1, I NSTITUT C AMILLE J ORDAN ,
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