Discrete Comput Geom 23:111–120 (2000)
DOI: 10.1007/s004549910006
Discrete & Computational
Geometry
©
2000 Springer-Verlag New York Inc.
Semistable Reduction in Characteristic Zero for Families
of Surfaces and Threefolds
K. Karu
Department of Mathematics, Boston University,
111 Cummington, Boston, MA 02215, USA
kllkr@math.bu.edu
Abstract. We consider the problem of extending the semistable reduction theorem of
[KKMS] from the case of one-parameter families of varieties to families over a base of
arbitrary dimension. Following [KKMS], semistable reduction of such families can be
reduced to a problem in the combinatorics of polyhedral complexes [AK]. In this paper we
solve it in the case when the relative dimension of the morphism is at most three, i.e., for
families of surfaces and threefolds.
1.
Introduction
One of the milestones in algebraic geometry is the semistable reduction theorem proved
in [KKMS]:
Theorem 1.1 [KKMS]. Let f : X → C be a flat morphism from a variety X onto a
nonsingular curve C, defined over an algebraically closed field k of characteristic zero.
Assume that 0 ∈ C is a point and the restriction f : X \ f −1 (0) → C\{0} is smooth. Then
there exist a finite morphism π: C 0 → C, with π −1 (0) = {00 }, and a proper birational
morphism (in fact, a blowup with center lying in the special fiber) p: X 0 → X ×C C 0 ,
X0
p
→
X ×C C 0
↓
C0
→
X
↓
π
→ C
so that the induced morphism f 0 : X 0 → C 0 is semistable; i.e.,
(i) both X 0 and C 0 are nonsingular, and
(ii) the special fiber f 0 −1 (00 ) is a reduced divisor with nonsingular components
crossing normally.
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K. Karu
To prove the theorem, Kempf et al. [KKMS] invented the theory of toroidal embeddings and reduced the geometric problem to the following purely combinatorial problem:
Theorem 1.2 [KKMS]. Let P ⊂ Rn be an n-dimensional polytope with vertices lying
in the integral points Zn ⊂ Rn . Then there exists an integer M and a projective subdivision {Pα }α of P such that every Pα has vertices in (1/M)Zn and the volume of Pα (in
the usual metric) is the minimal possible: vol(Pα ) = 1/M n n!.
Here a subdivision is called projective (or coherent) if it is defined by a continuous
piecewise linear convex function.
The main goal of [AK] was to extend the semistable reduction theorem to the case
where the base variety has arbitrary dimension. The problem can then be formulated as
follows:
Conjecture 1.3. Let f : X → B be a surjective morphism of projective varieties
with geometrically integral generic fiber, defined over an algebraically closed field of
characteristic zero. There exist a proper surjective generically finite morphism B 0 → B
and a proper birational morphism X 0 → X × B B 0 such that X 0 → B 0 is semistable;
i.e., for any closed point x 0 ∈ X 0 and b0 = f 0 (x 0 ) ∈ B 0 one can find formal coordinates
x1 , . . . , xn at x 0 and t1 , . . . , tm at b so that the morphism f is given by
ti =
li
Y
xj
j=li−1 +1
for some 0 = l0 < l1 < · · · < lm ≤ n.
Using the theory of toroidal embeddings, the geometric problem of semistable reduction can again be reduced to a combinatorial problem involving conical polyhedral
complexes. The aim of this paper is to solve the combinatorial problem for the case when
f has low relative dimension. First, we recall the definition of polyhedral complexes and
morphisms.
1.1.
Polyhedral Complexes
We consider (rational, conical) polyhedral complexes 1 = (|1|, ({σ, Nσ }) consisting
of a finite collection of lattices Nσ ∼
= Zn and rational full cones σ ⊂ Nσ ⊗ R with a
vertex. The cones σ are glued together to form the space |1| so that the usual axioms of
polyhedral complexes hold:
1. If σ ∈ 1 is a cone, then every face σ 0 of σ is also in 1, and Nσ 0 = Nσ | Span(σ 0 ) .
2. The intersection of two cones σ1 ∩ σ2 is a face of both of them.
A morphism f 1 : 1 X → 1 B of polyhedral complexes 1 X = (|1 X |, {σ, Nσ }) and
1 B = (|1 B |, {τ, Nτ }) is a compatible collection of linear maps f σ : (σ, Nσ ) → (τ, Nτ );
i.e., if σ 0 is a face of σ , then f σ 0 is the restriction of f σ . We only consider morphisms
f : 1 X → 1 B such that f σ−1 (0) ∩ σ = {0} for all σ ∈ 1 X .
Semistable Reduction in Characteristic Zero
113
Polyhedral complexes arise naturally in the theory of toroidal embeddings [KKMS].
They generalize the notion of fans of toric varieties. An open embedding of varieties
U X ⊂ X is said to be toroidal if it is locally formally isomorphic to a torus embedding
T ⊂ X σ ; a morphism of toroidal embeddings is a morphism of varieties that locally
formally comes from a toric morphism. To a toroidal embedding one associates a polyhedral complex, and a morphism of toroidal embeddings gives rise to a morphism of
polyhedral complexes. The condition of semistability, when applied to a toroidal embedding, translates into the following condition on the associated morphism of polyhedral
complexes.
Definition 1.4.
semistable if:
A surjective morphism f 1 : 1 X → 1 B such that f −1 (0) = {0} is
1. 1 X and 1 B are nonsingular.
2. For any cone σ ∈ 1 X , we have f (σ ) ∈ 1 B and f (Nσ ) = N f (σ ) .
We say that f is weakly semistable if it satisfies the two conditions except that 1 X may
be singular.
The following two operations are allowed on 1 X and 1 B :
1. Projective subdivisions 10X of 1 X and 10B of 1 B such that f induces a morphism
f 0 : 10X → 10B .
2. Lattice alterations: let 10X = (|1 X |, {σ, Nσ0 }), 10B = (|1 B |, {τ, Nτ0 }), for some
compatible collection of sublattices Nτ0 ⊂ Nτ , Nσ0 = f −1 (Nτ0 ) ∩ Nσ , and let
f 0 : 10X → 10B be the morphism induced by f .
Conjecture 1.5 (Combinatorial Semistable Reduction). Given a surjective morphism
f : 1 X → 1 B , such that f −1 (0) = {0}, there exists a projective subdivision f 0 : 10X →
10B followed by a lattice alteration f 00 : 100X → 100B so that f 00 is semistable.
1 X 00
↓ f 00
1 B 00
→ 1X 0
↓ f0
→ 1B0
→ 1X
↓ f
→ 1B
The importance of Conjecture 1.5 lies in the fact that it implies Conjecture 1.3 [AK].
Although we are concerned with the combinatorial version of semistable reduction in this
paper, we indicate briefly how the two conjectures are related. It is shown in [AK] that a
morphism f : X → B as in Conjecture 1.3 can be modified to a toroidal morphism, and so
we get a morphism of polyhedral complexes f 1 : 1 X → 1 B . Then one checks that if f 1 is
semistable according to Definition 1.4, then f is semistable as defined in Conjecture 1.3.
It remains to show that the two combinatorial operations on f 1 : 1 X → 1 B have
geometric analogues for f : X → B. Indeed, subdivisions of 1 X and 1 B correspond to
birational morphisms (see [KKMS]), and a lattice alteration corresponds to a finite base
change (see [AK]).
In the case when dim(1 B ) = 1, Conjecture 1.5 reduces to the combinatorial version
of the semistable reduction theorem proved in [KKMS]. In [AK] the conjecture was
proved with semistable replaced by weakly semistable. The main result of this paper is
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K. Karu
Theorem 1.6. Conjecture 1.5 is true if f 1 has relative dimension ≤ 3. Hence, Conjecture 1.3 is true if f has relative dimension ≤ 3.
The relative dimension of a linear map f σ : σ → τ of cones σ, τ is dim(σ ) −
dim( f (σ )). The relative dimension of f 1 : 1 X → 1 B is by definition the maximum of
the relative dimensions of f σ : σ → τ over all σ ∈ 1 X . To see that the second statement of
the theorem follows from the first, consider a surjective morphism of affine toric varieties
f : X σ → X τ defined by a linear map of cones and lattices f 1 : (σ, Nσ ) → (τ, Nτ ).
A general fiber of this morphism has dimension equal to the rank of the kernel of
f 1 : Nσ → Nτ , and this is at least the relative dimension of f 1 : σ → τ . Therefore,
if a toroidal morphism f : X → B has relative dimension ≤ d, then the associated
morphism of polyhedral complexes f 1 : 1 X → 1 B also has relative dimension ≤ d.
We remark that semistable reduction for families of curves over a base of an arbitrary
dimension was proved by de Jong [dJ]. Thus, the new result of Theorem 1.6 is semistable
reduction for families of surfaces and threefolds.
The rest of the paper is organized as follows. In Section 2 we use the construction of
[KKMS] to make f semistable over the edges of 1 B without increasing the multiplicity of
1 X . In Section 3 we modify the barycentric subdivision of 1 X so that we get a morphism
to the barycentric subdivision of 1 B . It is shown in Section 4 that in certain situations we
can choose a modified barycentric subdivision that decreases the multiplicity of 1 X . The
conditions when this happens are then verified for relative dimension ≤ 3 in Section 5.
2.
Notation and Preliminaries
2.1. Notation
We use notation from [KKMS] and [F]. For a cone σ ∈ N ⊗R we write σ = hv1 , . . . , vn i
if the points v1 , . . . , vn lie on the one-dimensional edges of σ and generate the cone.
If vi are the first lattice points along the edges we call them primitive points of σ . An
n-dimensional cone is simplicial if it has exactly n primitive points. For a simplicial cone
σ with primitive points v1 , . . . , vn , the multiplicity of σ is
m(σ, Nσ ) = [Nσ : Zv1 ⊕ · · · ⊕ Zvn ].
A polyhedral complex 1 is nonsingular if and only if m(σ, Nσ ) = 1 for all σ ∈ 1. To
compute the multiplicity of σ we can count the representatives w ∈ Nσ of classes of
Nσ /Zv1 ⊕ . . . ⊕ Zvn of the form
X
αi vi ,
0 ≤ αi < 1.
w=
i
The set of all such points is denoted by W (σ ). For cones σ1 , σ2 ∈ 1 we write σ1 ≤ σ2
if σ1 is a face of σ2 . Notice that if σ1 ≤ σ2 , then the multiplicity of σ1 is at most the
multiplicity of σ2 . Hence, to compute the multiplicity of a polyhedral complex 1, it
suffices to consider maximal cones only.
Let f 1 : 1 X → 1 B be a morphism of polyhedral complexes, and assume that 1 B
is simplicial. Let u 1 , . . . , u m be the primitive points of 1 B , and let M1 , . . . , Mm be
Semistable Reduction in Characteristic Zero
115
positive integers. By taking the (M1 , . . . , Mm )-sublattice at u 1 , . . . , u m we mean the
lattice alteration Nτ0 = Z{m i1 u i1 , . . . , m il u il } for all cones τ ∈ 1 B with primitive points
u i 1 , . . . , u il .
A subdivision 10 of 1 is called projective if there exists a homogeneous continuous
piecewise linear function ψ: |1| → R, convex on each cone σ ∈ 1, and taking rational
values on the lattice points Nσ such that the maximal cones of 10 are exactly the maximal
pieces in which ψ is linear.
2.2.
Applying the Result of [KKMS]
Let σ1 ⊂ Rn 1 and σ2 ⊂ Rn 2 be two cones. We consider σ1 × σ2 as a cone in Rn 1 +n 2 . If
{σ1,α }α is a subdivision of σ1 , and {σ2,β }β is a subdivision of σ2 , then {σ1,α × σ2,β }α,β
gives us a subdivision of σ1 × σ2 .
If 1 X and 1 B are simplicial, we say that f : 1 X → 1 B is simplicial if f (σ ) ∈ 1 B for
all σ ∈ 1 X . Assume that f 1 : 1 X → 1 B is a simplicial map of simplicial complexes. Let
u i , i = 1, . . . , m, be the primitive points of 1 B , and let vi j , i = 1, . . . , m, j = 1, . . . , Ji ,
be the primitive points of 1 X such that vi j is mapped to an integer multiple of u i . For each
i = 1, . . . , m we denote by 1 X,i the subcomplex of 1 X lying over the cone hu i i of 1 B :
1 X,i = f 1−1 (hu i i).
Note that if we forget the lattices of 1 X , then by the assumption that f 1−1 (0) = {0} we
get that 1 X = 1 X,1 × · · · × 1 X,m . If 10X,i are subdivisions of 1 X,i , we get a subdivison
10X of 1 X by taking the product
10X = 10X,1 × · · · × 10X,m .
Lemma 2.1. If 10X,i are projective subdivisions of 1 X,i , then 10X is a projective subdivision of 1 X .
Proof. Let ψi be a convex piecewise linear function defining the subdivison |10X,i |.
Extend
Pψi linearly to the entire |1 X | by setting ψi (|1 X, j |) = 0 for j 6= i. Clearly,
ψ = i ψi is a convex piecewise linear function defining the subdivision 10X .
Consider the restriction f 1 |1 X,i : 1 X,i → R+ u i . By the Main Theorem of Chapter 2
in [KKMS] there exist a subdivision 10X,i of 1 X,i and a positive integer Mi such that
after taking the Mi -sublattice at u i the induced morphism f 10 |10X,i is semistable. We let
10X be the product of the subdivisions 10X,i , and we take the (M1 , . . . , Mm )-sublattice
at (u 1 , . . . , u n ). Then f 10 : 10X → 10B is a simplicial map and f 10 |10X,i is semistable for
all i.
Lemma 2.2.
The multiplicity of 10X is not greater than the multiplicity of 1 X .
Proof. Let σ ∈ 1 X have primitive points vi j and let σ 0 ⊂ σ be a maximal cone in the
subdivision with primitive points vi0 j . The multiplicity of σ 0 is the number of points in
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K. Karu
W (σ 0 ). We show that W (σ 0 ) can be mapped injectively to W (σ ), hence the multiplicity
of σ 0 is not greater than the multiplicity of σ .
If w0 ∈ W (σ 0 ), we write
X
(βi j + bi j )vi j ,
0 ≤ βi j < 1, bi j ∈ Z+ .
w0 =
i, j
P
Then w = i j βi j vi j ∈ Nσ is in W (σ ). If two points w10 , w20 ∈ W (σ 0 ) give the same
w, then their difference w10 − w20 is an integral linear combination of vi j . However, then
w10 − w20 is also an integral linear combination of vi0 j because Z{vi0 j }i, j = Z{vi j }i, j ∩ Nσ 0 .
Hence w10 − w20 = 0.
3.
Modified Barycentric Subdivisions
Let f 1 : 1 X → 1 B be a simplicial morphism. Consider the barycentric subdivision
BS(1P
B ) of 1 B . The one-dimensional cones of BS(1 B ) are of the form R+ τ̂ where
τ̂ = u i is the barycenter of a cone τ ∈ 1 B with primitive points u 1 , . . . , u m . A cone
τ 0 ∈ BS(1 B ) is spanned by τ̂1 , . . . , τ̂k , where τ1 ≤ τ2 ≤ · · · ≤ τk is a chain of cones in
1B .
In general, f 1 does not induce a morphism BS(1 X ) → BS(1 B ). For example, if
σ = hv11 , v12 , v21 i, τ = hu 1 , u 2 i, and f 1 : vi j 7→ u i , then f 1 does not induce a morphism
of barycentric subdivisions. To get a morphism we need to modify the barycenters σ̂ of
cones σ ∈ 1 X so that they map to (multiples of) barycenters of 1 B .
Definition 3.1.
The data of modified barycenters consists of:
1. A subset of cones 1̃ X ⊂ 1 X .
2. For each cone σ ∈ 1̃ X a lattice point bσ ∈ int(σ ) ∩ Nσ such that f 1 (bσ ) ∈ R+ τ̂
for some τ ∈ 1 B .
Recall that for any total order ≺ on the set of cones in 1 X refining the partial order ≤,
the barycentric subdivision BS(1 X ) can be realized as a sequence of star subdivisions
at the barycenters σ̂ for all cones σ ∈ 1 X in the descending order ≺.
Definition 3.2. Given modified barycenters (1̃ X , {bσ }) and a total order ≺ on 1 X
refining the partial order ≤, the modified barycentric subdivision MBS1̃ X ,{bσ },≺ (1 X )
is the sequence of star subdivisions at bσ for all σ ∈ 1̃ X in the descending order ≺.
Example 3.3. Let f 1 : hv11 , v12 , v21 i → hu 1 , u 2 i be the morphism defined by
f 1 : vi j 7→ u i . Let 1̃ X consist of the two cones 1̃ X = {hv11 , v21 i, hv12 , v21 i}, and
let the modified barycenters be {bσ } = {v11 + v21 , v12 + v21 }. Depending on whether
hv11 , v21 i ≺ hv12 , v21 i or vice versa, we get two modified barycentric subdivisions as
shown in Fig. 1.
To simplify notation, we write MBS1̃ X (1 X ) or simply MBS(1 X ) instead of
MBS1̃ X ,{bσ },≺ (1 X ). By definition, MBS(1 X ) is a projective simplicial subdivision of
Semistable Reduction in Characteristic Zero
117
Fig. 1. Two modified barycentric subdivisions from Example 3.3
1 X . Next, we show that, as in the case of the ordinary barycentric subdivision, the cones
of MBS(1 X ) can be characterized by chains of cones in 1 X . We may assume that the
zero- and one-dimensional cones of 1 X are all in 1̃ X , and they precede all other cones
in the order ≺. For a cone σ ∈ 1 X let σ̃ be the maximal face of σ (with respect to ≺) in
1̃ X . Given a chain of cones σ1 ≤ · · · ≤ σk in 1 X , the cone hbσ̃1 , . . . , bσ̃k i is a subcone
of σk . Let C(1 X ) be the set of all such cones corresponding to chains σ1 ≤ · · · ≤ σk
in 1 X .
Proposition 3.4. C(1 X ) = MBS(1 X ).
Proof. We do induction on the number of cones in 1̃ X of dimension at least 2. If 1̃ X
contains only zero- or one-dimensional cones, then the statement is trivial. So, assume
that 1̃ X = 1̃ X,0 ∪ {σ0 }, where σ ≺ σ0 for any σ ∈ 1̃ X,0 , and assume that the proposition
is proved for 1̃ X,0 .
Without loss of generality we may assume that 1 X consists of cones containing σ0
and their faces only. We get MBS1̃ X (1 X ) from 1 X if we first subdivide at bσ0 and then at
bσ for σ ∈ 1̃ X,0 in the descending order ≺. If 1 X,0 is the subcomplex of 1 X consisting
of cones not containing σ0 , then the star subdivision of 1 X at bσ0 is 1 X,0 × hbσ0 i. Since
σ0 is greater than any σ ∈ 1̃ X,0 with respect to ≺, all bσ ∈ 1 X,0 , and we see that
MBS1̃ X (1 X ) = MBS1̃ X,0 (1 X,0 ) × hbσ0 i.
A cone in MBS1̃ X,0 (1 X,0 ) × hbσ0 i is of the form σ × ρ, where ρ is a face of hbσ0 i, i.e.
either {0} or hbσ0 i itself, and where σ is a cone in MBS1̃ X,0 (1 X,0 ). Applying induction
hypothesis to MBS1̃ X,0 (1 X,0 ), we get that σ = hbσ̃1 , . . . , bσ̃l i for a chain of cones σ1 ≤
· · · ≤ σl in 1 X,0 . Now if ρ = {0}, then σ × ρ = hbσ̃1 , . . . , bσ̃l i ∈ C(1 X ). If ρ = hbσ0 i,
we let σl+1 be a cone in 1 X that contains both σl and σ0 . Then σ̃l+1 = σ0 , and σ × ρ =
hbσ̃1 , . . . , bσ̃l , bσ̃l+1 i ∈ C(1 X ).
Conversely, let hbσ̃1 , . . . , bσ̃l i be a cone in C(1 X ) for some chain σ1 ≤ · · · ≤ σl in
1 X . Then for some k ≤ l we have that σ̃1 , . . . , σ̃k ∈ 1̃ X,0 , and σ̃k+1 = · · · = σ̃l = σ0 .
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K. Karu
By induction hypothesis, the cone hbσ̃1 , . . . , bσ̃k i coming from the chain σ1 ≤ · · · ≤ σk
in 1 X,0 is in MBS1̃ X,0 (1 X,0 ). Hence the cone hbσ̃1 , . . . , bσ̃l i is of the form σ × ρ, where
σ = hbσ̃1 , . . . , bσ̃k i ∈ MBS1̃ X,0 (1 X,0 ), and ρ = hbσ0 i if k < l, and ρ = {0} if k = l.
Corollary 3.5. Assume that f 1 : 1 X → 1 B is a simplicial morphism. If f 1 (σ̃ ) =
f 1 (σ ) for all σ ∈ 1 X , then f 1 induces a simplicial morphism f 10 : MBS(1 X ) →
B S(1 B ).
Proof. Let σ 0 ∈ MBS(1 X ) correspond to a chain σ1 ≤ · · · ≤ σk in 1 X . Since f 1 is
simplicial, we have a chain of cones f 1 (σ1 ) ≤ · · · ≤ f 1 (σk ) in 1 B . Recall that bσ̃i is
mapped to a multiple of a barycenter: f 1 (bσ̃i ) = R+ τ̂ for some τ ∈ 1 B . The assumption
f 1 (σi ), hence the cone hbσ̃1 , . . . , bσ̃k i
that f 1 (σ̃i ) = f 1 (σi ) implies that f 1 (bσ̃i ) ∈ R+ \
\
is mapped onto the cone h f\
1 (σ1 ), . . . , f 1 (σk )i ∈ BS(1 B ).
Example 3.6. Assume that f 1 : 1 X → 1 B is a simplicial morphism taking primitive
'
points of 1 X to primitive points of 1 B . Then for a cone σ ∈ 1 X such that f 1 : σ → τ
for some τ ∈ 1 B , we have f 1 (σ̂ ) = τ̂ .
Let 1 X = {σ ∈ 1 X : f 1 |σ is injective}, bσ = σ̂ . Clearly, the hypothesis of the lemma
is satisfied, and we have a simplicial morphism f 10 : MBS1 X (1 X ) → BS(1 B ).
Conversely, if (1̃ X , {bσ }) is the data of modified barycenters such that f 1 induces a
morphism f 10 : MBS1̃ X (1 X ) → BS(1 B ), then 1 X ⊂ 1̃ X . Thus, we may always assume
that 1 X ⊂ 1̃ X .
4.
Reducing the Multiplicity of 1 X
Proposition 4.1. Let f 1 : 1 X → 1 B be a simplicial morphism taking primitive points
to primitive points. Assume that 1 B is nonsingular, 1 X is singular, and every singular
cone σ ∈ 1 X contains a point w ∈ W (σ )\{0} mapping to a barycenter in 1 B . Then
there exists a modified barycentric subdivision MBS(1 X ) of 1 X having multiplicity
strictly less than the multiplicity of 1 X such that f 1 induces a simplicial morphism
f 10 : MBS(1 X ) → BS(1 B ).
Proof. For every singular cone σ ∈ 1 X we choose a point wσ as follows. By assumption, there exists a point w ∈ W (σ )\{0} mapping to a barycenter of 1 B : f 1 (w) = τ̂ .
Then for a unique cone τ0 ∈ 1 B we have f 1 (σ ) = τ × τ0 . We choose a face σ0 ≤ σ
'
such that f 1 : σ0 → τ0 . Set wσ = w + σ̂0 ; then
f 1 (σ ).
f 1 (wσ ) = f 1 (w) + f 1 (σ̂0 ) = τ̂ + τ̂0 = \
Having chosen the set {wσ }, we may remove some of the points wσ if necessary so that
every simplex ρ ∈ 1 X contains at most one wσ in its interior. With 1 X as in Example 3.6,
let 1̃ X = 1 X ∪ {ρ ∈ 1 X |wσ ∈ int (ρ) for some singular σ }, bρ = ρ̂ if ρ ∈ 1 X , and
bρ = wσ if wσ ∈ int(ρ).
Semistable Reduction in Characteristic Zero
119
Next we specify the order ≺. We refine the partial order ≤ as follows: for two faces
σ1 and σ2 of a cone σ ∈ 1 X we set σ1 ≺0 σ2 if dim f 1 (σ1 ) < dim f 1 (σ2 ). Since
1 X ⊂ 1̃ X , this ensures that the condition f 1 (σ̃ ) = f 1 (σ ) of Corollary 3.5 is satisfied
for any refinement of ≺0 . Now if σ is singular, then the point wσ constructed above lies
in the interior of a face ρσ such that f 1 (ρσ ) = f 1 (σ ). We further refine the order ≺0
by setting σ1 ≺0 ρσ for any face σ1 of the singular cone σ . Then σ̃ = ρσ whenever σ is
singular. Finally we let ≺ be any refinement of ≺0 to a total order.
Let σ ∈ 1 X be a cone, and let a maximal cone σ 0 ∈ MBS(1 X ) be given by a maximal
chain of faces of σ : σ1 ≤ · · · ≤ σn . We have to show that m(σ 0 , Nσ 0 ) ≤ m(σ, Nσ ), and
if σ is singular, then the inequality is strict.
We can order the primitive points v1 , . . . , vn of σ so that σ1 = hv1 i, σ2 = hv1 , v2 i, . . . ,
σn = hv1 , . . . , vn i. Since bσ̃i ∈ σi , the primitive points of σ 0 = hbσ̃1 , . . . , bσ̃n i can be
written as
v10
=
1
bσ̃
µ1 1
= a11 v1 ,
v20
=
1
bσ̃
µ2 2
= a21 v1
+ a22 v2 ,
1
bσ̃
µn n
= an1 v1
+
···
vn0
=
···
+ ann vn
for some ai j ≥ 0 and integers µi ≥ 1. The multiplicity of σ 0 is a11 · a22 · · · ann times the
multiplicity of σ . By the choice of bρ above, all aii ≤ 1, hence m(σ 0 , Nσ 0 ) ≤ m(σ, Nσ ).
If σ is singular, let i be the smallest index such that the face σi is singular. Then, with
notation as above, bσ̃i = w + σ̂0 for some w ∈ W (σi )\{0}, and σ0 ≤ σi . Now if aii = 1,
then w ∈ hv1 , . . . , vi−1 i, and this gives a contradiction with the choice of i. Hence
aii < 1 and m(σ 0 , Nσ 0 ) < m(σ, Nσ ).
5.
Families of Surfaces and Threefolds
Proof of Theorem 1.6. Let f 1 : 1 X → 1 B be a surjective morphism of polyhedral
complexes such that f 1−1 (0) = {0}. It is shown in [AK] that there exist projective
simplicial subdivisions 10X of 1 X and 10B of 1 B such that 1 B is nonsingular and f 1
induces a simplicial morphism f 10 : 10X → 10B . To obtain these subdivisions, one first
subdivides 1 B such that the image of every cone in 1 X is a union of cones in 10B . The
convex piecewise linear function defining the subdivision 10B can then be composed
with f 1 to give a subdivision of 1 X . A sequence of star subdivisions centered at the onedimensional edges yields the required simplicial subdivision 10X . Thus, we may assume
that 1 X is simplicial, 1 B is nonsingular, and f 1 : 1 X → 1 B is a simplicial map.
Applying the construction of [KKMS] over the edges of 1 B (Section 2.2), we can
make f 1 |1 X,i semistable without increasing the multiplicity of 1 X . We show below that
every singular simplex σ ∈ 1 X contains a point w ∈ W (σ )\{0} mapping to a barycenter
of 1 B . By Proposition 4.1, there exists a modified barycentric subdivision such that
f 1 induces a simplicial morphism f 10 : MBS(1 X ) → BS(1 B ), with multiplicity of
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K. Karu
MBS(1 X ) strictly less than the multiplicity of 1 X . Since f 10 is simplicial and BS(1 B )
nonsingular, the proof is completed by induction on the multiplicity of 1 X .
Consider the restriction of f 1 to a singular simplex f 1 : σ → τ , where τ has primitive points u 1 , . . . , u m , σ has primitive points vi j , i = 1, . . . , m, j = 1, . . . , Ji , and
f 1 (vi j ) = u i . Since σ is singular, it contains a point w ∈ W (σ )\{0}:
X
X
αi j vi j ,
0 ≤ αi j < 1,
αi j > 0.
w=
i, j
Considering a face of σ if necessary, we may assume
P that w lies in the interior of σ ,
hence 0 < αi j . Since f 1 (w) ∈ Nτ , it follows that j αi j ∈ Z for all i. In particular, if
Ji0 = 1 for some i 0 , then αi0 1 = 0, and w lies in a face
P of σ . So we may assume that
dimension
of
f
is
Ji > 1 for all i. Since the relative
1
i (Ji − 1), we have to consider
P
all possible decompositions i (Ji − 1) ≤ 3, where Ji > 1 for all i.
The cases when the relative dimension of f 1 is 0 or 1 are trivial and left to the reader.
If the relative dimension of f 1 is 2, then either J1 = 3 or J1 = J2 = 2. In the first
case we have that hv11 , v12 , v13 i is singular, contradicting the semistability of f 1 |1 X,1 .
In the second case, α11 + α12 , α21 + α22 ∈ Z and 0 < αi j < 1 imply that α11 + α12 =
α21 + α22 = 1. Hence f 1 (w) = u 1 + u 2 is a barycenter.
In relative dimension 3, either J1 = 4, or J1 = 3, J2 = 2, or J1 = J2 = J3 = 2.
In the first case we get a contradiction with the semistability of f 1 |1 X,1 ; the third case
gives α11 + α12 = α21 + α22 = α31 + α32 = 1 as for relative dimension 2. In the
second case, either α11 + α12 + α13 = α21P+ α22 = 1 and w maps to a barycenter, or
α11 + α12 + α13 = 2, α21 + α22 = 1 and ( vi j ) − w maps to a barycenter.
Example 5.1. We show by an example that the previous construction of modified
barycentric subdivisions does not work in relative dimension ≥ 4. Let τ = hu 1 , u 2 i and
σ = hv11 , v12 , v13 , v14 , v21 , v22 i, with lattices Nτ = Z{u 1 , u 2 } and Nσ = Z{v11 , . . . , v22 ,
1
(v + · · · + v22 )}. Then W (σ )\{0} consists of a single point w = 12 (v11 + · · · + v22 ),
2 11
and if f 1 maps vi j to u i , then w is mapped to 2u 1 + u 2 , which is not a barycenter.
Acknowledgments
The suggestion to write up the proof of semistable reduction for low relative dimensions
came from Dan Abramovich. I also wish to thank the anonymous reviewers for useful
suggestions.
References
[AK] D. Abramovich and K. Karu, Weak semistable reduction in characteristic 0, preprint. alggeom/9707012.
[dJ] A. J. de Jong, Smoothness, semistability, and alterations, Publ. Math. I.H.E.S. 83 (1996), 51–93.
[F] W. Fulton, Introduction to Toric Varieties, Princeton University Press, Princeton, NJ, 1993.
[KKMS] G. Kempf, F. Knudsen, D. Mumford, and B. Saint-Donat, Toroidal Embeddings I, LNM 339,
Springer-Verlag, Berlin, 1973.
Received March 18, 1998, and in revised form October 7, 1998.