Semistable Reduction in Characteristic Zero for Families of Surfaces and Threefolds

Discrete & Computational Geometry, Jan 2000

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.

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Semistable Reduction in Characteristic Zero for Families of Surfaces and Threefolds

Discrete Comput Geom Semistable Reduction in Characteristic Zero for Families of Surfaces and Threefolds K. Karu 0 0 Department of Mathematics, Boston University , 111 Cummington, Boston, MA 02215 , USA 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. (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. - 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 2 C is a point and the restriction f : X n f ¡1.0/ ! C nf0g is smooth. Then there exist a finite morphism ¼ : C 0 ! C , with ¼ ¡1.0/ D f00g, and a proper birational morphism (in fact, a blowup with center lying in the special fiber) p: X 0 ! X £C C 0, X 0 p ! X £C C 0 # C 0 ! ¼ ! X # C so that the induced morphism f 0: X 0 ! C 0 is semistable; i.e., 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 ½ Rn be an n-dimensional polytope with vertices lying in the integral points Zn ½ Rn. Then there exists an integer M and a projective subdivision f P®g® of P such that every P® has vertices in .1=M /Zn and the volume of P® (in the usual metric) is the minimal possible: vol. P®/ D 1=M nn!. 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 B0 ! B and a proper birational morphism X 0 ! X £B B0 such that X 0 ! B0 is semistable; i.e., for any closed point x 0 2 X 0 and b0 D f 0.x 0/ 2 B0 one can find formal coordinates x1; : : : ; xn at x 0 and t1; : : : ; tm at b so that the morphism f is given by ti D Yli jDli¡1C1 xj for some 0 D 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 D .j1j; .f¾; N¾ g/ consisting of a finite collection of lattices N¾ D» Zn and rational full cones ¾ ½ N¾ ­ R with a vertex. The cones ¾ are glued together to form the space j1j so that the usual axioms of polyhedral complexes hold: 1. If ¾ 2 1 is a cone, then every face ¾ 0 of ¾ is also in 1, and N¾ 0 D N¾ jSpan.¾ 0/. 2. The intersection of two cones ¾1 \ ¾2 is a face of both of them. A morphism f1: 1X ! 1B of polyhedral complexes 1X D .j1X j; f¾; N¾ g/ and 1B D .j1B j; f¿; N¿ g/ is a compatible collection of linear maps f¾ : .¾; N¾ / ! .¿; N¿ /; i.e., if ¾ 0 is a face of ¾ , then f¾ 0 is the restriction of f¾ . We only consider morphisms f : 1X ! 1B such that f¾¡1.0/ \ ¾ D f0g for all ¾ 2 1X . 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 UX ½ X is said to be toroidal if it is locally formally isomorphic to a torus embedding T ½ X¾ ; 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 f1: 1X ! 1B such that f ¡1.0/ D f0g is 1. 1X and 1B are nonsingular. 2. For any cone ¾ 2 1X , we have f .¾ / 2 1B and f .N¾ / D N f .¾ /: We say that f is weakly semistable if it satisfies the two conditions except that 1X may be singular. The following two operations are allowed on 1X and 1B : 1. Projective subdivisions 10X of 1X and 10B of 1B such that f induces a morphism f 0: 10X ! 10B . 2. Lattice alterations: let 10X D .j1X j; f¾; N¾0 g/; 10B D .j1B j; f¿; N¿0 g/, for some compatible collection of sublattices N¿0 ½ N¿ , N¾0 D f ¡1.N¿0 / \ N¾ , and let f 0: 10X ! 10B be the morphism induced by f . Conjecture 1.5 (Combinatorial Semistable Reduction). Given a surjective morphism f : 1X ! 1B , such that f ¡1.0/ D f0g, there exists a projective subdivision f 0: 10X ! 10B followed by a lattice alteration f 00: 10X0 ! 10B0 so that f 00 is semistable. 1X00 # f 00 1B00 ! ! 1X0 # f 0 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 f1: 1X ! 1B . Then one checks that if f1 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 f1: 1X ! 1B have geometric analogues for f : X ! B. Indeed, subdivisions of 1X and 1B correspond to birational morphisms (see [KKMS]), and a lattice alteration corresponds to a finite base change (see [AK]). In the case when dim.1B / D 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 Theorem 1.6. Conjecture 1.5 is true if f1 has relative dimension · 3. Hence, Conjecture 1.3 is true if f has relative dimension · 3. The relative dimension of a linear map f¾ : ¾ ! ¿ of cones ¾; ¿ is dim.¾ / ¡ dim. f .¾ //. The relative dimension of f1: 1X ! 1B is by definition the maximum of the relative dimensions of f¾ : ¾ ! ¿ over all ¾ 2 1X . To see that the second statement of the theorem follows from the first, consider a surjective morphism of affine toric varieties f : X¾ ! X¿ defined by a linear map of cones and lattices f1: .¾; N¾ / ! .¿; N¿ /. A general fiber of this morphism has dimension equal to the rank of the kernel of f1: N¾ ! N¿ , and this is at least the relative dimension of f1: ¾ ! ¿ . Therefore, if a toroidal morphism f : X ! B has relative dimension · d, then the associated morphism of polyhedral complexes f1: 1X ! 1B 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 1B without increasing the multiplicity of 1X . In Section 3 we modify the barycentric subdivision of 1X so that we get a morphism to the barycentric subdivision of 1B . It is shown in Section 4 that in certain situations we can choose a modified barycentric subdivision that decreases the multiplicity of 1X . 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 ¾ 2 N ­R we write ¾ D hv1; : : : ; vni if the points v1; : : : ; vn lie on the one-dimensional edges of ¾ and generate the cone. If vi are the first lattice points along the edges we call them primitive points of ¾ . An n-dimensional cone is simplicial if it has exactly n primitive points. For a simplicial cone ¾ with primitive points v1; : : : ; vn, the multiplicity of ¾ is m.¾; N¾ / D [N¾ : Zv1 © ¢ ¢ ¢ © Zvn]: A polyhedral complex 1 is nonsingular if and only if m.¾; N¾ / D 1 for all ¾ 2 1. To compute the multiplicity of ¾ we can count the representatives w 2 N¾ of classes of N¾ =Zv1 © : : : © Zvn of the form w D X ®i vi ; i 0 · ®i < 1: The set of all such points is denoted by W .¾ /. For cones ¾1; ¾2 2 1 we write ¾1 · ¾2 if ¾1 is a face of ¾2. Notice that if ¾1 · ¾2, then the multiplicity of ¾1 is at most the multiplicity of ¾2. Hence, to compute the multiplicity of a polyhedral complex 1, it suffices to consider maximal cones only. Let f1: 1X ! 1B be a morphism of polyhedral complexes, and assume that 1B is simplicial. Let u1; : : : ; um be the primitive points of 1B , and let M1; : : : ; Mm be positive integers. By taking the .M1; : : : ; Mm /-sublattice at u1; : : : ; um we mean the lattice alteration N¿0 D Zfmi1 ui1 ; : : : ; mil uil g for all cones ¿ 2 1B with primitive points ui1 ; : : : ; uil . A subdivision 10 of 1 is called projective if there exists a homogeneous continuous piecewise linear function à : j1j ! R, convex on each cone ¾ 2 1, and taking rational values on the lattice points N¾ such that the maximal cones of 10 are exactly the maximal pieces in which à is linear. 2.2. Applying the Result of [KKMS] Let ¾1 ½ Rn1 and ¾2 ½ Rn2 be two cones. We consider ¾1 £ ¾2 as a cone in Rn1Cn2 . If f¾1;® g® is a subdivision of ¾1, and f¾2;¯ g¯ is a subdivision of ¾2, then f¾1;® £ ¾2;¯ g®;¯ gives us a subdivision of ¾1 £ ¾2. If 1X and 1B are simplicial, we say that f : 1X ! 1B is simplicial if f .¾ / 2 1B for all ¾ 2 1X . Assume that f1: 1X ! 1B is a simplicial map of simplicial complexes. Let ui , i D 1; : : : ; m, be the primitive points of 1B , and let vi j , i D 1; : : : ; m, j D 1; : : : ; Ji , be the primitive points of 1X such that vi j is mapped to an integer multiple of ui . For each i D 1; : : : ; m we denote by 1X;i the subcomplex of 1X lying over the cone hui i of 1B : 1X;i D f1¡1.hui i/: Note that if we forget the lattices of 1X , then by the assumption that f1¡1.0/ D f0g we get that 1X D 1X;1 £ ¢ ¢ ¢ £ 1X;m . If 10X;i are subdivisions of 1X;i , we get a subdivison 10X of 1X by taking the product 10X D 10X;1 £ ¢ ¢ ¢ £ 10X;m : Lemma 2.1. If 10X;i are projective subdivisions of 1X;i , then 10X is a projective subdivision of 1X . Proof. Let Ãi be a convex piecewise linear function defining the subdivison j10X;i j. Extend Ãi linearly to the entire j1X j by setting Ãi .j1X; j j/ D 0 for j 6D i . Clearly, à D Pi Ãi is a convex piecewise linear function defining the subdivision 10X . Consider the restriction f1j1X;i : 1X;i ! RCui . By the Main Theorem of Chapter 2 in [KKMS] there exist a subdivision 10X;i of 1X;i and a positive integer Mi such that after taking the Mi -sublattice at ui the induced morphism f 10j10X;i is semistable. We let 10X be the product of the subdivisions 10X;i , and we take the .M1; : : : ; Mm /-sublattice at .u1; : : : ; un /. Then f 10: 10X ! 10B is a simplicial map and f 10j10X;i is semistable for all i . Lemma 2.2. The multiplicity of 10X is not greater than the multiplicity of 1X . Proof. Let ¾ 2 1X have primitive points vi j and let ¾ 0 ½ ¾ be a maximal cone in the subdivision with primitive points vi0j . The multiplicity of ¾ 0 is the number of points in W .¾ 0/. We show that W .¾ 0/ can be mapped injectively to W .¾ /, hence the multiplicity of ¾ 0 is not greater than the multiplicity of ¾ . If w0 2 W .¾ 0/, we write w0 D X.¯i j C bi j /vi j ; i; j 0 · ¯i j < 1; bi j 2 ZC: Then w D Pi j ¯i j vi j 2 N¾ is in W .¾ /. If two points w10; w20 2 W .¾ 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 vi0j because Zfvi0j gi; j D Zfvi j gi; j \ N¾ 0 . Hence w10 ¡ w20 D 0. 3. Modified Barycentric Subdivisions Let f1: 1X ! 1B be a simplicial morphism. Consider the barycentric subdivision BS.1B / of 1B . The one-dimensional cones of BS.1B / are of the form RC¿O where ¿O D P ui is the barycenter of a cone ¿ 2 1B with primitive points u1; : : : ; um . A cone ¿ 0 2 BS.1B / is spanned by ¿O1; : : : ; ¿Ok , where ¿1 · ¿2 · ¢ ¢ ¢ · ¿k is a chain of cones in 1B . In general, f1 does not induce a morphism BS.1X / ! BS.1B /. For example, if ¾ D hv11; v12; v21i, ¿ D hu1; u2i, and f1: vi j 7! ui , then f1 does not induce a morphism of barycentric subdivisions. To get a morphism we need to modify the barycenters ¾O of cones ¾ 2 1X so that they map to (multiples of) barycenters of 1B . Definition 3.1. The data of modified barycenters consists of: 1. A subset of cones 1QX ½ 1X . 2. For each cone ¾ 2 1QX a lattice point b¾ 2 int.¾ / \ N¾ such that f1.b¾ / 2 RC¿O for some ¿ 2 1B . Recall that for any total order Á on the set of cones in 1X refining the partial order ·, the barycentric subdivision BS.1X / can be realized as a sequence of star subdivisions at the barycenters ¾O for all cones ¾ 2 1X in the descending order Á. Definition 3.2. Given modified barycenters .1Q X ; fb¾ g/ and a total order Á on 1X refining the partial order ·, the modified barycentric subdivision MBS1Q X ;fb¾ g;Á.1X / is the sequence of star subdivisions at b¾ for all ¾ 2 1QX in the descending order Á. Example 3.3. Let f1: hv11; v12; v21i ! hu1; u2i be the morphism defined by f1: vi j 7! ui . Let 1QX consist of the two cones 1Q X D fhv11; v21i; hv12; v21ig, and let the modified barycenters be fb¾ g D fv11 C v21; v12 C v21g. Depending on whether hv11; v21i Á hv12; v21i or vice versa, we get two modified barycentric subdivisions as shown in Fig. 1. To simplify notation, we write MBS1Q X .1X / or simply MBS.1X / instead of MBS 1QX ;fb¾ g;Á.1X /. By definition, MBS.1X / is a projective simplicial subdivision of 1X . Next, we show that, as in the case of the ordinary barycentric subdivision, the cones of MBS.1X / can be characterized by chains of cones in 1X . We may assume that the zero- and one-dimensional cones of 1X are all in 1Q X , and they precede all other cones in the order Á. For a cone ¾ 2 1X let ¾Q be the maximal face of ¾ (with respect to Á) in 1QX . Given a chain of cones ¾1 · ¢ ¢ ¢ · ¾k in 1X , the cone hb¾Q1 ; : : : ; b¾Qk i is a subcone of ¾k . Let C .1X / be the set of all such cones corresponding to chains ¾1 · ¢ ¢ ¢ · ¾k in 1X . Proposition 3.4. C .1X / D MBS.1X /. Proof. We do induction on the number of cones in 1Q X of dimension at least 2. If 1Q X contains only zero- or one-dimensional cones, then the statement is trivial. So, assume that 1QX D 1QX;0 [ f¾0g, where ¾ Á ¾0 for any ¾ 2 1QX;0, and assume that the proposition is proved for 1QX;0. Without loss of generality we may assume that 1X consists of cones containing ¾0 and their faces only. We get MBS1Q X .1X / from 1X if we first subdivide at b¾0 and then at b¾ for ¾ 2 1QX;0 in the descending order Á. If 1X;0 is the subcomplex of 1X consisting of cones not containing ¾0, then the star subdivision of 1X at b¾0 is 1X;0 £ hb¾0 i. Since ¾0 is greater than any ¾ 2 1QX;0 with respect to Á, all b¾ 2 1X;0, and we see that MBS1Q X .1X / D MBS 1QX;0 .1X;0/ £ hb¾0 i: A cone in MBS 1QX;0 .1X;0/ £ hb¾0 i is of the form ¾ £ ½, where ½ is a face of hb¾0 i, i.e. either f0g or hb¾0 i itself, and where ¾ is a cone in MBS 1QX;0 .1X;0/. Applying induction hypothesis to MBS 1QX;0 .1X;0/, we get that ¾ D hb¾Q1 ; : : : ; b¾Ql i for a chain of cones ¾1 · ¢ ¢ ¢ · ¾l in 1X;0. Now if ½ D f0g, then ¾ £ ½ D hb¾Q1 ; : : : ; b¾Ql i 2 C .1X /. If ½ D hb¾0 i, we let ¾lC1 be a cone in 1X that contains both ¾l and ¾0. Then ¾QlC1 D ¾0, and ¾ £ ½ D hb¾Q1 ; : : : ; b¾Ql ; b¾QlC1 i 2 C .1X /. Conversely, let hb¾Q1 ; : : : ; b¾Ql i be a cone in C .1X / for some chain ¾1 · ¢ ¢ ¢ · ¾l in 1X . Then for some k · l we have that ¾Q1; : : : ; ¾Qk 2 1QX;0, and ¾QkC1 D ¢ ¢ ¢ D ¾Ql D ¾0. By induction hypothesis, the cone hb¾Q1 ; : : : ; b¾Qk i coming from the chain ¾1 · ¢ ¢ ¢ · ¾k in 1X;0 is in MBS1Q X;0 .1X;0/. Hence the cone hb¾Q1 ; : : : ; b¾Ql i is of the form ¾ £ ½, where ¾ D hb¾Q1 ; : : : ; b¾Qk i 2 MBS 1QX;0 .1X;0/, and ½ D hb¾0 i if k < l, and ½ D f0g if k D l. Corollary 3.5. Assume that f1: 1X ! 1B is a simplicial morphism. If f1.¾Q / D f1.¾ / for all ¾ 2 1X , then f1 induces a simplicial morphism f 10: MBS.1X / ! B S.1B /. Proof. Let ¾ 0 2 MBS.1X / correspond to a chain ¾1 · ¢ ¢ ¢ · ¾k in 1X . Since f1 is simplicial, we have a chain of cones f1.¾1/ · ¢ ¢ ¢ · f1.¾k / in 1B . Recall that b¾Qi is mapped to a multiple of a barycenter: f1.b¾Qi / D RC¿O for some ¿ 2 1B . The assumption that f1.¾Qi / D f1.¾i / implies that f1.b¾Qi / 2 RC \f1.¾i /, hence the cone hb¾Q1 ; : : : ; b¾Qk i is mapped onto the cone h f\1.¾1/; : : : ; f\1.¾k /i 2 BS.1B /. Example 3.6. Assume that f1: 1X ! 1B is a simplicial morphism taking primitive points of 1X to primitive points of 1B . Then for a cone ¾ 2 1X such that f1: ¾ !' ¿ for some ¿ 2 1B , we have f1.¾O / D ¿O. Let 1X D f¾ 2 1X : f1j¾ is injectiveg, b¾ D ¾O . Clearly, the hypothesis of the lemma is satisfied, and we have a simplicial morphism f 10: MBS1X .1X / ! BS.1B /. Conversely, if . 1QX ; fb¾ g/ is the data of modified barycenters such that f1 induces a morphism f 10: MBS1Q X .1X / ! BS.1B /, then 1X ½ 1QX . Thus, we may always assume that 1X ½ 1QX . 4. Reducing the Multiplicity of 1X Proposition 4.1. Let f1: 1X ! 1B be a simplicial morphism taking primitive points to primitive points. Assume that 1B is nonsingular, 1X is singular, and every singular cone ¾ 2 1X contains a point w 2 W .¾ /nf0g mapping to a barycenter in 1B . Then there exists a modified barycentric subdivision MBS.1X / of 1X having multiplicity strictly less than the multiplicity of 1X such that f1 induces a simplicial morphism f 10: MBS.1X / ! BS.1B /. such that f1: ¾0 !' ¿0. Set w¾ D w C ¾O0; then Proof. For every singular cone ¾ 2 1X we choose a point w¾ as follows. By assumption, there exists a point w 2 W .¾ /nf0g mapping to a barycenter of 1B : f1.w/ D ¿O. Then for a unique cone ¿0 2 1B we have f1.¾ / D ¿ £ ¿0. We choose a face ¾0 · ¾ f1.w¾ / D f1.w/ C f1.¾O0/ D ¿O C ¿O0 D \f1.¾ /: Having chosen the set fw¾ g, we may remove some of the points w¾ if necessary so that every simplex ½ 2 1X contains at most one w¾ in its interior. With 1X as in Example 3.6, let 1QX D 1X [ f½ 2 1X jw¾ 2 int .½/ for some singular ¾ g, b½ D ½O if ½ 2 1X , and b½ D w¾ if w¾ 2 int.½/. Next we specify the order Á. We refine the partial order · as follows: for two faces ¾1 and ¾2 of a cone ¾ 2 1X we set ¾1 Á0 ¾2 if dim f1.¾1/ < dim f1.¾2/. Since 1X ½ 1QX , this ensures that the condition f1.¾Q / D f1.¾ / of Corollary 3.5 is satisfied for any refinement of Á0. Now if ¾ is singular, then the point w¾ constructed above lies in the interior of a face ½¾ such that f1.½¾ / D f1.¾ /. We further refine the order Á0 by setting ¾1 Á0 ½¾ for any face ¾1 of the singular cone ¾ . Then ¾Q D ½¾ whenever ¾ is singular. Finally we let Á be any refinement of Á0 to a total order. Let ¾ 2 1X be a cone, and let a maximal cone ¾ 0 2 MBS.1X / be given by a maximal chain of faces of ¾ : ¾1 · ¢ ¢ ¢ · ¾n. We have to show that m.¾ 0; N¾ 0 / · m.¾; N¾ /, and if ¾ is singular, then the inequality is strict. We can order the primitive points v1; : : : ; vn of ¾ so that ¾1 D hv1i; ¾2 D hv1; v2i; : : : ; ¾n D hv1; : : : ; vni. Since b¾Qi 2 ¾i , the primitive points of ¾ 0 D hb¾Q1 ; : : : ; b¾Qn i can be written as for some ai j ¸ 0 and integers ¹i ¸ 1. The multiplicity of ¾ 0 is a11 ¢ a22 ¢ ¢ ¢ ann times the multiplicity of ¾ . By the choice of b½ above, all aii · 1, hence m.¾ 0; N¾ 0 / · m.¾; N¾ /. If ¾ is singular, let i be the smallest index such that the face ¾i is singular. Then, with notation as above, b¾Qi D w C ¾O0 for some w 2 W .¾i /nf0g, and ¾0 · ¾i . Now if aii D 1, then w 2 hv1; : : : ; vi¡1i, and this gives a contradiction with the choice of i . Hence aii < 1 and m.¾ 0; N¾ 0 / < m.¾; N¾ /. 5. Families of Surfaces and Threefolds Proof of Theorem 1.6. Let f1: 1X ! 1B be a surjective morphism of polyhedral complexes such that f1¡1.0/ D f0g. It is shown in [AK] that there exist projective simplicial subdivisions 10X of 1X and 10B of 1B such that 1B is nonsingular and f1 induces a simplicial morphism f 10: 10X ! 10B . To obtain these subdivisions, one first subdivides 1B such that the image of every cone in 1X is a union of cones in 10B . The convex piecewise linear function defining the subdivision 10B can then be composed with f1 to give a subdivision of 1X . A sequence of star subdivisions centered at the onedimensional edges yields the required simplicial subdivision 10X . Thus, we may assume that 1X is simplicial, 1B is nonsingular, and f1: 1X ! 1B is a simplicial map. Applying the construction of [KKMS] over the edges of 1B (Section 2.2), we can make f1j1X;i semistable without increasing the multiplicity of 1X . We show below that every singular simplex ¾ 2 1X contains a point w 2 W .¾ /nf0g mapping to a barycenter of 1B . By Proposition 4.1, there exists a modified barycentric subdivision such that f1 induces a simplicial morphism f 10: MBS.1X / ! BS.1B /, with multiplicity of MBS.1X / strictly less than the multiplicity of 1X . Since f 10 is simplicial and BS.1B / nonsingular, the proof is completed by induction on the multiplicity of 1X . Consider the restriction of f1 to a singular simplex f1: ¾ ! ¿ , where ¿ has primitive points u1; : : : ; um , ¾ has primitive points vi j ; i D 1; : : : ; m; j D 1; : : : ; Ji , and f1.vi j / D ui . Since ¾ is singular, it contains a point w 2 W .¾ /nf0g: w D X ®i j vi j ; 0 · ®i j < 1; X ®i j > 0: Considering a face of ¾ if necessary, we may assume that w lies in the interior of ¾ , hence 0 < ®i j . Since f1.w/ 2 N¿ , it follows that Pj ®i j 2 Z for all i . In particular, if Ji0 D 1 for some i0, then ®i01 D 0, and w lies in a face of ¾ . So we may assume that Ji > 1 for all i . Since the relative dimension of f1 is Pi . Ji ¡ 1/, we have to consider all possible decompositions Pi . Ji ¡ 1/ · 3, where Ji > 1 for all i . The cases when the relative dimension of f1 is 0 or 1 are trivial and left to the reader. If the relative dimension of f1 is 2, then either J1 D 3 or J1 D J2 D 2. In the first case we have that hv11; v12; v13i is singular, contradicting the semistability of f1j1X;1 . In the second case, ®11 C ®12; ®21 C ®22 2 Z and 0 < ®i j < 1 imply that ®11 C ®12 D ®21 C ®22 D 1. Hence f1.w/ D u1 C u2 is a barycenter. In relative dimension 3, either J1 D 4, or J1 D 3; J2 D 2, or J1 D J2 D J3 D 2. In the first case we get a contradiction with the semistability of f1j1X;1 ; the third case gives ®11 C ®12 D ®21 C ®22 D ®31 C ®32 D 1 as for relative dimension 2. In the second case, either ®11 C ®12 C ®13 D ®21 C ®22 D 1 and w maps to a barycenter, or ®11 C ®12 C ®13 D 2; ®21 C ®22 D 1 and .P 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 ¿ D hu1; u2i and ¾ D hv11; v12; v13; v14; v21; v22i, with lattices N¿ D Zfu1; u2g and N¾ D Zfv11; : : : ; v22; 12 .v11 C ¢ ¢ ¢ C v22/g. Then W .¾ /nf0g consists of a single point w D 2 1 .v11 C ¢ ¢ ¢ C v22/, and if f1 maps vi j to ui , then w is mapped to 2u1 C u2, 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. [AK] D. Abramovich and K. Karu , Weak semistable reduction in characteristic 0, preprint . alg- geom=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 .


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K. Karu. Semistable Reduction in Characteristic Zero for Families of Surfaces and Threefolds, Discrete & Computational Geometry, 2000, 111-120, DOI: 10.1007/PL00009484