GEOMETRY OF CHOW QUOTIENTS OF GRASSMANNIANStevelev/chow.pdf · GEOMETRY OF CHOW QUOTIENTS OF...

GEOMETRY OF CHOW QUOTIENTSOF GRASSMANNIANS

SEAN KEEL and JENIA TEVELEV

AbstractWe consider Kapranov’s Chow quotient compactification of the moduli space of or-dered n-tuples of hyperplanes in P r−1 in linear general position. For r = 2, this iscanonically identified with the Grothendieck-Knudsen compactification of M0,n whichhas, among others, the following nice properties:(1) modular meaning: stable pointed rational curves;(2) canonical description of limits of one-parameter degenerations;(3) natural Mori theoretic meaning: log-canonical compactification.We generalize (1) and (2) to all (r, n), but we show that (3), which we view as thedeepest, fails except possibly in the cases (2, n), (3, 6), (3, 7), (3, 8), where we con-jecture that it holds.

Contents1. Introduction and statement of results . . . . . . . . . . . . . . . . . . . . 2592. Various toric quotients of the Grassmannian . . . . . . . . . . . . . . . . 2633. Singularities of (X(r, n), B) . . . . . . . . . . . . . . . . . . . . . . . . 2734. The membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2835. Deligne schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2876. Global generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2957. Bubble space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3018. Limit variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3039. The bundle of relative log differentials . . . . . . . . . . . . . . . . . . . 308References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

1. Introduction and statement of resultsLet P(r, n) be the space of ordered n-tuples of hyperplanes in Pr−1. Let P◦(r, n) bethe open subset that is in linear general position. PGLr acts freely on P◦(r, n) with thequotient X(r, n). The space X(2, n) is usually denoted M0,n and has the Grothendieck-Knudsen compactification M0,n with many remarkable properties.

DUKE MATHEMATICAL JOURNALVol. 134, No. 2, c© 2006Received 20 August 2004. Revision received 17 November 2005.2000 Mathematics Subject Classification. Primary 14E; Secondary 14D, 52C35.Keel’s work partially supported by National Science Foundation grant DMS-9988874.

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260 KEEL and TEVELEV

1.1. Properties (of M0,n ⊂ M0,n)(1) M0,n has a natural moduli interpretation; namely, it is the moduli space of

stable n-pointed rational curves.(2) M0,n ⊂ M0,n is canonically associated to the open variety M0,n; namely, it is

the log canonical compactification (see [KM, Lemma 3.6]). In particular, thepair (M0,n, ∂M0,n) has log canonical singularities.

(3) Given power series f1(z), . . . , fn(z) in one variable which we think of as a one-parameter family in M0,n, one can ask: What is the limiting stable n-pointedrational curve in M0,n as z → 0? There is a beautiful answer, due to Kapranov[K2], in terms of the Bruhat-Tits tree for PGL2.

Question 1.2Is there a compactification X(r, n) ⊂ X(r, n) which satisfies any or all of the propertiesof Section 1.1?

In this article we consider Kapranov’s Chow quotient compactification X(r, n), intro-duced in [K1]. This carries a flat family of pairs of schemes with boundary

p : (S, B) → X(r, n), (1.2.1)

the so-called family of visible contours, generalising the universal family over M0,n.Lafforgue in [L2] gave a precise description of the fibres (S, B), showing in particularthat each pair has toroidal singularities.

1.3. Moduli interpretation as in Section 1.1(1)X(r, n) is a natural moduli space of semi log canonical pairs (the higher-dimensionalMori-theoretic generalisation of stable pointed curves). This is a recent result ofHacking [H]. We observed the same result independently. Together with Hacking, wegive a proof in [HKT].

1.4. Log canonical modelThe initial motivation for this article was the elementary observation (see Proposi-tion 2.18) that X(r, n) is minimal of log general type and thus (assuming generalconjectures of Mori theory) has a log canonical compactification. Unfortunately, thisis not, in general, the Chow quotient:

Let X(r, n) → X(r, n) be the normalisation.

THEOREM 1.5X(3, n) with its boundary fails to be log canonical for n ≥ 9 (for n ≥ 7 in characteristic2). X(4, n) is not log canonical for n ≥ 8.

GEOMETRY OF CHOW QUOTIENTS OF GRASSMANNIANS 261

We give a proof in §3. In the positive direction we speculate that the two compactifi-cations agree in the remaining cases.

CONJECTURE 1.6X(r, n) is the log canonical model of X(r, n) in the cases

(2, n), (3, 6), (3, 7), (3, 8)

and those obtained by the canonical duality X(r, n) = X(n − r, n) (see [K1, Theo-rem 2.3]). Moreover, in these cases the pair (X(r, n), B) has toroidal singularities.

For (2, n), the conjecture is true by Section 1.1(2). Together with Hacking, we haveproved it for (3, 6) using Theorem 2.19, and we expect the remaining two cases tofollow in the same way.

Question 1.7What is the log canonical compactification X(r, n) ⊂ Xlc(r, n)?

We believe that this compactification is of compelling interest, as it gives a birationalmodel with reasonable boundary singularities of a compactification of X(r, n) whoseboundary components meet in absolutely arbitrary ways.

1.8. Mnev’s universality theoremThe boundary

P(r, n)\P◦(r, n)

is a union of(n

r

)Weil divisors. The components have only mild singularities; however,

they meet in very complicated ways. Let Y be an affine scheme of finite type overSpec Z. By [L2, Theorem 1.8], there are integers n, m and an open subset

U ⊂ Y × Am

such that the projection U → Y is surjective, and U is isomorphic to a boundarystratum of P(3, n). (A boundary stratum for a divisorial boundary means the locallyclosed subset of points that lie in each of a prescribed subset of the irreduciblecomponents, but no others.)

We prove in Theorem 3.13 that singularities of the pair (X(3, n), B) are also,in general, arbitrary. Now by Proposition 2.18 and the finite generation conjectureof Mori theory, X(3, n) ⊂ Xlc(3, n) gives a canonical compactification in which theboundary has log canonical singularities. We do not know whether Xlc(3, n) maps toX(r, n); if it does, Xlc would give an absolutely canonical way of (partially) resolvinga boundary whose strata include all possible singularities.


1.9. Realization via Bruhat-Tits buildings as in Section 1.1(3)Throughout the article k is a fixed algebraically closed field. Let R = k[[z]], and letK be its quotient field. V = kr and VT = V ⊗k T for a k-algebra T . We write P forprojective spaces of quotients (or, equivalently, hyperplanes), P for spaces of lines.

Begin with a collection

F := {f1, . . . , fn} ⊂ VK, (1.9.1)

any r of which are linearly independent, and thus give a K-point of X(r, n). We thinkof F as the equations of n one-parameter families in Pr−1. Following [K2] (where thecase r = 2 is treated), we ask the following.

Question 1.10What is the limit as z → 0; that is, in the pullback of the family (1.2.1) along theassociated R-point of X(r, n), what is the special fibre?

We give a canonical solution, in terms of the Bruhat-Tits building B. Proofs andfurther related results are given in §§4 – 7. Recall that B is a set of equivalence classesof R-lattices in VK . We assume that the reader is familiar with basic notions associatedwith B, all of which are reviewed in §4.

Definition 1.11For a finite Y ⊂ B, let SY be the join of the projective bundles P(M), [M] ∈ Y ; thatis, fix one, and take the closure of the graph of the product of the birational mapsfrom this projective bundle to all the others. Let SY ⊂ SY be the special fibre ofp : SY → Spec(R).

By [M, Theorem 2.2], if Y is convex, then p is semistable (i.e., SY is nonsingular) andSY has smooth irreducible components and normal crossings. Following [F], we referto SY as the Deligne scheme. It represents a natural functor (see Theorem 5.2).

Definition 1.12We define the membrane [F] ⊂ B to be classes of lattices M which have a basisgiven by scalar multiples of some r elements from F.

For a lattice M and a nontrivial element f ∈ VK , there is a unique a > 0 suchthat zaf ∈ M\zM . We define f M := zaf ⊂ M and define f M its image underM → M := M/zM . For a subset � ⊂ VK , let �M := {f M | f ∈ �}.

We call [M] stable if FM contains r + 1 elements in linear general position. LetStab ⊂ [F] be the set of stable classes. Stab is finite (see Lemma 5.19). Let Y ⊂ B beany finite convex subset containing Stab (e.g., the convex hull [Stab], which is finite;


see Lemma 4.8). Let Bi ⊂ SY be the closure of the hyperplane

{fi = 0} ⊂ P(VK ) ⊂ SY

on the generic fibre of p : SY → Spec(R). Let B = ∑Bi . The pair (SY , B) represents

natural functors (see Definition-Lemma 5.21).

THEOREM 1.13SY , Bi are nonsingular, and the divisor SY + B has normal crossings. The 1-formsdlog(fi/fj ) define globally generating sections of the vector bundle �1

S/R(log B). Theimage of the associated map to the Grassmannian

SY → Spec(R) × G(r − 1, n − 1)

is S → Spec(R), the pullback of the family (1.2.1) along the R-point of X(r, n)defined by F. In particular, the relative log canonical bundle KSY

+ B is relativelyglobally generated and big, and SY → S is the relative minimal model and is crepant.

For the definition and basic properties of bundles of relative log differentials, see §9.We note that the crepant semistable model (SY , B) is in many ways preferable to itsminimal model S → Spec(R). For example, dropping the last hyperplane induces anatural regular birational map

S[Stab(F)] → S[Stab(F′)]

for F′ = F\{fn}, but for r ≥ 3 the associated rational map between minimal modelsis not, in general, regular. There are examples where regularity fails already with(r, n) = (3, 5).

The special fibre SY can be read off directly from Y ⊂ [F]. There is onecomponent P(M), a certain blowup of the projective space P(M), for each [M] ∈Y , glued in a canonical way given by the simplicial complex structure of B (seeRemark 5.14). The semistable model (SY , B) depends on the choice of the finiteconvex subset Y ⊂ [F], but there is a canonical infinite version, with no boundary,which dominates all finite SY (see §7).

1.14. Relation to tropical geometryThere are several connections between this work and tropical geometry. For example,[F] is naturally homeomorphic to the tropicalisation of the r-dimensional subspacein Kn defined by the rows of the matrix with columns fi (see Theorem 4.11).

2. Various toric quotients of the GrassmannianIn this section we fix notation and provide a background on quotients of Grassmanians,due mostly to Kapranov and Lafforgue. The new results are Theorem 2.3, Lemma 2.4,Theorem 2.17, Proposition 2.18, and Theorem 2.19.


2.1. Chow quotients (see [KSZ], [GKZ])Let H be an algebraic group acting on a projective variety P (which in our applicationsis a projective space, whence the choice of notation). Let P0 ⊂ P be a (sufficiently)generic open H -invariant subset. There is a natural map to the Chow variety,

P0/H → Chow(P), x �→ Hx.

The Chow quotient P//H is defined to be the closure of the image of this map.We begin by describing this in the case where H is an algebraic torus acting on a

projective space P. Let P ⊂ XR be a convex polytope with vertices in X, the characterlattice of H . We denote vertices of P by the same letter. Let V be the k-vector spacewith a basis {zp |p ∈ P }. The torus H acts on V by the formula h · zp = p(h)zp.Let P := P(V ). For any S ⊂ P, let Supp(S) ⊂ P be the set of coordinates that do notvanish on S. Let P0 = {p ∈ P | Supp(p) = P }.

Take the big torus H = GPm with its obvious action on V . We can assume, without

loss of generality, that H ⊂ H. This is equivalent to 〈P 〉Z = X, where for any S ⊂ X

we denote by 〈S〉Z the minimal sublattice containing S. P//H is a projective H-toric variety with open orbit P0/H ⊂ P//H and canonical H-equivariant Chowpolarisation. By a toric variety, we mean a variety with an action of a torus havinga dense open orbit. We do not assume that the action is effective or that the varietyis normal. The fan of the normalisation � : P//nH → P//H can be described asfollows. A triangulation T of P (with all vertices in the set of vertices of P ) is calledcoherent if there exists a concave piecewise affine function on P whose domainsof affinity are precisely maximal simplices of T . It gives rise to a polyhedral coneC(T ) ⊂ RP of the maximal dimension which consists of all functions ψ : P → R

such that ψT : P → R is concave, where ψT is given by affinely interpolating ψ

inside each simplex of T . Cones C(T ) (and their faces) for various T give a completefan F(P ). Lower-dimensional faces of F(P ) correspond to (coherent) polyhedraldecompositions P of P . More precisely, C(P ) is the set of concave functions affineon each polytope of P .

On occasion, we abuse notation and refer to the collection of maximal-dimensionalpolytopes of a polyhedral decomposition as a polyhedral decomposition itself. We havethe orbit decomposition

P//H =⊔P

(P//H )P

(and a similar one for P//nH ) indexed by polyhedral decompositions. A cycle X ∈(P//H )P is the union of toric orbits with multiplicities

X =∑P ′∈P

mP ′XP ′, Supp(XP ′) = P ′, mP ′ = [X : 〈P ′〉Z]. (2.1.1)


If mP ′ = 1 for any P ′ ∈ P , then we say that X is a broken toric variety. If

〈P ′′〉Z = X ∩ 〈P ′′〉Q

for any face P ′′ of a polytope P ′ ∈ P , then we call P unimodular.

Definition 2.2A decomposition P is called central if P = {C, S1, . . . , Sr}, where Si ∩Sj ⊂ ∂P . Wecall C the central polytope. Let UP ⊂ P//nH be an affine open toric subset with fanC(P ). It contains (P//nH )P as the only closed orbit.

THEOREM 2.3If P is unimodular and central, then �(UP ) is quasi-smooth (i.e., �|UP

is bijective,and UP is smooth).

ProofTo show that UP is smooth, we have to show that C(P ) = (Z≥0)r up to global affinefunctions. Let f ∈ C(P ). Then f is a concave locally affine function. Let f ′ bean affine function equal to f on C. So f − f ′ is a concave locally affine functionthat vanishes on C. Let fi , 1 ≤ i ≤ r , be a primitive (i.e., not divisible by aninteger) concave locally affine function that vanishes on P \Si . Then f −f ′ is a linearcombination of fi’s with nonnegative coefficients.

To show that �|UPis bijective, it suffices to prove that �|(P//nH )P is bijective.

Indeed, other strata in UP correspond to decompositions coarser than P which areautomatically unimodular and central, so we can use the same argument.

For a moment, let P be any unimodular decomposition. The following construc-tion is a variation of the Ishida complex (see [O]). Let P i be the set of i-codimensionalfaces of polytopes in P which do not belong to the boundary ∂P . Fix some orientationof each Q ∈ P i . Let A be an abelian group. Consider the complex C•

Aff(P , A)with Ci

Aff(P , A) = ⊕Q∈P i Aff(Q,A), where Aff(Q, A) is the group of affine maps

Q → A. The differential di : CiAff(P , A) → Ci+1

Aff (P , A) is a direct sum of differen-tials dQ,R for Q ∈ P i , R ∈ P i+1. If R is not a face of Q, then dQ,R = 0. Otherwise,dQ,R is the restriction map Aff(Q,A) → Aff(R, A) taken with a negative sign if thefixed orientation of R is opposite to the orientation induced from Q. Let H •

Aff(P , A)be the cohomology of C•

Aff(P , A). It is clear that H 0Aff(P , A) is the set of piecewise

affine functions P → A.

LEMMA 2.4If H 1

Aff(P , Z) = 0, then �|(P//nH )P is bijective.


ProofWe identify H with maps P → Gm. Elements of H of order N are maps P → µN ,and any map a : P → Z gives a one-parameter subgroup (1-PS) z �→ {p �→ za(p)}.

Let X ∈ (P//H )P be as in (2.1.1). Let x ∈ �−1(X). We claim that Hx → HX

is bijective. Since Hx ⊂ HX, it suffices to prove that the stabilizer HX is connected.Let h ∈ HX. Then h ∈ HXP ′ for any P ′ ∈ P . But if e is a generic point of XP ′ ,

then

HXP ′ = {h ∈ H |h · e ∈ XP ′ } = {h ∈ H | ∃hP ′ ∈ H, h · e = hP ′ · e}.

It follows that h(p) = hP ′(p), and hence h is affine on each P ′. We see that HX =H 0

Aff(P , Gm).It is enough to show that any element h ∈ HX of a finite order N embeds in a

1-PS γ ⊂ HX. So let h ∈ H 0Aff(P , µN ). We have the exact sequence

0 → C•Aff(P , Z)

·N−→ C•Aff(P , Z) → C•

Aff(P , µN ) → 0.

Since H 1Aff(P , Z) = 0, there exists an element of H 0

Aff(P , Z) which maps to h. Thecorresponding 1-PS contains h and belongs to the stabilizer HX. �

Back in our situation, it is clear that P 1 = {F1, . . . , Fp} is the set of codimension-onefaces of C which are not on the boundary of P . We want to use Lemma 2.4. Letc ∈ C1

Aff(P , Z), so c = (f1, . . . , fp), where fi is affine on Fi . For each i, we haveFi = C ∩Sj for some j . We can choose gj ∈ Aff(Sj , Z), which restricts on fi (takinginto account the orientation). Then c is equal to the differential of the cochain c, wherec(C) = 0 and c(Si) = gi . �

2.5. Matroid polytopes (see [GGMS], [L2])Let H = Gn

m be the torus acting on P = P( ∧r

kn). The weights ei1 + · · · + eir ∈ Rn

are the vertices of the hypersymplex

�(r, n) :={

(x1, . . . , xn) ∈ Rn∣∣∣

∑xi = r, 1 ≥ xi ≥ 0

}.

It has 2n faces {xi = 0} and {xi = 1}. The Plucker embedding G(r, n) ⊂ P induces aclosed embedding G(r, n)//H ⊂ P//H . For a subset I ⊂ N , we write xI = ∑

i∈I xi

and consider xI as a function on �(r, n); in particular, xI = r − xIc .A matroid C of rank r on the set N = {1, . . . , n} gives rise to a matroid polytope

P ⊂ �(r, n), a convex hull of vertices ei1 + · · · + eir for any base {i1, . . . , ir} ⊂ N . Itis known that P is defined by inequalities xI ≤ rank I . In almost all our applications,P = �(r, n), and so we adopt the following notational convention throughout thearticle. If we drop the polytope P from notation, it is assumed to be �(r, n) for a pair


(r, n) clear from the context. Let

PP = {x ∈ P

∣∣ Supp(x) ⊂ P}, PP,0 = {

x ∈ PP∣∣ Supp(x) = P

}.

In [L2, Theorem 2.1] Lafforgue defines a subfan of F(P ) whose cones are in one-to-one correspondence with matroid decompositions P of P (i.e., tilings of P bymatroid polytopes). This is a fan because a polyhedral decomposition coarser than amatroid decomposition is a matroid decomposition. (Moreover, if a convex polytopeQ ⊂ �(r, n) admits a tiling by matroid polytopes, then Q itself is a matroid polytope;see [L1].) The associated toric variety is denoted AP . Just by definition, AP is thetoric open subset in the normalisation of the Chow quotient:

AP ⊂ PP //nH. (2.5.1)

Orbits in AP correspond to matroid decompositions. Notice that the action of GPm on

AP is not effective: the kernel (GPm)∅ ⊂ GP

m is the subtorus of affine maps P → Gm.Let AP

∅ := GPm/(GP

m)∅.For any face Q of P , Lafforgue defines a toric face map AP → AQ. The

corresponding map of fans is given by the restriction of piecewise affine functionsfrom P to Q. In particular, the image of the orbit AP

P is AQ

P ′ , where P ′ is the matroiddecomposition of Q obtained by intersecting polytopes in P with Q.

Lafforgue introduces a second normal toric variety AP for the torus AP∅ :=

GPm/Gm and a map of toric varieties

AP → AP (2.5.2)

extending the natural quotient map AP∅ � AP

∅ . The torus orbits of AP are in one-to-one correspondence with (P ,P ′) for P , a matroid decomposition, and P ′ ∈ P , oneof the matroid polytopes. By [L2, Proposition IV.3], (2.5.2) is projective and flat, withgeometrically reduced fibres, and there exists an equivariant closed embedding

AP ⊂ AP × PP . (2.5.3)

The fibre of (2.5.2) over a closed point of APP = (PP //nH )P is a broken toric

variety (2.1.1) in PP . All multiplicities are equal to 1 because of the following funda-mental observation (see [GGMS]):

any matroid decomposition is unimodular. (2.5.4)

In fact, (2.5.2) is the pullback of the universal Chow family over the Chow quotientPP //H along the map AP ⊂ PP //nH → PP //H .

For each maximal face P ′ of P , the pair (∅, P ′), where ∅ denotes the trivialdecomposition ( just P and its faces), corresponds to an irreducible boundary divisor


of AP . Denote the union of these boundary divisors as BP ⊂ AP . In the case whereP = �(r, n), there are 2n such boundary divisors, corresponding to the maximal faces{xi = 0}, {xi = 1} of �(r, n). We indicate by Bi the divisor corresponding to {xi = 1}.Boundary divisors of AP induce boundary divisors B on fibres of (2.5.2) for eachmaximal face of P . For P = �(r, n), we write Bi for the divisor corresponding to Bi .

2.6. Realizable matroidsLet C = {Li}i∈N be a configuration of n hyperplanes in Pr−1. Then independentsubsets of the corresponding realizable matroid (denoted by the same letter C) aresubsets of linearly independent hyperplanes. C has rank r if there is at least oneindependent r-tuple. By the multiplicity of a point p ∈ Pr−1 with respect to C wemean the number of hyperplanes in C which contain p (i.e., the usual geometricmultp C if we view C as a divisor in Pr−1). Recall that a hyperplane arrangement iscalled connected if its group of automorphisms is trivial. It is well known that C is aconnected arrangement if and only if PC has a maximal dimension.

Let XC(r, n) be the corresponding moduli space (i.e., N-tuples of hyperplaneswith incidence as specified by C modulo PGLr). We recall the Gelfand-MacPhersoncorrespondence. Consider an (r × n)-matrix MC with columns given by equations ofhyperplanes of C (defined up to a scalar multiple). The row space of MC gives a pointx ∈ G(r, n) ⊂ P

( ∧rkn

). The convex hull of Supp(x) ⊂ �(r, n) is PC . The locally

closed subscheme

GP,0(r, n) = G(r, n) ∩ PP,0

is called a thin Schubert cell. Of course, G�(r,n),0(r, n) = G0(r, n). Thus XC(r, n) isidentified with the quotient of (the reduction of ) GPC,0(r, n)/H .

Next, we consider Lafforgue’s main object, �P

, which we consider only in thecase P = �(r, n). We use a different construction from his, as it is a quicker way ofdescribing the scheme structure: � is the subscheme of A over which the fibres of(2.5.2) are contained in G(r, n).

PROPOSITION 2.7The Lafforgue space � ⊂ A is ϕ−1

(Hilb(G(r, n))

), where ϕ : A → Hilb

(P( ∧r (kn)

))is the map induced by (2.5.3).

ProofAs Lafforgue pointed out to us, this follows from [L2, Theorems 4.4, 4.22]. �

2.8. Structure mapWe have the composition

� ⊂ A → A/A∅


(where the last map is the stack quotient), which Lafforgue calls the structure map.In particular, this endows � with a stratification by locally closed subschemes, �P

(the restriction of the corresponding toric stratum of A), parameterized by matroiddecompositions P of �(r, n). The Lafforgue stratum �P is empty if the matroiddecomposition P contains nonrealizable matroids. The stratum for the trivial decom-position, ∅ (meaning the only polytope is �(r, n)), is an open subset

�∅ = X(r, n) ⊂ �,

which Lafforgue calls the main stratum. Lafforgue proves that � is projective andthus gives a “compactification” of X(r, n)—quotes are used because his space is, ingeneral, reducible, as we observe in Proposition 3.10.

2.9We denote the closure of �∅ in � by XL(r, n). There are immersions

X(r, n) ⊂ XL(r, n) ⊂ A ⊂ P//nH

(the first and last open, the middle one closed) and

X(r, n) ⊂ X(r, n) ⊂ P//H

(open followed by closed). It follows that there exists a finite birational map

XL(r, n) → X(r, n). (2.9.1)

In particular, X(r, n) and XL(r, n) have the same normalisation X(r, n).

2.10. Toric familyWe denote the pullback of A → A to � by T → � (T denotes toric). By definition,T ⊂ � × G(r, n).

Kapranov [K1, Theorem 1.5.2] shows that the natural Chow family

T → X(r, n), T ⊂ X(r, n) × G(r, n),

is flat. The family T → XL(r, n) is the pullback of T → X(r, n) along (2.9.1).Let B, Bi ⊂ T be the restrictions of the boundary divisors B, Bi ⊂ A.

2.11. Family of visible contoursLet Ge(r −1, n−1) ⊂ G(r, n) be the subspace of r-planes containing the fixed vectore = (1, . . . , 1). Kapranov defines the family of visible contours

S = T ∩ (X(r, n) × Ge(r − 1, n − 1)

) ⊂ X(r, n) × G(r, n).


Kapranov shows that the family S is flat and that the associated map

X(r, n) → Hilb(Ge(r − 1, n − 1)

)(2.11.1)

is a closed embedding. There is a similar family over �. (Lafforgue calls it P(E).)

Definition 2.12Let S ⊂ T be the scheme-theoretic intersection

S := T ∩ [� × Ge(r − 1, n − 1)] ⊂ [� × G(r, n)].

H acts on A, trivially on A, and A → A is H -equivariant. Thus H acts on T (andtrivially on �), so that T → � is equivariant.

Let B, Bi ⊂ S indicate the restriction of B, Bi ⊂ T. We note that B ⊂ S isthe union of Bi , as the n components of B ⊂ T corresponding to the faces xi = 0 of�(r, n) are easily seen to be disjoint from Ge(r − 1, n − 1).

The fibres of (S, B) → � have singularities like (or better than) those of (T, B),as follows from the following transversality result.

PROPOSITION 2.13 (see [L2, page xv])The natural map S → T/H to the quotient stack (or, equivalently, S × H → T)is smooth.

ProofFor the reader’s convenience, we recall Lafforgue’s elegant construction. Let

E ⊂ G(r, n) × An

be the universal rank r subbundle, and let◦

E ⊂ E be the inverse image under thesecond projection of the open subset H ⊂ An (i.e., the subset with all coordinates

nonzero). H obviously acts freely on◦

E, and the quotient is canonically identified withGe(r − 1, n − 1). This gives a smooth map

Ge(r − 1, n − 1) = ◦E /H → G(r, n)/H.

Now for any H -equivariant T → G(r, n), the construction pulls back. �

Note, in particular, that this shows the following.

COROLLARY 2.14S ⊂ T is regularly embedded, with normal bundle the pullback of the universalquotient bundle of Ge(r − 1, n − 1).


2.15A precise description of the fibres of S is given in [L2, Chapter 5]. Let S ⊂ T be aclosed fibre of S ⊂ T over a point of �P . We have by Proposition 2.13 a smoothstructure map S → T/H , and so S inherits a stratification from the orbit stratificationof T/H , parameterized by P ∈ P . In particular, the facets (maximal-dimensionalpolytopes) of P correspond to irreducible components, and the stratum SP (which arethe points of S which lie only on the irreducible component corresponding to P ) isthe complement in Pr−1 to a connected arrangement of n hyperplanes with associatedmatroid polytope P (see §2.6). The irreducible component itself is the log canonicalcompactification of SP , as follows, for example, from Theorem 2.17. For r = 3, thiscompactification is smooth, and it is described by Lemma 8.8.

2.16. Log structures and toric stacksFor basic properties of log structures and toric stacks, we refer to [O11, §5]. Anylog structure we use in this article is toric; that is, the space comes with an evidentmap to a toric variety, and we endow the space with the pullback of the toric logstructure on the toric variety. In fact, we do not make any use of the log structure itself,only the bundles of log (and relative log) differentials, all of which are computed bythe following basic operation. (Our notation is chosen with an eye to its immediateapplication.)

Let q : A → A be a map of toric varieties so that the map of underlying tori isa surjective homomorphism with kernel H . We have the smooth map

A → A/H

(where the target is the stack quotient) and, in particular, its relative cotangent bundle,which is canonically identified with a trivial bundle with fibre the dual of the Liealgebra to H . We denote the bundle

�1q(log) = �1

A/(A/H ) (2.16.1)

as q is log smooth, and this is its bundle of relative log differentials, as follows from[O11, Proposition 5.14] and [O12, §3.7].

For a map � → A, consider the pullback

T := A ×A � → �.

Then (2.16.1) pulls back to the relative cotangent bundle for

T → T/H.

T → � is again log smooth, with this (trivial) bundle of relative log differentials.


Now suppose that S ⊂ T is a closed subscheme, so that the map S × H → T(or, equivalently, S → T/H ) is smooth. Then the relative cotangent bundle for

S → T/H

is a quotient of the pullback of �1T/(T/H ) (p : S → � is log smooth) with bundle of

relative log differentials

�1p(log) = �1

S/(T/H ).

THEOREM 2.17The visible contour family p : S → � is log smooth. Its bundle of log differentials

�1p(log) = �1

S/(T/H )

is a quotient of the pullback of �1A/(A/H )

, which is the trivial bundle A × Vn. Fibres(S, B) are semi log canonical, and the restriction of the Plucker polarisation toS ⊂ Ge(r − 1, n − 1) is O(KS + B).

ProofLet (S, B) ⊂ (T , B) be closed fibres of (S, B) ⊂ (T, B). (T ,B) is semi log canoni-cal, and O(KT +B) is canonically trivial (e.g., by [A, Lemma 3.1]). (S, B) is now semilog canonical by (2.13), and by adjunction O(KS +B) is the determinant of its normalbundle, which is the Plucker polarisation by (2.14). The other claims are immediatefrom Proposition 2.13 and the general discussion in §2.16. �

The initial motivation for this article was the following elementary observation.

PROPOSITION 2.18X(r, n) is minimal of log general type. Its first log canonical map is a regular immer-sion.

ProofFixing the first r + 1 hyperplanes identifies X(r, n) with an open subset of Un−(r+1),where U ⊂ Pr−1 is the complement to B, the union of r + 1 fixed hyperplanes inlinear general position. Since KPr−1 + B = O(1), the first log canonical map on U isjust the open immersion U ⊂ Pr−1. �

We have the following criterion to guarantee that X(r, n) ⊂ XL(r, n) is a log minimalmodel. Let Tp(log) be the dual bundle to �1

p(log) on S (i.e., the relative tangent bundleto S → T/H ). The sheaf �1

XL(r,n)/k(log B) is defined in Definition 9.1.


THEOREM 2.19If R2p∗(Tp(log)) vanishes at a point of XL(r, n) ⊂ �, then � → A/A∅ is smooth,XL(r, n) = �, � is normal, and the pair (XL(r, n), B) has toroidal singularities, nearthe point.

If R2p∗(Tp(log)) vanishes identically along XL(r, n), then �1XL(r,n)/k

(log B) islocally free, globally generated, and its determinant O(KXL

+B) is globally generatedand big. In particular, X(r, n) ⊂ XL(r, n) is a log minimal model.

ProofBy [L2, Theorems 4.25(ii), 5.15], vanishing of R2 implies that the structure map issmooth. Now suppose that the structure map is smooth along XL(r, n). The bundleof log differentials for the toric log structure on a normal toric variety is preciselythe bundle in Definition 9.1, which implies the analogous statement for (XL(r, n), B).The bundle of differentials is the cotangent bundle of the structure sheaf, and thusa quotient of the cotangent bundle to A → A/A∅ which by §2.16 is canonicallytrivial, whence the global generation. Now KXL(r,n) + B is big by Proposition 2.18. �

3. Singularities of (X(r, n), B)In this section we prove Theorem 1.5. The very simple idea is as follows. The notionof log canonical pair (X,

∑Bi) generalises normal crossing. In particular, if all the

irreducible components Bi are Q-Cartier, then log canonical implies at least that theintersection of the Bi has the expected codimension (see Proposition 3.18). We proveTheorem 1.5 by observing that well-known configurations give points of X(r, n) lyingon too many boundary divisors. The main work is to show that these points are actuallyin the closure of the generic stratum and that the boundary divisors are Cartier nearthese points.

3.1. Divisor BI

Let I ⊂ N . It is easy to see that {xI ≤ k} ⊂ �(r, n) is a realizable matroid polytopefor any 0 < k < r . The corresponding hyperplane arrangement {L1, . . . , Ln} is asfollows: the only condition we impose is

codim⋂i∈I

Li = k.

This polytope has full dimension if and only if |I | > k.It follows that if |I | > k and |I c| > r−k, then there is a matroid decomposition of

�(r, n) with two polytopes {xI ≥ k} and {xI ≤ k}. The corresponding stratum of A isobviously maximal among boundary strata. We denote its closure (and correspondingsubschemes of �, XL(r, n), etc.) by BI .


For example, let r = 3. In the configuration with polytope {xIc ≤ 1}, lines of I c

are identified, and lines of I are generic. In the configuration {xI ≤ 2}, the lines of I

have a common point of incidence, and lines of I c are generic.

3.2. Central configurations and matroidsPROPOSITION 3.3Let I be an index set, and for each α ∈ I, let Iα ⊂ N be a subset such that |Iα| ≥ r

and

|Iα ∩ Iβ | ≤ r − 2 for α �= β. (3.2.1)

Let us call S ⊂ N independent if |S| < r or |S| = r and S �⊂ Iα for any α ∈ I. Thisgives a structure of a matroid on N .

ProofWe only have to check that for any S ⊂ N , all maximal independent subsets in S havethe same number of elements. It suffices to prove that if S contains an independent setT , |T | = r , then any independent subset R ⊂ S can be embedded in an independentsubset with r elements. We can assume that |R| = r − 1. If R �⊂ Iα for any α ∈ I,then we can just add any element to R. If R ⊂ Iα for some α, then this α is uniqueby (3.2.1), and we add to R an element of T which is not contained in Iα . �

Definition 3.4We call matroids of this form central. A polytope PC of a central matroid C is givenby inequalities xIα

≤ r − 1 for all α ∈ I. Let Pα ⊂ �(r, n) be the matroid polytopexIα

≥ r − 1. Let I = {PC} ∪ {Pα}α∈I.

LEMMA 3.5I is a central decomposition of �(r, n) (see Definition 2.2 for the definition) withcentral polytope PC . For each subset I′ ⊂ I, I′ is a matroid decomposition, coarserthan I, and all matroid decompositions coarser than I occur in this way.

ProofTo show that I is a central decomposition, it suffices to check that Pα ∩ Pβ is on theboundary for any α �= β. (This implies, in particular, that any interior point of any wall{xIα

= r −1} belongs to exactly two polytopes, PC and Pα .) Assume that x ∈ Pα ∩Pβ

is an interior point of �(r, n). Then xIα∩Iβ< r − 2 by (3.2.1). (Otherwise, xi = 1 for

any i ∈ Iα∩Iβ , and therefore x is on the boundary.) Therefore xIα\Iβ= xIα

−xIα∩Iβ> 1

and xIα∩Iβ= xIα\Iβ

+ xIβ> r , a contradiction.


Any matroid decomposition coarser than I is obviously central and can be ob-tained by combining PC with several Pα’s. This has the same effect as taking theseα’s out of I. �

PROPOSITION 3.6Let UI be the affine open toric subset of A, as in Definition 2.2. Then UI is smoothand bijective to �(UI) ⊂ P(�rkn)//H . Let U = UI ∩�. U ⊂ XL(r, n) maps finitelyand homeomorphically onto its image in X(r, n).

ProofThe proof follows from Theorem 2.3. �

Definition 3.7We say that a hyperplane arrangement C is central if a pair (Pr−1, C) has normalcrossings on the complement to a zero-dimensional set. If r = 3, it simply meansthat there are no double lines. Let I ⊂ Pr−1 be the set of points of multiplicity atleast r . Then a matroid of C is a central matroid that corresponds to subsets Iα ⊂ N

of hyperplanes containing α ∈ I.

Definition 3.8Fix a hyperplane L ⊂ Pr−1. For each subset J ⊂ N , |J | ≥ r , let QJ be the modulispace of J -tuples of hyperplanes, Lj , j ∈ J , in Pr−1 such that the entire collection ofhyperplanes, together with L, is in linear general position modulo automorphism ofPr−1 preserving L.

Note that QJ is a smooth variety of dimension (r − 1)(|J | − r). Intersecting with thefixed hyperplane L gives a natural smooth surjection

QJ → X(r − 1, |J |).

LEMMA 3.9Let C, I, I be as in Definition 3.7. For each α ∈ I, we have a natural mapXC(r, n) → X(r − 1, |Iα|) taking the hyperplanes through a. Let

M =∏α∈I

X(r − 1, |Iα|) and Q =∏α∈I

QIα.

There is a natural identification

�I = XC(r, n) ×M Q.


m

m

Figure 1

In particular,

dim(�I) = dim(XC(r, n)

) +∑α∈I

(|Ia| − r).

ProofThe proof is immediate from [L2, §3.6]. �

Next, we demonstrate that Lafforgue’s space � is reducible.

PROPOSITION 3.10Let C be the following configuration of 6m − 2 lines in R2 (see Figure 1).

Let I be its multiple points, as in Definition 3.7. Then dim �I ≥ m2. The

Lafforgue space ��(3,n)

is not irreducible for large n.

ProofThe configuration C has at least m2 points of multiplicity 4, so the inequality is imme-diate from Lemma 3.9. The final remark follows as the main component XL(3, 6m−2)

of ��(3,6m−2)

has dimension 12m − 12. �

However, for a large class of central configurations, the stratum �I belongs to theclosure of the main stratum.

3.11. Lax configurationsWe say that a central hyperplane arrangement C is lax if there is a total orderingon N such that for each i ∈ N , points on Li of multiplicity strictly greater thanr with respect to N≤i are linearly independent. For example, the configuration in


Figure 2

Proposition 3.10 is not lax for m ≥ 4. On the other hand, any configuration of lines inP2 without points of multiplicity 4 or more is lax.

THEOREM 3.12The notation is as in Definition 3.7. Assume that C is lax.(1) The stratum �I is contained (set theoretically) in XL(r, n) ⊂ �.(2) Let U = UI ∩ �, where UI ⊂ A is the smooth toric affine open set of

Proposition 3.6. Let U → U be the normalisation. Then U is an irreducibleopen factorial subset of XL(r, n) ⊂ �, smooth in codimension one. More-over, the boundary strata BIa

are Cartier, generically smooth, and irreducibleon U , their union is the boundary, and their scheme-theoretic intersection isthe stratum �I.

(3) Let U → U be the normalisation, and let B ⊂ U be the reduction of theinverse image of B. If KU + B is log canonical at a point in the inverse imageof p ∈ �I, then the stratum has pure codimension |I| in U near p; that is,

∑α∈I

(|Iα| − r + 1) + dim XC(r, n) = n(r − 1) − r2 + 1.

We postpone the proof until the end of this section.

Proof of Theorem 1.5Consider the Brianchon-Pascal configuration (see [HC], [D]) of nine lines with |I| = 9and |Iα| = 3 for all α (see Figure 2).

It is easy to compute that dim XC(3, 9) = 2. Now apply Theorem 3.12: the left-hand side in Theorem 3.12(3) is equal to 11, but the right-hand side is 10. If n ≥ 10,add generic lines.

There is an even better configuration of nine lines with |I| = 12 and |Iα| = 3 forall α. Namely, fix a smooth plane cubic. Every line containing two distinct inflectionpoints contains exactly three. This gives a configuration of 12 lines. Furthermore,each inflection point lies on exactly three lines, and these are all the intersection pointsof the configuration. This is the famous Hesse Wendepunkt-configuration (see [HC],


[D]). Let C be the dual configuration. Now apply Theorem 3.12: the left-hand side inTheorem 3.12(3) is equal to 12, but the right-hand side is 10.

For the characteristic 2, use the Fano configuration (see [GGMS, Theorem 4.5]),and argue as above: the left-hand side in Theorem 3.12(3) is equal to 7, but theright-hand side is 6.

In the (4, 8) case, take the configuration of eight planes in P3 given by the faces ofthe octahedron. There are 12 points of multiplicity 4 (i.e., lying on four of the planes),while X(4, 8) is 9-dimensional. If n ≥ 9, add generic planes. �

THEOREM 3.13The boundary strata of (X(3, n), B) for lax configurations have arbitrary singular-ities; that is, their reductions give reductions of all possible affine varieties definedover Z (up to products with A1).

ProofBy Theorem 3.12, it suffices to prove that �

�(3,n)I for lax configurations C, I satisfies

Mnev’s theorem (see [L2, Theorem 1.14]); that is, given affine variety Y over Z, thereare integers n, m, an open set U ⊂ Y × Am with U → Y surjective, and a laxconfiguration C with n lines such that U is isomorphic to the reduction of the Lafforguestratum �

�(3,n)I . One can follow directly the proof of Mnev’s theorem: Lafforgue

constructs an explicit configuration that encodes the defining equations for Y , and itis easy to check that this configuration is lax. The ordering of lines (in Lafforgue’snotation) should be as follows: lines [0, 1α, Pα, ∞α] and the infinite line should gofirst (at the end of the process there will be many points of multiplicity greater than 3along them); then take all auxiliary lines in the order of their appearance in Lafforgue’sconstruction. �

Now we proceed with the proof of Theorem 3.12.

3.14. Face maps and cross-ratiosThe collection of X(r, n) has a hypersimplicial structure: there are obvious mapsBi : X(r, n) → X(r, n − 1) (dropping the ith hyperplane) and Ai : X(r, n) → X(r −1, n − 1) (intersecting with the ith hyperplane). These maps extend to maps of Chowquotients (see [K1, Theorem 1.6]) and to maps of Lafforgue’s varieties XL ⊂ � ⊂ A(see [L2, Theorem 2.4]). For A, these maps are just restrictions of face maps in §2.5corresponding to faces {xi = 0} � �(r, n − 1) and {xi = 1} � �(r − 1, n − 1).

In particular, let V, W ⊂ N be subsets such that |V | = 4, |W | = r−2, V ∩W �= ∅.Then dropping all hyperplanes not in V ∪ W and intersecting with all hyperplanes inW give cross-ratio maps

CRV,W : X(r, n) → X(2, 4) = M0,4 = P1\{0, 1, ∞} (3.14.1)


and

CRV,W : � → X(2, 4) = M0,4 = P1.

It follows that CRV,W (�P ) ⊂ P1\{0, 1, ∞} if and only if P does not break

�V,W (2, 4) =⋂

i �∈V ∪W

{xi = 0} ∩⋂i∈W

{xi = 1}.

�(2, 4) is an octahedron, and values {0, 1, ∞} correspond to three decompositions of�(2, 4) into two pyramids.

To write (3.14.1) as a cross-ratio, let V = {i1i2i3i4} with i1 < i2 < i3 < i4. LetL1, . . . , Ln be a collection of hyperplanes in X(r, n). Consider an (r × n)-matrix M

with columns given by equations of these hyperplanes. Then

CRV,W (L1, . . . , Ln) = Deti1i2W Deti3i4W

Deti1i3W Deti2i4W

,

where each DetT is an (r × r)-minor of M with columns given by T .Let C = {L1, . . . , Ln} be any configuration as in §3, and let x0 ∈ G(r, n) be a point

that corresponds to C under the Gelfand-MacPherson transform. Let (X, x0) ⊂ G(r, n)be a pointed curve such that X ∩ G0(r, n) �= ∅. Let F : G(r, n)0 → X(r, n) be thecanonical H -torsor. Then

p0 = limx→x0

F (x) ∈ XL(r, n)

belongs to �P , where P is a matroid decomposition of �(r, n) containing PC . Indeed,it is clear that x0 is contained in the fibre of the universal family (2.5.2) over p0, soPC = Supp(x0) is in P .

PROPOSITION 3.15Let C be central, as in Definition 3.7. If

limx→x0

CRV,W (x) �∈ {0, 1, ∞}

for any W ⊂ Iα , |V ∩ I cα | = 1, α ∈ I, then P = I.

ProofAny decomposition containing PC is a refinement of I. So it remains to prove thefollowing combinatorial statement: any realizable matroid decomposition P refiningI is equal to I, provided that P ∩ �(2, 4) = �(2, 4) for any face �(2, 4) ⊂ �(r, n)which belongs to the boundary of some Pα and such that exactly one face of thisoctahedron �(2, 4) belongs to the wall xIα

= r − 1. (This is a condition equivalent toW ⊂ Iα , |V ∩ I c

α | = 1.)


Restrictions of P and I to the faces of �(r, n) have the same form. Also, ifr = 2, then the claim follows, for example, from the explicit description of matroiddecompositions of �(2, n) (see [K1, Theorem 1.3]), so we can argue by induction, andit remains to prove the following: any realizable matroid decomposition P refining Iis equal to I, provided that P |F = I|F for any face F = {xi = 1} of �(r, n), r > 2.

Assume, on the contrary, that a certain Pα ∈ I is broken into pieces. Choosea polytope Q ⊂ P ∩ Pα such that the boundary of Q contains the face F = {xl =1} ∩ Pα , l �∈ Iα . A polytope Q is realizable. In the corresponding configuration D,the hyperplane Ll is multiple (of multiplicity |I c

α |), and intersections of hyperplanesLj , j ∈ Iα , with Ll are in general position (because F ⊂ Q). It follows that D iscentral (except that Ll is multiple). If Q �= Pα , then there is at least one degeneracy;hyperplanes Lj , j ∈ J ⊂ Iα , |J | = r , pass through a point β �∈ Ll . Since not allhyperplanes Li , i ∈ Iα pass through β, there exist indices k, k′ ∈ J and i ∈ Iα suchthat a line

⋂i∈J\{k,k′} Li intersects Lk and Lk′ at β, Ll at Li at two other distinct points.

It follows that �{k,k′,l,i},J\{k,k′}(2, 4) is broken by P . �

Proof of Theorem 3.12(1)Let MC be as in §2.6 for a fixed lax hyperplane arrangement C. Let Z ⊂ �I be thefibre over the point of XC(r, n) given by C, in the product decomposition Lemma 3.9.

We consider lines xM : A1 → M(r, n), xM (z) = MC + zM for M ∈ M(r, n),and the induced regular map (which we abusively denote by the same symbol) xM :A1 → XL(r, n). We consider the limit of xM as z → 0.

We assume that N has the lax order of §3.11, so for any l, points on Ll ofmultiplicity greater than r with respect to L1, . . . , Ll−1 are linearly independent. Letpl be the number of these points. A moment’s thought, and Lemma 3.9 yields theequality

∑α∈I

(|Ia| − r) =∑

i

pi = dim(Z).

We now construct the columns of M . Suppose that the first l − 1 columns of M

are already constructed, and consider column l. Let ei ∈ Ll , i = 1, . . . , pl , be pointsof multiplicity greater than r . We include these ei’s in the basis e1, . . . , er and writethe lth column in the dual basis. Let Vi, Wi ⊂ N , i = 1, . . . , pn, be any choice ofsubsets as in §3.14 such that Vi = {i1, i2, i3, l}, |Vi ∩ I c

ei| = 1, Wi ⊂ Iei

.

CLAIM 3.16For i = 1, . . . , pl ,

limz→0

CRVi,Wi

(xM (z)

)


does not depend on Mjl for j �= i and depends nontrivially on Mil (i.e., we can makethis limit any general value by varying Mil).

We can assume, without loss of generality, that i1 �∈ Iei. (Otherwise, take an appropriate

automorphism of a cross-ratio function.) Note that the claim implies the result. First,it is clear that any single choice of subsets W, V as in Proposition 3.15 can be chosenas Wi, Vi for some i and l. So all these cross-ratios are generic (i.e., take on valuesother than {0, 1, ∞}) for general M . Now by Proposition 3.15, the limit point is in Z.Now by the claim, we can vary dim(Z) of the cross-ratios completely independentlyby varying M . Since Z is smooth and connected by Lemma 3.9, it thus follows thatZ ⊂ XL(r, n) (set theoretically), and so since C was arbitrary, this completes theproof.

Let W := Wi . Then

limz→0

CRVi,Wi

(xM (z)

) = limz→0

Deti1i2W Deti3lW

Deti1i3W Deti2lW

.

Notice that limz→0 Deti1i2W and Deti1i3W are not zero; by assumption, Li1 does notpass through ei , but projections of any r − 1 hyperplanes in Iei

from ei are linearlyindependent.

So we have to demonstrate that

limz→0

Deti3lW

Deti2lW

does not depend on Mjl for j �= i and depends not trivially on Mil . Indeed, the constantterms of Deti3lW and Deti2lW vanish; let us find coefficients at z. The ith rows of thecorresponding submatrices of MC are trivial, so we can expand both determinantsalong this row and get

limz→0

Deti3lW

Deti2lW

= Mii3Rii3 + MilRil + · · ·Mii2Rii2 + MilRil + · · · ,

where Rij are cofactors of the corresponding submatrices of MC . These cofactors arenot trivial because projections of any r − 1 hyperplanes in Iei

from ei are linearlyindependent. So we see that the limit indeed does not depend on Mjl for j �= i andis a Mobius function in Mil . This function can be made nontrivial by adding an opencondition Mii3Rii3 �= Mii2Rii2 . �

PROPOSITION 3.17Let C be a lax configuration with multiple points I. If |I| ≥ 2, then codimXL(r,n)

�I ≥ 2.


ProofWe proceed by induction on

∑ |Iα| using Lemma 3.9 and the following observation:a configuration near a connected configuration is connected. We compare quantitiesdim XC(r, n) + ∑

α∈I(|Ia| − r) for various configurations. Since all of them willbe connected, we can replace XC(r, n) by its PGLr-torsor PC(r, n), the space of allconfigurations with prescribed multiplicities.

Assume first that there are some points of multiplicity greater than r . Take thelast hyperplane L in the lax order which contains such a point. Move L a little bit totake it off this point, but keep all other points of multiplicity greater than r on L. (Thisis possible because they are linearly independent.) If there are no such points, keepsome point of multiplicity r (if there are any of them on L). Then the dimension ofthe configuration space will increase by at least one, the sum

∑ |Iα| − r will decreaseby at most one, and |I| is still at least 2. At the end, there will be at least two pointsA, B of multiplicity r and no points of higher multiplicity. Now take a hyperplanethrough A and move it, keeping B if it belongs to this hyperplane. This will increasethe dimension of the configuration space by at least one, but the result will still not begeneric and thus having codimension at least one. �

Proof of Theorem 3.12(2)From Lemma 3.9, the generic stratum of BIa

is smooth and connected and codi-mension one in U . By Proposition 3.17, all other boundary strata of U are lowerdimensional. The boundary of U is (by definition) the scheme-theoretic inverse imageof the boundary of UI, and so it is Cartier and, in particular, pure codimension one,by Theorem 2.3. It follows that the BIa

are irreducible and Cartier, and their union isthe full boundary. They are generically smooth by Lemma 3.9. The proof of Propo-sition 2.18 shows that their complement, the main stratum X(r, n), is isomorphic toan open subset of affine space and thus has a trivial divisor class group. Thus U

is factorial. Now it is smooth generically along the Cartier divisors BIaby Lemma

3.9. In the open set UI ⊂ A, the stratum AI is the scheme-theoretic intersec-tion of the boundary divisors that contain it. (This is true in any toric variety.)Thus �I is scheme-theoretically the intersection of the boundary divisors of U thatcontain it. �

The proof of Theorem 3.12(3) now follows from Lemma 3.9 and Proposition 3.18.

PROPOSITION 3.18Let X be a normal variety. Let Bi be irreducible Q-Cartier Weil divisors. If KX + ∑

Bi

is log canonical, then the intersection

B1 ∩ B2 ∩ · · · ∩ Bn

is (either empty or) pure codimension n.


ProofWe can intersect with a general hyperplane to reduce to the case when n is thedimension of X and then apply [Ko, Theorem 18.22]. �

4. The membraneWe begin with some background on buildings; for more details, see [Br]. We follow thenotation of §1.9. We follow [S] for elementary definitions and properties of simplicialcomplexes.

Definition 4.1The total Grassmannian Gr(V ) is a simplicial complex of dimension r −1. Its verticesare nontrivial subspaces of V . (In particular, V itself is a vertex.) A collection ofdistinct subspaces forms a simplex if and only if they are pairwise incident (i.e., one iscontained in the other), from which it easily follows that m − 1 simplices correspondto flags of nontrivial subspaces

0 = U0 ⊂ U1 ⊂ U2 ⊂ · · · ⊂ Um. (4.1.1)

The compactified Bruhat-Tits building B is a set of equivalence classes ofnontrivial free R-submodules of VK , where M1, M2 are equivalent if and only ifM1 = cM2 for some c ∈ K∗. The Bruhat-Tits building B ⊂ B is the subsetof equivalence classes of lattices (i.e., free submodules of rank r). As a set, B =PGLr(K)/PGLr(R∗).

B is a simplicial complex of dimension r − 1. [M1], . . . , [Mm] ∈ B span an(m − 1)-simplex σ if and only if we can choose representatives so

zMm = M0 ⊂ M1 ⊂ M2 ⊂ · · · ⊂ Mm. (4.1.2)

LEMMA 4.2 (see [Br])For any lattice [�] ∈ B, there exists a canonical isomorphism

Star� B � Gr(�),

where � = �/z�. More generally, for any simplex σ (see (4.1.2)) in B,

Starσ B � Gr(Mm/Mm−1) ∗ · · · ∗ Gr(M1/M0),

where ∗ denotes the join of simplicial complexes.

ProofIndeed, lattices between z� and � are obviously in incidence-preserving bijectionwith subspaces of �. Similarly, lattices that fit in the flag σ correspond bijectively tosubspaces in one of Mi/Mi−1. It is worth mentioning that for any simplex σ of Gr(V )as in (4.1.1), Starσ Gr(V ) = Gr(Um/Um−1) ∗ · · · ∗ Gr(U1/U0). �


Definition 4.3Let [�] ∈ B. We denote by

R� : B → Star� B

the map that sends a submodule M to M� + z�. More generally, for any simplex(4.1.2) in B there is a map Rσ : B → Starσ B defined as follows: Rσ (M) = M�k +�i ,where i is maximal such that M�k �⊂ �i . Finally, we denote by Res� and Resσ

compositions of R� and Rσ with isomorphisms of Lemma 4.2.

LEMMA 4.4R� and Rσ are retraction maps of simplicial complexes.

ProofIt is clear that the Rσ -map preserves incidence and is the identity on Starσ B. �

4.5. Convex structureA subset of B is called convex if it is closed under finite R-module sums. For acollection of free R-submodules {Mα}, their convex hull in B, denoted [Mα], is thesubcomplex with vertices (with representatives) of form

∑cαMα , cα ∈ K . This is

obviously the smallest convex subset that contains all the [Mα]. Similarly, a subset ofGr(V ) is called convex if it is closed under Minkowski sums. We write [Uα] for theconvex hull of subspaces {Uα} ⊂ Gr(V ).

It is clear that stars of simplices in B and Gr(V ) are convex and that isomorphisms(see Lemma 4.2) preserve the convex structure.

Example 4.6For T = {g1, . . . , gr} a basis of VK , the convex hull [T ] is called an apartment. It isthe set of equivalence classes [M] such that M has an R-basis c1g1, . . . , crgr for someci ∈ K .

Retractions commute with convex hulls.

PROPOSITION 4.7Let {Mα} ⊂ B be a subset, and let σ ⊂ B be a simplex. Then

[RσMα] = Rσ [Mα].

If σ ⊂ [Mα], then both sides are also equal to Starσ [Mα].

We leave the proof of this proposition as an exercise for the reader.


LEMMA 4.8 (see [F])The convex hull of a finite subset of B is finite.

LEMMA 4.9The membrane [F] is the convex hull of {Rf1, . . . , Rfn} and the union of apart-ments [T ] for subsets T ⊂ F, |T | = r .

ProofThe proof is immediate from the definitions and Nakayama’s lemma. �

4.10. Membranes as tropical subspacesWe begin by recalling the construction of the tropical variety (also called a non-Archimedean amoeba or a Bieri-Groves set; see [SS] for details). Let H = Gn

m be analgebraic torus. Let K be the field of generalised Puiseux series

∑α∈I⊂R cαzα , where

I is a locally finite subset of R, bounded below (and is allowed to vary with the series).There is an evaluation map

ord : H (K) → H (K)/H (R) = Rn,

where R ⊂ K is the subring of series for which I ⊂ R≥0. For any subvariety Z ⊂ H ,ord(Z) is called the tropicalisation of Z. It is a polyhedral complex of dimension dim Z.If Z is invariant under dilations, then ord(Z) is invariant under diagonal translations,and we consider

Ord(Z) = ord(Z) mod R(1, . . . , 1).

Let F := {Rf1, . . . , Rfn} be as in Lemma 4.9. Consider the map

: V ∨K → Kn, F �→ (

F (f1), . . . , F (fn)).

Let Z = (V ∨K ) ∩ H . Then Z is of course the intersection with H of the r-plane

spanned by the rows of the (r × n)-matrix with columns given by fi’s. Its tropicalisa-tion Ord(Z) ⊂ Rn−1 is called a tropical projective subspace.

For any simplicial complex C, we denote by |C| the corresponding topologicalspace (obtained by gluing physical simplices). Recall that |B| can be identified withthe space of equivalence classes of additive norms on VK , where an additive norm N

is a map VK (K) → R ∪ {∞} such that

N(cv) = ord(c) + N(v) for any c ∈ K, v ∈ VK (K),

N(u + v) ≥ min(N(u), N(v)

)for any u, v ∈ VK (K),


and

N(u) = ∞ iff u = 0.

Two additive norms are equivalent if they differ by a constant. For a norm N , let�(N) = (N(f1), . . . , N(fn)) ∈ Rn. Now consider

� : |B| → Rn−1, �([N]) = �(N) mod R(1, . . . , 1).

The map is continuous because the topology on |B| is exactly the topology of pointwiseconvergence of norms. The following theorem is our version of the tropical Gelfand-MacPherson transform.

THEOREM 4.11� induces a homeomorphism |[F]| � Ord(Z).

ProofLet � be Drinfeld’s symmetric domain, the complement to the union of all K-rationalhyperplanes in V ∨

K (K). There is a surjection (see [Dr])

D : � → |B|, F �→ [v → ord F (v)] for v ∈ VK (K);

here we interpret |B| as the set of equivalence classes of norms. The following diagramis obviously commutative:

It follows that Im(�) ⊂ Ord(Z).For any lattice � ∈ B, the corresponding norm N� is

N�(v) = {−a | zav ∈ �\z�} ∈ Z.

In particular, �(B) ⊂ Zn−1. Also, it follows easily from definitions that � is affineon simplices of |B| and uniformly continuous. Since Ord(Z) is a polyhedral complex,it remains to check that for any Q-point of Ord(Z), there exists a unique Q-pointof |[F]| which maps onto it. (A Q-point of |B| means a point of some simplexwith rational barycentric coordinates.) Now we can pass from K to Puiseux seriesk[[z1/m]] with sufficiently large m (this does not change either Ord(Z) or |[F]|; seealso Proposition 6.4 for another version of this barycentric trick), and it remains tocheck the latter statement for Z-points. Replacing fi’s by zai fi’s, we can assume that


this point is O = (0, . . . , 0). Now we claim that if O ∈ ord(Z), then �(�) = O for[�] ∈ [F] if and only if � = Rf1 + · · · + Rfn.

Suppose that �(Rf1+· · ·+Rfn) �= O; that is, suppose that fj ∈ z(Rf1+· · ·+Rfn)for some j . By Nakayama’s lemma, we can assume, without loss of generality, thatRf1 +· · ·+Rfn = Rf1 +· · ·+Rfr . Therefore fj ∈ z(Rf1 +· · ·+Rfr ). But then forany F ∈ V ∨

K (K), if ord F (fi) = 0 for i ≤ r , then ord F (fj ) > 0. But this contradictsO ∈ ord(Z).

Now take any lattice � = Rza1f1 +· · ·+Rzanfn. We can assume, without loss ofgenerality, that � = Rza1f1 + · · · + Rzar fr . Let �(�) = O. Then fi ∈ � for any i;therefore � = Rf1 + · · · + Rfn. �

5. Deligne schemesNow we turn to the proof of Theorem 1.13. We follow the notation of the introductionand §4. Here we prove that the pair (SY , SY +B) of Theorem 1.13 has normal crossings(see Theorem 5.23). Global generation is considered in §6.

5.1. Deligne functor (see [F])Let Y ⊂ B be a finite set. The Deligne functor SY is the functor from R-schemes tosets, a T -valued point q which consists of a collection of equivalence classes of linebundle quotients

qM : MT � L(MT )

for each lattice [M] ∈ Y , where MT := T ×R M , where two quotients are equivalentif they have the same kernel, satisfying the following compatibility requirements.� For each inclusion i : N ↪→ M , there is a commutative diagram

NT

qN−−−−→ L(NT )

iT

��

MT

qM−−−−→ L(MT )

� Multiplication by c ∈ K∗ gives an isomorphism

ker qM·c−−−−→ ker qcM.

It is clear from this definition that SY is represented by a closed subscheme

SY ⊂∏

[M]∈Y

P(M),

(SY )K = P(VK ), and SY contains the Mustafin join (see Definition 1.11).


THEOREM 5.2 (see [F])Assume that Y is finite and convex. Then SY is smooth and irreducible. (In particular, itis isomorphic to the Mustafin join.) Its special fibre SY = (SY )k has normal crossings.

We begin by explaining Faltings’s proof of Theorem 5.2, recalling and expanding onthe three paragraphs of [F, page 167]. This is the substance of §5.3 – Remark 5.18.For this, Y ⊂ B is an arbitrary finite convex subset. Beginning with Lemma 5.19,our treatment diverges from [F]. We specialize to convex subsets Y ⊂ [F] as inTheorem 1.13 and consider singularities of the natural boundary.

5.3. Maximal lattices (see [F])Consider a k-point of SY ; that is, consider a compatible family of one-dimensionalk-vector space quotients

qM : M � L(M), [M] ∈ Y.

This gives a partial order on Y : [N] ≤q [M] if and only if the composition

N = NM → M � L(M)

induced by inclusion NM ⊂ M is surjective and thus, by compatibility, canonicallyidentified with qN . In this case, we also say that qM does not vanish on N .

A lattice [M] ∈ Y is called maximal for q if it is maximal with respect to theorder ≤q . In other words, qN vanishes on M for any [N] ∈ Y , [N] �= [M].

LEMMA 5.4 (see [F])Maximal lattices form a nonempty simplex σ as in (4.1.2). The corresponding k-pointof SY is equivalent to a collection of hyperplanes

Hi ⊂ Mi/Mi−1, m ≥ i ≥ 1,

which do not contain Resσ [M] for any [M] ∈ Y .

Definition 5.5Let [M] ∈ Y . We let P(M) ⊂ SY be the subfunctor of compatible quotients suchthat for each [N] ∈ Y , N = NM , the quotient qN : NT � L(NT ) vanishes on(N ∩ zM)T ⊂ NT .

It is clear that P(M) is represented by a closed subscheme of SY .

LEMMA 5.6The k-points of P(M) ⊂ SY are precisely the set of k-points of SY for which M is amaximal lattice.


ProofConsider a k-point q of P(M). Suppose that M � N , [N] ∈ Y . Then NM = zkN

for some k > 0. By the definition of P(M), qzkN vanishes on zkN ∩ zM . So bycompatibility of quotients with scaling, qN vanishes on N ∩z1−kM , which contains M .Thus M is a maximal lattice.

Conversely, suppose that M is maximal for a k-point q. Take [N] ∈ Y such thatN = NM . By maximality, qz−1N+M vanishes on M . Thus by compatibility, qN+zM

vanishes on zM; thus, again by compatibility, qN vanishes on N ∩ zM . So the pointlies in P(M). �

Definition 5.7Let V be a finite-dimensional k-vector space, and let W ⊂ Gr(V ) be a finite convexcollection of subspaces that includes V . Let BL(P(V ), W) be the functor from k-schemes to sets which assigns to each T the collection of line bundle quotientsWT � L(WT ), W ∈ W, WT the pullback, compatible with the inclusion mapsbetween the W ; that is, the composition

AT → BT

qB−−−−→ L(BT )

factors through qA : AT � L(AT ) for A ⊂ B, A,B ∈ W.

PROPOSITION 5.8There is a canonical identification

P(M) = BL(P(M), ResM (Y )

).

ProofThe proof is immediate from the definitions. �

PROPOSITION 5.9BL(P(V ), W) is represented by the closure of the graph of the product of canonicalrational maps P(V ) �� P(W ), W ∈ W. Furthermore, BL(P(V ), W) is smooth.

ProofWe induct on the number of subspaces in W. When W = {V }, the result is obvious.In any case, it is clear that the functor is represented by a certain closed subscheme

X ⊂∏

W∈WP(W ).

Let P0(V ) ⊂ P(V ) be an open subset of quotients that do not vanish on any W ⊂ W.Then P0(V ) is an open subset of X; its closure X′ in X is the closure of the graph inthe statement.


Take a fixed closed point q0. We show that X = X′ near q0. Let W ∈ W be amaximal subspace such that q0

V vanishes on W . If there are none, then q0 ∈ P0(V ),and so X = X′ near q0.

Let D ⊂ X be the subscheme of compatible quotients so that qV vanishes on W ,and let D0 ⊂ D be a sufficiently small neighbourhood of q0. Let WW ⊂ W be thosesubspaces contained in W ; clearly, WW is convex. Take any E ∈ W and q ∈ D0.If E �∈ WW , then q0

V (and hence qV ) does not vanish on E, from which it followsthat E → L(V ) is surjective and thus identified with qE . It follows easily that D0

is represented by an open subset of BL(P(W ), WW ) × P(V/W ). In particular, byinduction, D0 is connected and smooth of dimension dim V − 2.

We claim that D0 ⊂ X′. As D0 is integral, it is enough to check this on someopen subset of D0. We consider the open subset where qW does not vanish on anyE ∈ WW , and qV does not vanish on any E �∈ WW . This is naturally identified withan open subset for W = {V,W }, and so we reduce to this case. But in this case itis easy to see that X = X′ is the blowup of P(V ) along P(V/W ), and so obviously,D0 ⊂ X′.

By the claim, X has dimension at least dim V − 1 along D0. D ⊂ X is locallyprincipal, defined by the vanishing of a map between the universal quotient linebundles for W and V . In order to prove that X is smooth and equal to X′ along D0, itsuffices to use the following well-known fact. Let (A, m) be a local Noetherian ringof Krull dimension at least d. Assume that A/f is regular of dimension at most d − 1for f ∈ m. Then A is regular of dimension d. �

Definition 5.10Define the depth of W ∈ W to be the largest d ≥ 0 so that there is a proper flag

W = W0 ⊂ W1 ⊂ · · · ⊂ Wd = V

with Wi ∈ W. Let W≤m ⊂ W be the subset of subspaces of depth at most m. LetWm ⊂ W≤m be the subset of subspaces of depth exactly m.

Notice that BL(P(V ), W0) = P(V ) and BL(P(V ), W≤N ) = BL(P(V ), W) forN � 0. Thus the next proposition shows that the canonical map BL(P(V ), W) →P(V ) is an iterated blowup along smooth centers.

PROPOSITION 5.11A forgetful functor

p : BL(P(V ), W≤m+1

) → BL(P(V ), W≤m

)(3.14.1)

is represented by the blowup along the union of the strict transforms of P(V/W ) ⊂P(V ) for W ∈ Wm+1 (which are pairwise disjoint).


ProofLet W ∈ Wm+1. We claim that the strict transform of P(V/W ) represents the subfunc-tor XW of BL(P(V ), W≤m) of compatible quotients such that qE vanishes on E ∩ W

for all E ∈ W≤m. This subfunctor is naturally identified with BL(P(V/W ), WW≤m),

where WW≤m is the (obviously convex) collection of subspaces (E + W )/W ⊂ V/W

for E ∈ W≤m. By Proposition 5.9, it is smooth and connected and thus the stricttransform. For disjointness: if W ′, W ′′ ∈ Wm+1, then W := W ′ + W ′′ ∈ W≤m, and itis not possible for qW to vanish both on W ′ and on W ′′; thus the strict transforms aredisjoint.

The map (3.14.1) is obviously an isomorphism outside the union of subfunctorsXW . Take W ∈ Wm+1. The inverse image

p−1(XW ) ⊂ BL(P(V ), W≤m+1

)

is naturally identified with P(W ) × BL(P(V/W ), WW≤m). In particular, by Proposi-

tion 5.9 it is a smooth connected Cartier divisor. It follows that the exceptional locusof p is the disjoint union of these divisors. Thus p factors through the proscribedblowup, and the induced map to the blowup will have no exceptional divisors and isthus an isomorphism (as domain and image are smooth). �

Definition 5.12For a subset σ ⊂ Y , define the intersection

P(σ ) :=⋂M∈σ

P(M) ⊂ SY .

PROPOSITION 5.13P(σ ) is nonempty if and only if σ is a simplex (4.1.2). Consider the convex subsetResσ (Y ), a collection of convex subsets Wi ⊂ Gr(Mi/Mi−1). There is a canonicalidentification

P(σ ) =∏

m≥i≥1

BL(P(Mi/Mi−1), Wi

) =: BL(P(σ ), Resσ Y

).

ProofBy Lemma 5.6, the k-points of the intersection are exactly those for which all M ∈ σ

are maximal. Thus if it is nonempty, σ is a simplex by Lemma 5.4. The expression forthe intersection is immediate from the definition of Res (see Definition 4.3) and thefunctorial definitions of P(M) and BL. �

Remark 5.14Observe by Proposition 5.8 – Proposition 5.13 that the special fibre SY has normalcrossings. Moreover, it can be canonically defined purely in terms of the subcomplex


Y ⊂ B. Indeed, by Proposition 5.8 its irreducible components and their intersectionsare encoded by the BL(P(σ ), Resσ (Y )) for simplices σ ⊂ Y , and by Proposition 4.7we have canonical identifications

Resσ Y = Starσ Y ⊂ Starσ B = Starσ Gr(�m).

Definition 5.15Let σ ⊂ Y be a simplex. Let U (σ ) ⊂ SY be the open subset whose complement is theclosed subset of the special fibre given by the union of P(N), [N] ∈ Y\σ .

LEMMA 5.16U (σ ) is the union of the generic fibre together with the open subset of the special fibreconsisting of all k-points whose simplex of maximal lattices (see §5.3) is contained inσ . It represents the following subfunctor. Let σ be the simplex (4.1.2). For [M] ∈ Y ,choose minimal i so that MMm ⊂ Mi . A T -point of SY is a point of U (σ ) if and onlyif the composition

MT → (Mi)T � L((Mi)T

)

is surjective for all [M] ∈ Y .

ProofThe proof is immediate from Lemma 5.6 and the definitions. �

Note by Lemma 5.4 that the U (σ ) for σ ⊂ Y give an open cover of SY . Faltings provesthat U (σ ) is nonsingular, and semistable over Spec(R), by writing down explicit localequations (see [F, page 167]). This can also be seen from the following.

PROPOSITION 5.17Let σ ⊂ Y be the simplex (4.1.2). Let U ⊂ P(Mm) be the open subset of quo-tients Mm → L such that NMm → L is surjective for all [N] ∈ Y\σ . Letq : BL(P(Mm), σ ) → P(Mm) be the iterated blowup of P(Mm) along the flag ofsubspaces of its special fibre

P(Mm/Mm−1) ⊂ P(Mm/Mm−2) ⊂ · · · ⊂ P(Mm/M1) ⊂ P(Mm/M0) = P(Mm);

that is, blow up first the subspace

P(Mm/Mm−1) ⊂ P(Mm) ⊂ P(Mm),

then the strict transform of P(Mm/Mm−2), and so on. There is a natural isomorphism

U (σ ) → q−1(U ).


Remark 5.18When σ = [M], Proposition 5.17 is immediate from Lemma 5.16. As this is the onlycase of Proposition 5.17 which we need, we omit the proof, which in any case isanalogous to (and simpler than) those of Propositions 5.9 and 5.11. Proposition 5.17can also be deduced from the claim on [F, page 168] that for any [N] ∈ Y , the naturalmap SY → P(N) is a composition of blowups with smooth centers (which Faltingsdescribes).

Now fix F as in (1.9.1).

LEMMA 5.19Let Z ⊂ N be a subset with |Z| = r + 1. There is a unique stable lattice [�Z] ∈ [F]such that the limits f �

Z are generic (i.e., any r of them is an R-basis). In particular,there are finitely many stable lattices, and Stab is finite. If we reorder so that Z ={0, 1, . . . , r} and express

f0 = za1p1f1 + · · · + zar prfr

with ai ∈ Z, pi ∈ R∗, then �Z = Rza1f1 + · · · + Rzar fr .

ProofIt is clear that for �Z as given, the limits F�Z are in general position, so �Z isstable. For uniqueness, assume that the limits F�

Z are in general position. Then f �i ,

r ≥ i ≥ 1, are an R-basis of �. Define bi ∈ Z by zbi fi = f �i . Scaling �, we may

assume that bi ≥ ai with equality for some r ≥ i ≥ 1. Thus f0 = f �0 . Then bi = ai

for all i; otherwise, f �0 is in the span of some proper subset of the f �

i , r ≥ i ≥ 1. So� = �Z . �

5.20. NotationFor a subset I ⊂ N , let VI ⊂ VK be the vector subspace spanned by fi, i ∈ I , andlet V I := V/VI . For each lattice M ⊂ V , let MI be its image in V I ; that is, letMI := M/M ∩ VI .

Let Y ⊂ [F] be a finite convex collection containing Stab. One checks immediatelythat the collection of equivalence classes

Y I := {[MI ]

}[M]∈Y

is convex.


DEFINITION-LEMMA 5.21Let Bi ⊂ SY be the subfunctor of compatible quotients such that qM vanishes on f M

i

for all [M] ∈ Y . Then Bi is the nonsingular Deligne scheme for Y {i} and is the closureof the hyperplane on the generic fibre

{fi = 0} ⊂ P(VK ) ⊂ SY .

ProofClearly, M ∩ VI = f M

i R, so Bi = SY {i} . The rest follows from Theorem 5.2. �

PROPOSITION 5.22Let [M] ∈ [F] be a maximal lattice for a k-point of

⋂i∈I

Bi ⊂ SY .

Then the limits f Mi , i ∈ I , are independent over R; that is, they generate an R-direct

summand of M of rank |I |.

ProofWe consider the corresponding simplex of maximal lattices

zM = M0 ⊂ M1 ⊂ · · · ⊂ Mm = M.

For each m ≥ s ≥ 1, let Is ⊂ I be those i such that f Mi ∈ Ms\Ms−1. Clearly, it is

enough to show that the images of f Mi , i ∈ Is , in Ms/Ms−1 are linearly independent.

By scaling (which allows us to move any of the Mi to the M1 position), it is enoughto consider s = 1 and show that the images of fi , i ∈ I1, are linearly independent inM/zM . Or, suppose not. Choose a minimal set whose images are linearly dependent,which after reordering we may assume are f0, f1, . . . , fp. With further reordering,we may assume that

f M1 , f M

2 , . . . , f Mr

are an R-basis of M or, equivalently, that their images give a basis of M/zM . Renamingthe fi , we can assume that f M

i = fi . Now consider the unique expression

f0 =r∑

i=1

zai pifi (5.22.1)

with ai ∈ Z, pi ∈ R∗. By construction, ai ≥ 0. Since the images of f0, . . . , fp inM/zM are a minimal linearly dependent set, it follows that ai ≥ 1 for i > p, and


ai = 0 for p ≥ i ≥ 1. Now let

� := Rf1 + · · · + Rfp + Rzap+1fp+1 + · · · + Rzar fr .

Note that fM1i = fi for p ≥ i ≥ 0, so by assumption qM1 vanishes on these fi .

Notice that zat ft ∈ zMk = M0 ⊂ M1 for t ≥ p + 1. Thus by Theorem 5.2, qM1

vanishes on these zat ft . Thus qM1 vanishes on �. But by (5.22.1) and Lemma 5.19, �

is stable. In particular, [�] ∈ Y . Clearly, � ⊂ M1, but � �⊂ M0. So the vanishing ofqM1 |� violates Lemma 5.4. �

THEOREM 5.23Let Y ⊂ [F] be a finite convex set containing all the stable lattices. For any subsetI ⊂ N , the scheme-theoretic intersection

⋂i∈I

Bi ⊂ SY

is nonsingular and (empty or) of codimension |I | and represents the Deligne functorSY I . The divisor SY + B ⊂ SY has normal crossings.

ProofBy Theorem 5.2, it is enough to show that the intersection I represents the Delignefunctor. It is obvious from the definitions and Definition-Lemma 5.21 that I representsthe subfunctor of SY of compatible quotients qM which vanish on f M

i , i ∈ M , whileSY I is the subfunctor where qM vanishes on M ∩ VI . Since f M

i ∈ M ∩ VI , clearly

SY I ⊂ I

is a closed subscheme. Since by Theorem 5.2, SY I is flat over R, it is enough to showthat they have the same special fibres. And so it is enough to show that the subfunctorsagree on T = Spec(B) for B a local (k = R/zR)-algebra, with residue field k. Bymaximal for such a T -point we mean maximal for the closed point. We take a familyof compatible quotients vanishing on all f M

i and show that they actually vanish onVI ∩M . By Lemma 5.4, it is enough to consider maximal M . But by Proposition 5.22,f M

i , i ∈ I , are independent over R, so in fact they give an R-basis of VI ∩ M . �

6. Global generationHere we complete the proof of Theorem 1.13.

6.1. BubblingWe choose an increasing sequence of finite convex subsets Yi ⊂ B whose union isthe membrane [F]; the existence, for example, follows from Lemma 4.8. We assume


SYi

y x

SYj

Pij

Figure 3

that Stab ⊂ Yi . We have the natural forgetful maps

pi,j : SYi→ SYj

, i ≥ j.

By Theorem 5.23, we have the vector bundle �1p(log B) on SYi

(see §9).

PROPOSITION 6.2Given a closed point x ∈ SYj

for all i sufficiently large, there is a closed pointy ∈ SYi

\B, so that pi,j (y) = x (see Figure 3).

ProofSay that x lies on PYj

(M), [M] ∈ Yj . Choose i sufficiently big so that

[M + z−1f MR] ∈ Yi

for all f ∈ F. Clearly, pi,j (PYi(M)) = PYj

(M). It remains to show that PYj(M) is

disjoint from B ⊂ SYj. Suppose that q ∈ PYj

(M) ∩ Bk . Let N = M + z−1f Mk R.

Clearly, z−1f Mk = f N

k , so by definition of Bk , qN vanishes on z−1f Mk . But qN vanishes

on M by definition of a maximal lattice. So qN = 0, a contradiction. �

6.3. Barycentric subdivision trickNext, we introduce a convenient operation. Let R′ = k[[z1/m]], and let Spec(R′) →Spec(R) be the associated finite map. Let Mv be a collection of lattices in VK , and letY be their convex hull. Let M ′

v := Mv ⊗R R′, and let Y ′ be their convex hull.


SY �b

yy�

SY

Figure 4

PROPOSITION 6.4There is a commutative diagram

SY ′b−−−−→ SY

p′� p

�Spec(R′) −−−−→ Spec(R)

with all arrows proper. If m ≥ r , then given a k-point y ∈ SY there is a k-pointy ′ ∈ SY ′ in its inverse image which lies on a unique irreducible component of thespecial fibre SY ′ (see Figure 4).

ProofFor any R-object X, we denote by X′ the base change to R′. It is clear that S′

Y

represents the functor SY defined as in Theorem 5.2 but for the nonconvex collectionY = {M ′ | M ∈ Y }. Since Y ⊂ Y ′, there is a forgetful map SY ′ → S′

Y sending compat-ible collections of quotients to compatible collections. This implies the commutativediagram above.

Now let m ≥ r , let ω = z1/m, and let y ∈ SY be a k-point with the simplex ofmaximal lattices

σ = {M0 = zMk ⊂ M1 ⊂ · · · ⊂ Mk}.

By Lemma 5.4, y is determined by a collection of hyperplanes Hi ⊂ Mi/Mi−1 whichdo not contain any Resσ M for M ∈ Y . Let N be the R′-lattice

N = M ′1 + ωM ′

2 + ω2M ′3 + · · · + ωk−1M ′

k.


Observe that ωiM ′i ⊂ ωN , k ≥ i ≥ 0. (M ′

0 = ωm−kωkM ′k , and for k > 0 the inclusion

is clear.) Thus we have a map

k⊕i=1

M ′i/M

′i−1

⊕·ωi−1−−−−→ N/ωN. (6.4.1)

This map is clearly surjective; thus it is an isomorphism, as its domain and range arer-dimensional k-vector spaces. By the injectivity of the map, we have

ωi−1M ′i ∩ ωN = ωi−1Mi−1. (6.4.2)

Now let y ′ ∈ P(N/zN) be given by any hyperplane H ′ ⊂ N/zN which restricts to Hi

on Mi/Mi−1 under (6.4.1). It is enough to show y ′ ∈ U ([N]); then, clearly, y ′ is sentto y and N is the only maximal lattice of y ′. (Or equivalently, by Lemma 5.6, y ′ lies ona unique irreducible component of the special fibre SY ′ .) By Lemma 5.4, it is enoughto show that H ′ does not contain Res[N][�] for [�] ∈ Y ′, and by Proposition 4.7, it isenough to check this for [�] = [M ′], [M] ∈ Y .

We can assume that M = MMk , M ⊂ Mi , M �⊂ Mi−1. Then Resσ [M] =(M + Mi)/Mi−1, and it follows from (6.4.2) that (M ′)N = ωi−1M ′; thus we have

(·ωi−1)(Resσ [M]) = Res[N][M′],

by (6.4.1). Thus H ′ does not contain Res[N][M ′] since Hi �⊃ Resσ [M]. �

THEOREM 6.5Under the natural maps pi,j : SYi

→ SYjfor j ≤ i, there is an induced isomorphism

of vector bundles

p∗i,j

(�1

p(log B)) → �1

p(log B).

Furthermore, the global sections

dlog(f

g

), f, g ∈ F,

generate �1p(log B) globally. In particular, ωp(B) is globally generated.

ProofSince the sections are pulled back from SYj

, the last remark implies the first. Further-more, to prove that the given sections generate at some point y ∈ SYj

, it is enough toprove that they generate at some inverse image point on SYi

. Thus by Proposition 6.2,it is enough to prove that they generate at a point y ∈ Si\B.


By Proposition 6.4, we have the proper (generically finite) map

SY ′i→ SYi

.

Clearly, the dlog(f/g) pull back to the analogous forms on the domain, so by Proposi-tion 6.4, we may assume that y has a unique maximal lattice M . Now by Remark 5.18,the natural map

SYi→ P(M)

is an isomorphism of y ∈ U ([M]) onto an open subset of P(M) which misses all ofthe hyperplanes

f Mi = 0.

Note that FM contains an R-basis of M . Reordering, say, f Mk , r ≥ k ≥ 1, gives such

a basis. Then it is enough to show that the (regular) forms

dlog(f M

k

f M1

), r ≥ k ≥ 2,

give trivialisation of the ordinary cotangent bundle over the open set in question, whichis obvious. �

6.6. Minimal modelBy Theorem 6.5, the line bundle ωp(B) is globally generated. We consider the p-relative minimal model, π : SY → S; that is,

S := Proj⊕

m

p∗(ωp(B)⊗m

). (6.6.1)

Note by Theorem 6.5 that S is independent of Y . Let π∗(Bi) =: Bi ⊂ S.

THEOREM 6.7Let Spec(R) → X(r, n) be the unique extension of the map which sends the genericfibre to

[Pr−1, L1 + · · · + Ln] ∈ P0(r, n)/PGLr = X(r, n) ⊂ X(r, n).

The pullback of the universal visible contour family (S, B) (see Definition 2.12) is(S, B).


42 3 5 2 4

3

11=5

blow up

and glue

P2 P2

Figure 5

ProofBy Theorem 6.5, we have a natural surjection

Vn ⊗ OSY� �1

p(log B)

inducing a regular map SY → G(r−1, n−1), which on the general fibre is Kapranov’svisible contour embedding of Pr−1 given by the bundle of log forms with poles on then general hyperplanes. This induces a regular map SY → S, where S → Spec(R)is the pullback of the visible contour family (see Definition 2.12); ωp(B) is pulledback from a relatively very ample line bundle (the Plucker polarisation) on S, byTheorem 2.17. Thus SY → S factors through a finite map

S → S.

The map is birational, an isomorphism on the generic fibre. By Proposition 2.13, Sis normal. Thus it is an isomorphism. �

6.8. IllustrationLet us look at the first nontrivial example for r = 3,

F = {f1, f2, f3, f4 = f1 + f2 + f3, f5 = z−1f1 + f2 + f3}

for f1, f2, f3, the standard basis of k3 ⊂ K3. In this case there are two stable lattices,

M1 = Rf1 + Rf2 + Rf3 and M2 = Rz−1f1 + Rf2 + Rf3.

Figure 5 illustrates the limit surface (S, B). Note that zM2 ⊂ M1 ⊂ M2, so thestable lattices in this case form a 1-simplex, σ ; in particular, they are already convex.So we can take Y = σ = Stab. The two images in Figure 5 are the configurationsof limit lines. The components of S are P(M1) = P(M1) and P(M2), the blowup

of P(M2) at the intersection point fM32 = f

M33 = 0. The two components are glued


along P(σ ) = P(M2/M1), which embeds in P(M1) as the line fM11 = 0, and in

P(M2) as the exceptional curve.

7. Bubble spaceIn §6 we chose finite convex Y ⊂ [F] containing Stab. Though there is a canonicalchoice, namely, the convex hull of Stab, it is more aesthetic to take the infinite set[F]. Let Yi be an increasing sequence of finite convex subsets, containing Stab, withunion [F]. Call Yi full along the simplex σ ⊂ Yi if Star[M] Yi = Star[M][F] for all[M] ∈ σ . It is clear that if Yi is full along σ , then so is Yj for j ≥ i. Let Ui ⊂ Yi bethe union of all UYi

(σ ) such that Yi is full along σ . The next lemma shows that wemay view Ui as an increasing sequence of open sets. We define S = ⋃

i Ui.

LEMMA 7.1If Yi is full along σ , then p−1

ji (UYi(σ )) ⊂ UYj

(σ ). Moreover, p−1ji (Ui) ⊂ Uj , and the

map p−1ji (Ui) → Ui is an isomorphism.

ProofAssume that Yi is full along σ . Take x ∈ p−1

ji (UYj(σ )), and take a maximal lattice

[N] ∈ Yj for x. Maximal lattices form a simplex, so [N] is adjacent to a lattice in σ ,and therefore [N] ∈ Yi because Yi is full along σ . Now it is clear that [N] is maximalfor pji(x), so [N] ∈ σ . Thus x ∈ UYi

(σ ) ⊂ Ui .To show that p−1

ji (Ui) → Ui is an isomorphism, it suffices to check that

p−1ji (UYi

(σ )) → UYi(σ ) is an isomorphism for any σ ⊂ Yi . This map is proper

and birational, and its domain and range are nonsingular. So to show that it is anisomorphism, it is enough to check that there are no exceptional divisors and soto check that each irreducible component of the special fibre of the domain mapsonto an irreducible component of the special fibre for the image. These componentsare the (appropriate open subsets of the) PYj

(M), [M] ∈ σ , and it is obvious thatPYj

(M) � PYi(M). �

THEOREM 7.2S is nonsingular and locally of finite type. Its special fibre, S∞, has smooth projectiveirreducible components and normal crossings. Let Bi ⊂ S be the hyperplane fi =0 of the generic fibre. Bi ⊂ S is closed and disjoint from S∞. B = ∑

Bi hasnormal crossings. In particular, there is a natural vector bundle �1

S/R(log B) whosedeterminant is ωS/R(B). The bundle is globally generated.

There are natural surjective maps pi : S → SYifor all i, and there are isomor-

phisms

p−1i

(�1

p(log B)) = �1

S/R(log B),

H 0(SYi

, �1Si /R

(log B)) → H 0

(S, �1

S/R(log B)).


S� SY S[Stab] S

Figure 6

The differential forms dlog(f/g), f, g ∈ F, define a natural inclusion

Vn ⊂ H 0(S∞, �1S∞).

The sections generate the bundle. The associate map S∞ → G(r − 1, n − 1) factorsthrough SYi

for all i, and its image is the special fibre of the pullback of the familyS → X(r, n) for the map Spec(R) → X//H given by F.

Figure 6 illustrates the case where r = 2.

ProofArguing as in Proposition 6.2, one can show that the special fibre of Ui → Spec(R)is disjoint from B ⊂ SYi

. It follows that S∞ is disjoint from B.We check that S → SYi

is surjective for all i. The rest then follows easily fromTheorems 6.5 and 6.7. Take [M] ∈ Yi. It is clear from the definitions that P(M) ⊂ SYj

surjects onto P(M) ⊂ SYifor j ≥ i. Moreover, there are only finitely many simplices

of [F] which contain [M] by Lemma 4.4, and for j large, Yj contains them all, fromwhich it follows that P(M) ⊂ Uj . Thus the image of S → Yi contains P(M). Theunion of the P(M) is the full special fibre, so S → Yi is surjective. �

The Deligne functor for [F] is not represented by a scheme, but S represents a naturalsubfunctor. In particular, S is independent of the choice of a sequence Yi .

THEOREM 7.3S represents the subfunctor of the Deligne functor for [F] a T -valued point of whichis a collection of compatible quotients such that each k′-point of T admits a maximallattice.


ProofTake a k-point of Ui . By Lemma 7.1, any lattice [M] ∈ Yi maximal in Yi for k ismaximal in [F]. It follows that Ui is a subfunctor of the functor in the statement,and thus S is a subfunctor. For the other direction, it is enough to consider T thespectrum of a local ring, with residue field k′. Consider a T -point of the subfunctor inthe statement. Note that in the proof of Lemma 5.4, the only place finiteness of Y isused is to establish the existence of a maximal lattice which here we assume. So thek′-point has a simplex of maximal lattices, σ , satisfying Lemma 5.4. For i large, Yi

is full along σ , and now it is clear that the quotients define a T -point of Ui and thusof S. �

8. Limit varietyThe matroid decomposition corresponding to the fibre of the visible contour familycan be readily obtained from the power series, as we now describe. From the matroiddecomposition, one can describe the fibre using [L2, Theorem 5.3]. We assume thatthe reader is familiar with the variation of geometric invariant theory (VGIT) quotients(see, e.g., [DH]). We note, in particular, that �(r, n) parameterizes PGLr-ample linebundles on P(r, n) with nonempty semistable locus.

We call a lattice M connected if the configuration of limiting hyperplanes FM isconnected. This obviously implies [M] ∈ [F]. Of course, stable implies connected.For r ≤ 3, they are the same (see Lemma 8.6), but they are in general different.

Definition 8.1Call a polarisation L ∈ �(r, n) on P(r, n) generic if there are no strictly semistablepoints. For [M] ∈ B, let CM be the arrangement of limiting hyperplanes

{f M = 0} ⊂ P(M), f ∈ F,

and let PM ⊂ �(r, n) be the matroid polytope of CM (see §3).

LEMMA 8.2If L ∈ �(r, n) is a generic polarisation, then there is a unique [M] ∈ [F] such thatthe configuration CM is L-stable. Moreover, M is connected.

ProofLet Q be the GIT quotient of P(r, n) given by L. Q is a fine moduli space for L-stableconfigurations and carries a universal family, a smooth etale locally trivial Pr−1-bundle. F gives a K-point of Q which extends uniquely to an R-point. The pullbackof the universal family is trivial over R (as R is Henselian), and so is P(M) for somelattice M ⊂ VK . It is clear that the limit configuration (given by the image in Q ofthe closed point of R) is equal to CM . As CM has no automorphisms, it follows that


this configuration contains r hyperplanes in general position, and so [M] ∈ [F]. Theproof shows that if CN, CM are both L-stable, then the rational map P(N) → P(M)is a regular isomorphism (both CN and CM are pullbacks of the universal family overQ), which implies that [N] = [M]. CM has a trivial automorphism group, so the finalremark holds by definition. �

By the next result, there are only finitely many connected equivalence classes.

THEOREM 8.3Let x ∈ X(r, n) be the limit point for the one-parameter family given by F. Assumethat x belongs to a stratum given by the matroid decomposition P . Then the maximal-dimensional polytopes of P are precisely the PM for which M is connected.

ProofBy [K1] and the theory of VGIT, the matroid decomposition is obtained as follows.GIT equivalence determines a polyhedral decomposition of �(r, n). Chambers (inte-riors of maximal-dimensional polytopes in the decomposition) correspond to genericpolarisations. For each such chamber, there is a corresponding GIT quotient that is afine moduli space for configurations of hyperplanes stable for this polarisation. Theone-parameter family has a unique limit in each such quotient, and, in particular,associated to each chamber, we have a limiting configuration. Associated to the con-figuration is its matroid polytope, and the polytopes obtained in this way are preciselythe facets of P . Now by Definition 8.1, if L is a polarisation in a chamber, thenthe limiting configuration is CML

for a unique [ML] ∈ [F]. Conversely, if we take[M] ∈ [F] so that CM has no automorphisms, the polytope PM ⊂ �(r, n) is maximaldimensional (see [L2, Theorem 1.11]). General L ∈ PM will be generic (in the senseof Definition 8.1), and it is clear that CM is L-stable, so M = ML. �

8.4. StratificationThe membrane [F] is, by Lemma 4.9, a union of apartments. We have an apartmentfor each I ⊂ N , |I | = r , and thus for each vertex of �(r, n). We stratify [F] byapartments—with one stratum for each collection of vertices—those points that lie inthese, but no other, apartments. It follows easily from Theorem 8.3 that the stratum isnonempty if and only if the collection is the vertices of some P ∈ P , in which casethe stratum consists of those [M] ∈ [F] (or, more generally, rational points of therealization; see §4.10) with PM = P . The dimension of the stratum is the codimensionof the polytope P in �(r, n). We write [F]k for the union of k-strata. Note that [F]0

is precisely the union of connected lattices.It is easy to describe the stratification in terms of the power series F. For any

a1, . . . , ar , let S(a1, . . . , ar ) be the stratification of [f1, . . . , fr ] by cones spanned by


(a4, b4, c4)

(a5, b5, c5)

(a3, b3, c3)

(a1, b1, c1) (a2, b2, c2)

Figure 7

rays R0, . . . , Rr , where

Ri = [za1f1, . . . , zai−1fi−1, z

ai+pfi, zai+1fi+1, . . . , z

ar fr ], p ≥ 0.

LEMMA 8.5Let fi = p1

i za1

i f1 + · · · + pri z

ari fr for i = r + 1, . . . , n, where pk

i ∈ R∗. Then theinduced stratification of an apartment [f1, . . . , fr ] is defined by intersections of theS(a1

i , . . . , ari ).

Figure 7 illustrates the case r = 3, where (a2, b2, c2) = (a − 3, b, c), (a4, b4, c4) =(a, b + 1, c), and so on.

ProofThe proof is immediate from the definitions. �

Now consider the limit pair (S, B). The irreducible components correspond to [F]0;for [M] ∈ [F]0, the corresponding component is the log canonical model ofP(M)\C[M], the complement to the union of limiting hyperplanes. We note by [L2,Theorem 5.3] that a collection of components has a common point of intersection ifand only if the corresponding matroids have a common face, which is if and only ifthe points in [F]0 all lie on the boundary of the corresponding stratum. In particular,if they have a nonempty intersection, they all lie in a single apartment.

From now on, we assume that r = 3.


LEMMA 8.6A configuration of lines in P2 is connected if and only if it contains four lines in lineargeneral position. A configuration is not connected if and only if there is a point in theconfiguration which is in the complement of at most one of the lines. In this case, theautomorphism group is positive dimensional.

ProofThe proof is easy linear algebra. �

Remark 8.7By Lemma 8.6, stable is the same as connected if r = 3. This fails in higher dimen-sions: the configuration of planes

x1 = 0, x2 = 0, x3 = 0, x4 = 0, x1 + x2 + x3 = 0,

x2 + x3 + x4 = 0

in P3 is connected but not stable.

LEMMA 8.8Let C be a stable configuration of lines indexed by N . Let S → P2 be the blowup ofall points of multiplicity at least 3. Let B ⊂ S be the reduced inverse image of thelines. Then KS + B is ample, and

P2\C ⊂ S

is the log canonical compactification except in one case: if there are points a, b on aline L of C such that any other line of C meets L in either a or b. In this case, (thestrict transform of ) L ⊂ S is a (−1)-curve, the blowdown is P1 × P1, and

P2\C ⊂ P1 × P1

is the log canonical model with boundary a union of fibres for the two rulings.

We refer to this exceptional case as a special stable configuration (see Figure 8).

ProofWe induct on n. When n = 4, then S = P2 and the result is obvious. So we assumethat n > 4. If the configuration is special, the result is clear, so we assume that it is not.By Lemma 8.6, we can drop a line, M , so the resulting configuration C ′ is stable. If C ′

is special, with special line L, then since C is not special, it follows that if we insteaddrop L, the resulting configuration is stable and nonspecial. So we may assume that


L1

Figure 8

C ′ is not special. Add primes to the notation to indicate analogous objects for C ′. Wehave q : S → S ′, the blowup along the points of M where C has multiplicity exactly3. (Note that S ′ → P2 is an isomorphism around these points.) Thus

KS + B = q∗(KS ′ + B ′) + M

(where we use the same symbol for a curve and for its strict transform). It is clear thatKS + B is q-ample. As KS ′ + B ′ is ample, the only curve on which KS + B can havenonpositive intersection is M . But (KS + B) · M > 0 by adjunction since C is notspecial. It follows that KS + B is ample. �

8.9. The limit surfaceNow we describe the limit pair (S, B) precisely. The irreducible components aresmooth and described by Lemma 8.8. We write SM for the component corresponding to[M] ∈ Stab. Unbounded 1-strata—rays in some apartment—correspond to irreduciblecomponents of B, and bounded 1-strata correspond to irreducible components ofSing(S). For each [M] ∈ Stab, the 1-strata that bound [M] correspond to boundarycomponents of SM (components of the complement to P(M)\CM ). SM and SN havea one-dimensional intersection if and only if [M], [N] comprise the boundary of a 1-stratum, and in that case they are glued along the corresponding boundary component(a copy of P1). Triple points of S (points on three or more components) correspondto bounded 2-strata. The local analytic singularities of (S, B) are described by thefollowing theorem, which was discovered independently by Hacking.

THEOREM 8.10Let p ∈ S be a point where the pair (S,B) fails to have normal crossings. There aretwo possibilities for the germ of (S,B) in an analytic neighbourhood p ∈ Up:

Up = 〈e1, e2〉 ∪ 〈e2, e3〉 ∪ 〈e3, e4〉 ⊂ A4, (8.10.1)


B ∩ Up is the union of 〈e1〉 and 〈e4〉, and these are components of a single Bi;

Up = 〈e1, e2〉 ∪ 〈e2, e3〉 ∪ · · · ∪ 〈en, e1〉 ⊂ An, (8.10.2)

n = 3, 4, 5, 6, and B ∩ Up = ∅.Here e1, . . . , en are coordinate axes in An, and 〈·〉 is the linear span.

ProofIt is simple to classify bounded 2-strata using Lemma 8.5. Then the gluing amongcomponents is described by [L2, Theorem 5.3]. �

9. The bundle of relative log differentialsHere we recall a general construction that is used at various points throughout thearticle. Let

p : (S, B) → C, B =n∑

i=1

Bi

be a pair of a nonsingular variety with normal crossing divisor, semistable over thecurve C, in a neighbourhood of 0 ∈ C (i.e., (S, F +B) has normal crossings, where F

is the fibre over zero). We assume that the general fibre is projective but not necessarilythe special fibre. We define the bundle of relative log differentials �1

p(log B) by theexact sequence

0 −−−−→ �1C/k(log 0) → �1

S/k(log F + B) → �1p(log B) → 0. (9.0.1)

Assume on the generic fibre S that the restrictions of the boundary components,Bi, Bj , are linearly equivalent. Then we can choose a rational function f on S so that

(f ) = Bi − Bj + E

for E supported on F . Then dlog(f ) gives a global section of �1S/k(log F + B). Note

that f is unique up to multiplication by a unit on C\0, and thus the image of dlog(f )in �1

p(log B), which we denote by dlog(Bi/Bj ), is independent of f .From now on, we assume that for the general fibre H 0(S, �1

S) = 0, dlog(Bi/Bj )is characterized as the unique section whose restriction to the general fibre has residue1 along Bi , −1 along Bj , and is regular off of Bi + Bj .

In this way, we obtain a canonical map

Vn → H 0(S, �1

p(log B))

(9.0.2)

(Vn is the standard k-representation of the symmetric group Sn) which is easily seento be injective (e.g., by the description of the residues on the general fibre).


The restriction

�1S/k(log B) := �1

p(log B)|S (9.0.3)

for (S, B), the special fibre of (S, B), is canonically associated to (S, B), that is, isindependent of the smoothing (see, e.g., [Fr, §3] or [KN]: these authors treat normalcrossing varieties without boundary, but the theory extends to normal crossing pairsin an obvious way). Finally, there is a canonical residue map (e.g., induced via (9.0.1)by ordinary residues on S)

Res : �1SY /k(log B)|Bi

→ OBi. (9.0.4)

Together with (9.0.2), this gives a canonically split inclusion

Vn ⊂ H 0(S, �1S(log B)

). (9.0.5)

Definition 9.1Let (S,B) be a pair of a normal variety S with a reduced Weil divisor B ⊂ S. Assumefor an open subset i : U ⊂ S with complement of codimension at least two that U isnonsingular and B|U has normal crossings. Define

�1S/k(log B) := i∗

(�1

U/k(log B|U )).

Acknowledgments. We thank B. Hassett, J. McKernan, and F. Ambro for help withunderstanding material related to the article. W. Fulton pointed out to us a seriouserror in an earlier version of the introduction. M. Olsson helped us a great deal withlog structures; in particular, we learned the construction in §2.16 from him. P. Hackingshowed some impressive flexibility in agreeing to work jointly with us on his modularresult. We especially thank L. Lafforgue for a series of detailed e-mail tutorials on [L2]and M. Kapranov for several illuminating discussions and, in particular, for posingto us the question of whether his properties in Section 1.1(3) could be generalised tohigher dimensions.

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KeelDepartment of Mathematics, University of Texas at Austin, Austin, Texas 78712, USA;[email protected]

TevelevDepartment of Mathematics, University of Texas at Austin, Austin, Texas 78712, USA;[email protected]

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