making conical compactifications wonderful

Sel. math., New ser. 4 (1998) 125 – 1391022–1824/98/010125–15$1.50 + 0.20/0

c©Birkhauser Verlag, Basel, 1998

Selecta Mathematica, New Series

Making conical compactifications wonderful

R. MacPherson and C. Procesi

Mathematics Subject Classification (1991). 14, 18.

Key words. Compactifications, normal crossrings, conical.

Introduction

Let M be a complex manifold. Suppose we are given a compactification X of Mthat is “conical” (Def. 1 below), which means roughly that every point on X hasa neighborhood in which X −M has the structure of a cone. In this paper weconstruct a “minimal wonderful” compactification X of M . The compactificationX is wonderful ([D-P2]) in the sense that X−M is a divisor with normal crossingsin X . Furthermore, X is minimal among the wonderful compactifications obtainedfrom X by blowing up along strata.

Several interesting compactifications in the literature turn out to be examples ofthe minimal wonderful compactification X constructed in this paper. These includesome compactifications of symmetric varieties of [D-P2], [D-P3], the compactifica-tions of configuration spaces of [F-M], and the compactifications of arrangementsof complements of linear subspaces of [D-P1]. See §4 for these examples.

1. Conical stratifications

1.1. Let X be a smooth n−dimensional analytic manifold. A stratification of Xis a decomposition X =

⋃α Sα of X , where {Sα} is a locally finite family of locally

closed disjoint analytic submanifolds of X called the strata, such that the closureSα of each stratum is a union of strata. This condition implies that the set ofstrata forms a poset: Sα ≥ Sβ if and only if Sβ lies in the closure of Sα. If X

R. MacPherson was partially supported by NSF, C. Procesi was partially supported byM.U.R.S.T. 40.

126 R. MacPherson and C. Procesi Sel. math., New ser.

is connected, then there is a unique open dense stratum in X called the genericstratum.

If Y → X is a submersion of complex manifolds (i.e. a map with surjectivedifferential), a stratification of X determines an induced stratification of Y whosestrata are the inverse images of the strata of X . If X and Y are both stratified,then there is a product stratification of X × Y whose strata are products of strataof X with strata of Y . The trivial stratification of X is the stratification with onestratum, namely X itself.

Assumption. For convenience, we will assume that all of the strata Sα in X areconnected. Further, we will assume that all Sα are locally connected relative to X ,i.e. every point x ∈ X has some neighborhood U such that Sα ∩ U is connected.

The assumption that the strata are connected involves no loss of generality, aswe can always stratifyX by the connected components of a given stratification. Theassumption that the strata are locally connected relative to X is for convenienceof exposition. It may be dropped as explained in §3.2.

We want to define the notion of a conical stratification. Consider the caseX = Cn, with the group C∗ acting by scalar multiplication.

We say that Cn is stratified as a cone if {0} is a stratum and all strata are C∗stable.

A disk stratified as a cone is an open ball D of radius r centered at 0 in Cnwhose stratification is induced from a stratification of Cn as a cone.

More generally we will also need to consider a closed cone X in Cn, i.e. a closedC∗ stable analytic (hence algebraic by G.A.G.A.) subvariety of Cn.

A local cone stratified as a cone is the intersection of a closed cone with anopen ball D of radius r centered at 0 in Cn, whose stratification is induced from astratification of Cn as a cone.

Remarks.

(1) Stratifying Cn as a cone is equivalent to stratifying the complex projectivespace Pn−1. Given a stratification of Pn−1, stratify Cn −{0} by the strati-fication induced by projection Cn−{0} → Pn−1, and add the stratum {0}.We call the stratification of Pn the projectivization of the stratification ofCn.

(2) Similarly, if D is a disk stratified as a cone, D − {0} fibers over Pn−1 withfibers punctured disks. The conical stratification of D is again induced byone in Pn−1.

Definition. A stratification of a smooth variety X is conical if, given any pointx ∈ X, there exists an open neighborhood A of x and an analytic isomorphism ofA with a product of two disks DT , DN such that, x corresponds to (0, 0), and thestratification induced in A is the product of the trivial stratification of DT with a

Vol. 4 (1998) Making conical compactifications wonderful 127

stratification of DN as a cone. In this situation, we call DT a tangent disk and DN

a normal disk at x.For a possibly singular variety X a stratification is conical if, given any point

x ∈ X, there exists an open neighborhood A of x and an analytic isomorphism ofA with a product of a disk DT with a local cone CN such that the stratificationinduced in A is the product of the trivial stratification of DT with a stratificationof CN as a cone. In this situation, we call DT a tangent disk and CN a normalcone at x.

Warning. A stratification of Cn as a cone need not be conical! (cf. Prop. 1.3)

A prime example of a conical stratification is a wonderful stratification:

Definition. A stratification of X is wonderful if X is smooth and there is a divisorwith normal crossings D =

⋃Di, where Di are nonsingular irreducible divisors

meeting transversely, such that two points are in the same stratum if and only ifthey are contained in the same set of irreducible divisors.

More generally, one could consider stratifications that are locally wonderful.See §3.2.

The fact that a wonderful stratification is conical follows from the fact that theconical property is analytically invariant and that locally a divisor with normalcrossings is isomorphic to a union of coordinate hyperplanes in Cn, n = dimX .

Conical stratifications are natural and well behaved mathematical objects, asindicated in the following propositions:

Proposition.

(1) The product stratification on X×Y is conical if and only if the stratificationsof X and Y are both conical.

(2) A conical stratification is locally finite, i.e. any x ∈ X has a neighbor-hood U that is met by finitely many strata of X. So if X is compact, thestratification is finite.

(3) A stratification of Cn as a cone is conical if and only if the associatedstratification of the complex projective space Pn−1 is conical.

(4) The closure of each stratum of a conical stratification is an analytic variety(possibly singular), with an induced conical stratification. The strata of aconical stratification of Pn−1 are algebraic subvarieties.

Proof. For the “if” part of (1), observe that if Cn and Cm are both stratified ascones, then the product stratification of Cn×Cm is stable under the product actionof C∗ ×C∗ and thus under the diagonal action. For the converse, first we considerthe case where the stratification of Y is trivial, i.e. it has only one stratum. Assumethat the product stratification of X × Y is conical, and choose x ∈ X . We wantto find tangent and normal disks DT , DN with the defining property at x ∈ X .


Choose an arbitrary y ∈ Y . Let DT , DN be tangent and normal disks for (x, y).DT ∩ (X × y) is a complex manifold because the map to Y has maximal rank. Forthe same reasons the map from (DT ∩ (X × y))× DN to X induced by projectionis a local isomorphism. For the general case, one reduces to the previous case byconsidering the top stratum of the Y times the stratification of X .

(2) We use induction on the dimension. It is clear that the first part of thestatement implies the second. Applying the second part of the statement to Pk(which is compact) for k < n, we get the first part for varieties X of dimension n,using tangent and normal disks (where k+ 1 is the dimension of the normal disk).

(3) Follows from (1) and the fact that Cn − 0 is a C∗ bundle over Pn−1 withlocal trivializations near each point in Pn−1.

(4) We use induction on dimension. Suppose that we know the first part forvarieties of dimension n−1. Then the closures of the strata of a conical stratificationof Pn−1 are analytic subvarieties of Pn−1, so by GAGA they’re algebraic, so thestrata themselves are algebraic. Suppose we know the second part for Pn−1. Thecone over an algebraic variety is algebraic. Let X have dimension n. Using tangentand normal disks, we get that the closure of each stratum in X is algebraic in localcoordinates, so it’s analytic.

Remark. It should be clear to the reader that usually even algebraic stratificationsare not conical. For instance consider the cuspidal cubic C := {y2 = x3} (in affinecoordinates) and stratify P2 by

P2 − C, C − 0, 0

it is easily seen that this is not a conical stratification (in 0).

1.2. Normal bundles to strata

For any stratum S of a conical stratification of a smooth variety X , there is astratification of the normal bundle TSX called the normal stratification constructedas follows:

For x ∈ S, choose tangent and normal DT and DN disks for a neighborhoodA of x, and an embedding DN ⊂ Cm as a round disk centered at 0, where Cm isstratified as a cone. This determines an identification of the restriction of TSX toA ∩ S with

DT × T0DN = DT × T0Cm = DT × Cm.

We stratify the restriction of TSX to A∩S by the product of the trivial stratificationof DT with the stratification of Cm. We must show that it is independent of thechoices involved.


Proposition.

(1) The stratification of the restriction of TSX to A defined above is indepen-dent of the choice of the tangent and normal disks, and is independent ofthe choice of embedding of the normal disk in Cm.

(2) For any two points x and y in the stratum S, there is an isomorphism of thefibers of TSX at x and y as vector spaces that identifies their stratificationsinduced from the stratification of TSX.

(3) For any two points x and y in the same stratum S, if DxT and Dx

N aretangential and normal disks near x and Dy

T and DyN are tangential and

normal disks near y, then the stratifications of DxN and Dy

N are identical,when restricted to possibly smaller neighborhoods of 0 in each disk.

Proof. We will construct the stratification of TSX by a process that is independentof the choice of normal and tangential disks. Let Z be a closed union of strata ofX . Then the corresponding closed union of strata in TSX is the normal cone toS along Z determined by the following property: Let f : C → X be any analyticcurve parameterized by an open set C ⊂ C such that f maps 0 ∈ C to S and mapsC to Z. The derivative of f at 0 projects to a vector in the normal cone to Salong Z. The individual strata of the stratification of TSX can be reconstructedby a process of taking differences.

If X is not smooth the previous analysis extends but we have to replace thenormal bundle with a normal cone bundle.

Remark. Any conical stratification is Whitney. This is because a stratificationof a disk Dn conically induced from a Whitney stratification of Pn−1 is Whitney,and the product in DN × DT of a Whitney stratification of DN with a trivialstratification of DT is Whitney. The Whitney conditions will not be used explicitlyin this paper.

2. The minimal wonderful blowup

2.1. Irreducible strata

We come now to the main definition.

Definition. Suppose that X has a conical stratification. A stratum S is reducibleif for any (and hence every) point x ∈ S there is a neighborhood A of x with adecomposition A = DT ×D1

N ×D2N into disks (or local cones in the singular case),

where D1N and D2

N are of positive dimension, such that the induced stratificationof A is the product of the trivial stratification of DT with conical stratifications ofD1N and D2

N . Otherwise S is irreducible.


If 0 is reducible in one of the normal disks, say D2N , we can further factor a

smaller neighborhood of 0 in D2N as D2

N×D3N . Proceeding in this way, we can finally

find a neighborhood of x with an isomorphismD = DT×D1N×D2

N×· · ·×Dk−1N ×Dk

N

where the origin is irreducible in each DiN . We will call such neighborhood a

complete product neighborhood.We want to show that the list of stratified disks Di

N obtained in this way isunique. In fact, they are characterized as the normal disks to a list of strata calledthe factors of S:

Definition. Given any stratum S, the factors of S are the irreducible strata Asuch that A ≥ S and there is no irreducible stratum T with A > T ≥ S.

Lemma. If we have a product stratification X × Y then an irreducible stratum iseither of the form S×Y with S irreducible in X or X×T with T irreducible in Y .

As a consequence we have:

Proposition (Unique factorization of conical stratifications). Assume that D =DT × D1

N × D2N × · · · × Dk−1

N × DkN with the Di

N irreducible in the origin. Anyirreducible stratum of the induced stratification of D is of the form DT × D1

N ×D2N ×Di−1

N ×S×Di+1N . . . Dk−1

N ×DkN where S is an irreducible stratum in Di

N . Inparticular, the factors of S = DT × O1 ×O2 × · · · × Ok are the strata of the formDT ×D1

N ×D2N ×Di−1

N ×Oi ×Di+1N . . . Dk−1

N ×DkN where Oi is the origin in Di

N .

Remark. Let S be any stratum and let U be a neighborhood of S that onlyintersects strata ≥ S. Then within U , the closures of the factors of S are smooth,they intersect transversely, and their intersection is S.

2.2. Blowups

From now until 3.3 we shall discuss the case in which X is smooth.Recall the blow-up construction: If Y ⊂ X is a closed nonsingular analytic

subvariety, then the blow-up of X along Y is a smooth analytic variety BlY (X)equipped with a map π : BlY (X)→ X .

We first consider the case that X is Cn and Y is the origin. Then BlY (X) =Bl0(Cn) is nothing but the total space of the tautological bundle of Pn−1. Theprojection Bl0Cn → (Cn) takes the complement of the 0 section isomorphically toCn−0, and takes the 0 section to 0 ∈ C. The 0 section is itself isomorphic to Pn−1.

Definition. Suppose that Cn is conically stratified as a cone. Then the inducedstratification of Bl0(Cn) is constructed as follows:

Bl0(Cn)− π−1(0) = Cn − 0 is stratified the same way Cn − 0 is, and π−1(0) =Pn−1 is stratified by the projectivization of the stratification of Cn.


Lemma. The induced stratification of Bl0(Cn) is conical. The irreducible strataof this stratification are the irreducible strata in Cn − 0 together with the genericstratum of the 0 section of the line bundle.

Proof. Each point of the 0 section has a product neighborhood (by trivializing theline bundle) where the induced stratification is the product stratification. So theclaim follows.

Let D be the unit ball in Cn, stratified as a cone. Then Bl0D is the unitdisk bundle in the tautological bundle over Pn−1. It is stratified by restricting thestratification of Bl0Cn, we shall use the notation τD := Bl0D.

2.3. Blowing up an irreducible stratum

Definition. A stratum S is called minimal irreducible if it is irreducible, and everystratum < S is reducible.

Proposition. Suppose that S is a minimal irreducible stratum of a conical strati-fication of X.

(1) The closure Y of S is nonsingular.(2) Every point y ∈ Y has neighborhood D with a product stratification DY ×

DN , where DN is a complex analytic ball stratified as a cone, and Y ∩D =DY × 0.

(3) The blowup BlY X of X along Y has an induced stratification character-ized by the property that, over the product neighborhood D = DY × DN

it coincides with the product stratification BlY ∩DD = DY × τDN , whereτDN = Bl0DN is stratified as in Lemma 2.2.

(4) The induced stratification of BlY X is conical.(5) The irreducible strata of BlY X are the irreducible strata of BlY X−π−1Y =

X − Y together with the generic stratum of π−1Y .

Proof. Take a point y ∈ Y −S. Then y must lie in some reducible stratum T . Takea complete product neighborhood D of y, namely D = DT × D1

N × D2N × · · · ×

Dk−1N ×Dk

N with the DiN irreducible in the origin. Now, since S is irreducible, we

know that for some i, S ∩D is an irreducible stratum R in DiN times the generic

stratum in the other factors. We assert that R is the zero stratum in DiN . If not,

the 0 stratum in DiN times the generic stratum in the other factors would be an

irreducible stratum in the closure of S, contradicting the minimality of S. Nowtake DN to be Di

N and take DY to be the product of the other factors. This provesthe first two statements.

For the third statement, the difficulty is to show that locally the stratificationof the exceptional divisor of BlY X is independent of the product structure chosenon the neighborhood. We do this by giving an invariant description of the induced


stratification of the exceptional divisor, which is the projectivization of the bundleTYX to Y . The normal bundle TYX has a canonical stratification derived fromthat of DN . It also has the stratification obtained by taking inverse images ofthe strata of Y . The induced stratification is the common refinement of these twostratifications.

The fourth statement follows from the fact that conical stratifications are pre-served under products. The last statement follows from the unique factorization ofstratifications.

2.4. Construction of the minimal wonderful stratification

The result of the last section can be paraphrased as follows: Blowing upX along theclosure Y of a minimal irreducible stratum S preserves the number of irreduciblestrata, preserves the codimension of all of them except for S, and replaces S by anirreducible stratum of codimension 1. This represents progress towards producinga wonderful blowup for the following reason:

Proposition. A conical stratification is wonderful if and only if the irreduciblestrata are all of codimension 1.

Proof. It is clear that a wonderful stratification has the given property. Conversely,suppose that the irreducible strata are all of codimension 1. For any point x ∈ X ,choose a complete product neighborhood of x. All of the normal cones CiN areone dimensional hence smooth and X is smooth, by the unique factorization resultof §2.1. Therefore, the stratification is wonderful, at least locally. The global resultfollows from the fact that the strata are locally connected.

Definition. The minimal wonderful blowup X of X is the result of repeatedlyblowing up the irreducible strata of minimal dimension, until all irreducible strataare of codimension 1. (Irreducible strata of minimal dimension are, of course,minimal irreducible strata.)

More precisely, let m be the number of irreducible strata of X . Let X0 = X ,and for all 1 ≤ i ≤ m, let Xi be the result of blowing up of Xi−1 along a theclosure of an irreducible stratum of minimal dimension of Xi−1, with the inducedstratification. Then Xm = X .

Proposition. The minimal irreducible compactification is canonically independentof the choices involved: Suppose π1 : X1 → X and π2 : X2 → X are minimalirreducible compactifications constructed from two different sequences of choices.Then π−1

2 π1 takes the generic stratum of X1 isomorphically to the generic stratumof X2. This extends uniquely to an isomorphism X1 → X2.

Proof. Suppose that Y1, Y2, . . . , Yj are the closures of the k dimensional irreduciblestrata of X , and every other irreducible stratum has dimension > k. Let X be theresult of blowing up first along Y1, then along the proper transform of Y2, and so


on. More precisely BlY (X), is constructed iteratively as follows: let X0 be X itself.Suppose that Xi has been constructed, and that it maps to X via the compositionXi → Xi−1 → · · · → X0 = X . Then the proper transform of Yi+1 in Xi isnonsingular. Xi+1 is the blow up Xi along the proper transform of Yi+1. ThenXi+1 maps to Xi and hence to X via the compositionXi+1 → Xi → · · · → X0 = X .Then Xj = X .) We need to show that X is independent of the ordering of thevarieties Y1, Y2, . . . , Yj .

First, blowing up is a local notion, in the sense that if a smooth variety X hasan open covering Ui and Y is a smooth subvariety then the varieties BlY Ui forman open covering of BlY X , so it is enough to prove it locally.

Second, for x ∈ X , take a complete product neighborhoodD = DT×D1N×D2

N×· · ·×Dk−1

N ×DkN of x. Consider the varieties Yi which have a nonempty intersection

with D, we may assume (renumbering them) that they are the first h and that, aftera change of numbering of the normal disks, Yi∩D = DT ×D1

N×· · ·×0i×· · ·×DkN ,

where 0i is the zero point inside the i-th disk. This follows from §2.1. Then thepart of X over D is

DT × τD1N × τD2

N × · · · × τDhN ×Dh+1

N × . . . Dk−1N ×Dk

N

(remember that τD is the blowup of a disk at the origin). This is visibly indepen-dent of the ordering. Moreover the same argument implies that this independenceis also valid as stratified space in case the Yi are the closures of the minimal irre-ducibles since locally the stratification we obtain is always the product stratificationof this expression.

Remark. In fact it is not essential to blow up the irreducible strata in order ofincreasing dimension. They can be blown up in any order, provided that at everystep the next blowup is along the closure of a minimal irreducible stratum.

Question. For a sequence of positive integers V = v1, v2, . . . , vn, let IV be thesheaf of ideals of functions on X that vanish to degree ≥ vi on each irreduciblestratum Xi of codimension i. Then for v1 � v2 � · · · � vn, the blowup of X atthis IV is the wonderful blowup.

3. Formal properties of the wonderful blowup

The divisors (closures of codimension one strata) of the minimal wonderful com-pactification X of X are clearly in one to one correspondence with the irreduciblestrata of X . Since X is wonderful, any other stratum is an intersection of someset of divisors, so it will correspond to some subset of the set of irreducible strata.The subsets that occur are the nested subsets, as described below.


3.1. Nested subsets

A flag of strata or simply a flag, is a sequence S1, . . . , Sk of strata such that, foreach i, we have Si ≤ Si+1.

We now give two definitions of a nested set of irreducible strata.

Definition. A set S of irreducible strata is called nested if it satisfies one of thefollowing two equivalent properties:

(1) There exists a flag of strata F such that the elements in S are all irreduciblefactors of elements of F .

(2) Let A1, . . . , Ak be the minimal elements of S and let Si be the set ofelements in S that are > Ai. Then A1, . . . , Ak are all factors of some singlestratum, and Si is nested, as defined by induction.

Given a flag of strata, we can associate to this the set of all factors of all thestrata in the flag. We call this the factorization of the flag.

Theorem.

(1) The irreducible divisors of the boundary divisor of the wonderful blowup areindexed by the irreducible strata of X. The irreducible divisor is the closureof the inverse image of the corresponding irreducible stratum.

(2) The family of divisors indexed by some set T of irreducible strata hasnonempty intersection if and only if the set T is nested.

(3) The strata of the wonderful blowup are indexed by the nested sets. Theorder of strata is the opposite to the inclusion ordering of nested sets.

Proof. This proof is by induction based on the basic step of blowing up the closureof the minimal irreducibles.

A maximal set of divisors with nonempty intersection is the set of factors of amaximal flag.

Proposition. The number of factors of a maximal flag equals the length of theflag.

Proof. By induction, take a minimal stratum S0 in the flag and its decomposition ina point DT ×

∏iD

iN , a minimal stratum over S0, locally in this disk, is of the form

DT ×Oi×∏i6=j Xj where Xj is minimal in Dj

N . In particular S1 is of this form forsome j1. Of the factors of S0 there is only one which does not appear by inductionin the factorization of the flag starting at S1 – that is DT ×

∏i6=j1 D

iN ×Oj1 .


3.2. Strata that are not locally connected

If we drop the simplifying assumption that strata of the conical stratification of Xare locally connected, there will still be a minimal wonderful blowup X . However,the strata of the wonderful blowup will not necessarily be locally connected. (Sothe irreducible divisors in the wonderful blowup will not necessarily be smooth:they will themselves have singularities with normal crossings.)

Consider a complete product neighborhood D = DT ×D1N ×D2

N×· · ·×Dk−1N ×

DkN of a point x ∈ X . Let Si ⊂ D be the product of the zero stratum in Di

N withthe generic stratum in the other factors. Failure of local connectedness allows thepossibility that more than one of the Si may be the intersection of D with the samestratum of X . Now, as in the proof of Proposition 2.4, suppose that the smallestirreducible strata have dimension k. We want to blow up the closure of S1, thenthe proper transform of the closure of S2, and up to Sj, where D1

N , . . . , DjN are the

n− k dimensional disks. Globally, this is impossible since the stratum intersectingD in S1 may also intersect D in another Si. However locally within D, we can dothis, and the result is canonically independent of the choices. The result is obtainedby doing these blowups locally, and patching them together globally.

3.3. Singular varieties

The minimal wonderful blowup can also be constructed for a conical stratificationof a singular variety. It is again a nonsingular variety stratified by a divisor withnormal crossings. (So, in particular, it provides canonical resolution of singularitiesof a conical singular stratified variety.) We sketch the theory here.

The blowup construction can be applied whenever one has an ideal sheaf I inan analytic variety X . The blowup is defined as Proj(⊕∞k=0Ik).

Consider a closed coneC in Cn, which is the affine cone associated to a projectivevariety C, and blow up C at the origin, i.e. at the sheaf of functions vanishing atthe origin. The resulting variety is the restriction to C of the tautological bundle onthe projective space. For a local cone C we shall again denote by τC the resultingblow up. One should remark that now the stratification of τC is such that thenormal cones are all of smaller dimension than C.

Proposition 2.3 needs several changes. One has to use again the fact thatblowing up is a local notion and moreover that, if a variety is a product X×Y andthe ideal sheaf being blown up is the ideal sheaf of the subvariety X ×O with O apoint in Y , the resulting blow up is X × BlO Y .

This, when applied to a closure of an irreducible stratum produces, in the blownup variety, a stratification where the minimal irreducible strata have normal conesof smaller dimension. This fact implies that after a sequence of blow ups all normalcones become disks, and the final model is smooth and wonderful.


4. A wider class of wonderful compactifications

The previous notions are related to the ideas of building sets introduced in [D-P1]. These provide a finite family of wonderful models obtained by a sequenceof blowups along closures of strata, of which the one constructed before is in factreally minimal.

Definition. Let W be a set of strata ≥ S. We say that S is factored by W iffor any (and hence every) point x ∈ Sα there is a neighborhood A of x with adecomposition D = DT × D1

N × · · · ×DmN , where Di

N is stratified as a cone, andthe strata in W intersected with A are the products of 0 in one normal disk Di

N

with the open dense stratum in the other normal disks. Additionally, a stratum Sis factored by the set consisting of itself.

It follows that each stratum in W has a nonsingular closure in some neighbor-hood of S, and these closures intersect transversely in S.

Definition.

(1) A set G of strata is a building set if any stratum S is factored by the minimalelements in G which are ≥ S. (This condition is always satisfied if S is anelement of G.)

(2) Given a stratum S, a minimal element G ∈ G with G ≥ S is called a Gfactor of S.

(3) Let G be a building set. A set T ⊂ G of strata is called nested withrespect to G if it satisfies one of the following two equivalent properties:

a) There exists a flag of strata F such that the elements in T are allG-factors of elements of F .

b) Let A1, . . . , Ak be the minimal elements of T and let Ti be the setof elements in T that are > Ai. Then A1, . . . , Ak are all G- factorsof some single stratum, and Ti is nested, as defined by induction.

Examples. The set of all irreducible strata is a building set: it is the smallestone. The set of all strata is a building set: it is, quite naturally, the largest one.See [D-P1] for a procedure for enlarging a building set.

One can use building sets, rather than just the set of irreducible strata, toconstruct non minimal wonderful blowups. The procedure follows verbatim thatalready developed in 2.3, 2.4.

First prove that the closure Y of a minimal element S of a building set G issmooth. Next blow up Y and construct in the blow up BlY (X) the induced buildingset. This consists of all the strata in G different from S, and hence disjoint fromY , and the open stratum in the exceptional divisor of the blow up. One verifiesthat this is again a building set and proceeds until all the elements of the buildingset are divisors. We will denote by XG the wonderful model so obtained.


It is easy to see that, given two building sets G1, G2, there is a (unique) mapXG1 → XG2 , which is the identity on the generic stratum, if and only if G1 ⊃ G2and that every building set contains all the irreducible strata, thus

Corollary. The minimal wonderful blowup is the minimal blowup among the setof blowups constructed from building sets.

As in the case of the minimal wonderful model, one has that the irreducibledivisors inXG are indexed by the elements of G and that a set of them has nonemptyintersection if and only if the corresponding elements of G are G- nested.

5. Examples

In this section, we consider several examples of complex manifolds M that havean obvious “naive” compactification X , with an obvious stratification that is con-ical. In the examples considered here, the minimal wonderful compactification Xwas already known, and it turned out to be useful. The construction of the min-imal wonderful model is a first step in some geometric program for the manifoldunder consideration (for example, defining an intersection theory for enumerativegeometry purposes, or computing cohomology and its Hodge structure).

5.1. Configurations spaces

Let Z be a compact complex manifold and let M be the N−th configurationsspace of Z: a point in M is an ordered set of N distinct points in Z. A naivecompactification of M is the N -fold cartesian product {z1, z2, . . . , zN |zi ∈ Z}. Ithas a stratification X =

⋃α Sα, where α is a partition of the set {1, 2, . . . , N}

into subsets and Sα is defined by the condition that zi = zj if i and j are in thesame subset. This stratification is conical since being conical is a local propertyand locally this is a special case of the next example, which is clearly conical. Theminimal wonderful compactification X of M is the compactification constructedin [F-M].

5.2. Complements of configurations of linear subspaces

Let X be a complex projective space (or an open set in a projective space), letLi be a collection of proper distinct linear subspaces of X , and let M be thecomplement X −

⋃i Li. Then X itself is a naive compactification of M . It has

a stratification X =⋃α Sα, where α is a subset of the Li and Sα consists of

points in the intersection of all Li in α that do not lie in any smaller intersectionof the linear subspaces. This stratification is conical, and the minimal wonderfulcompactification X of M is the compactification constructed in [D-P1].


Of particular interest is the case of the projective space associated to a rootsystem and its root hyperplanes.

In particular for type An−2 the minimal blowup of this configuration of hyper-planes coincides with the compactification of the moduli space of stable N−pointedcurves of genus zero M0,n considered by Keel [K], (cf. also Kapranov [Ka]).

5.3. Complete quadrics, projectivities and null-correlations

Let M be the variety of nonsingular quadric hypersurfaces in Pn. It is compactifiedby the variety X of all quadric hypersurfaces in Pn, which is stratified by the rank ofthe symmetric matrix giving the quadratic equation. This is a conical stratification.The minimal wonderful compactification X is the variety of complete quadrics, asin [D-P2], [D-P3]. In this case, all strata of X are irreducible and all subsets arenested.

Similarly we can consider the projective space associated to the linear spaceof n × m matrices resp. n × n skew symmetric matrices stratified by rank. Theopen stratum is classically called the space of projectivities resp. the space ofnull correlations. In these cases as well the stratification is conical, all strataare irreducible and all sets of strata nested. The resulting minimal wonderfulcompactifications are the space of complete projectivities resp. complete null-correlations. The way to prove that the stratifications are conical is to considerthe group of symmetries of the stratification that for symmetric or skew symmetricn × n matrices is the general linear group GL(n,C) while for the n ×m matricesis the product GL(n,C) × GL(m,C). In every case the strata are the orbits. Soto verify the conical property we can do it in just one point p in each orbit. Onecan always then choose a normal slice that is stable under some subgroup of thestabilizer of p, which acts as scalars on this normal slice giving rise to the conicalstructure.

In these examples, it is interesting to analyze the action of the group of sym-metries also on the wonderful blowup. One sees that again the strata are orbits.

References

[D-P1] C. De Concini and C. Procesi. Wonderful models of subspace arrangements. SelectaMathematica 1 (1995), 459–494.

[D-P2] C. De Concini and C. Procesi. Complete symmetric varieties. CIME 82 Springer L. N.997 (1983), 1–44.

[D-P3] C. De Concini and C. Procesi. Complete symmetric varieties II. Intersection Theory.Adv. Stud. Pure Math. 6 (1985), 481–513.

[F-MP] W. Fulton and R. MacPherson. A compactification of configuration space. Ann. of Math.(2) 139 (1994), 183–225.

[Ka] M.M. Kapranov. Veronese curves and Grothendieck-Knudsen moduli space M0,n. J.Algebraic Geom. 2 (1993), 239–262.


[K] S. Keel. Intersection theory of moduli space of stable N−pointed curves of genus zero.T.A.M.S. 330 (1992), 545–574.

R. MacPhersonDept. of MathematicsThe Institute of Advanced StudiesPrinceton, NJ 08540USAe-mail: [email protected]

C. ProcesiDip. di MatematicaUniversita di Roma“La Sapienza”Piazzale Aldo Moro 2I-00185 RomaItalye-mail: [email protected]

making conical compactifications wonderful

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