Alignment correspondences

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izing

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Journal of Algebra 278 (2004) 553–570

www.elsevier.com/locate/jalgebr

Alignment correspondences

Heather Russell

Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA

Received 1 July 2003

Available online 5 June 2004

Communicated by Paul Roberts

Abstract

Given a smoothn-dimensional varietyX over a fieldK and a sequence ofr monomial ideals offinite colength in the ringK[[x1, . . . , xn]], we study corresponding variety inside the product or

punctual Hilbert schemes. We also study the closures of such varieties. 2004 Elsevier Inc. All rights reserved.

1. Introduction

The subschemes of a variety with a fixed Hilbert polynomial are parametrizea Hilbert scheme as constructed by Grothendieck in [7]. If the Hilbert polynomia constantd , then the Hilbert scheme, denoted Hilbd(X), parametrizes degreed zero-dimensional subschemes ofX. Such Hilbert schemes are called punctual Hilbert schemThe Hilbert scheme Hilbd(X) has a stratification with each stratum correspondinga partition ofd . In this paper, we are concerned only with the stratum parametrsubschemes ofX with a single point of support.

Our basic objects of study can be described as follows. LetX be a smooth, propevariety of dimensionn over an arbitrary fieldK. Let R be the ringK[[x1, . . . , xn]]and m its maximal ideal. LetI be an ideal of colengthd in R. The spaceU(I) ofsubschemes ofX isomorphic to Spec(R/I) has a natural embedding in the punctual Hilbscheme Hilbd(X) as a locally closed subvariety (Theorem 2.1). We study the more gespace

E-mail address:[email protected].

0021-8693/$ – see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.jalgebra.2004.04.010

554 H. Russell / Journal of Algebra 278 (2004) 553–570

es. Ane idealbient

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imarys. Thebe lessdencemost

g to ag to a

fncesah [4]acting

nton theima’s

finingnment

U(I1, . . . , Ir ) = {(a1, . . . , ar) ∈ U(I1) × · · · × U(Ir): ∃p ∈ X and

ϕ :R ∼−→ OX,p with ϕ(I1, . . . , Ir ) = (a1, . . . , ar )}

and its closureC(I1, . . . , Ir ) in the appropriate product of Hilbert schemes. If eachIj is amonomial ideal of finite colength, we will say that the spaceC(I1, . . . , Ir ) is analignmentcorrespondencewith interior U(I1, . . . , Ir ). We will call the sequence of idealsI1, . . . , Ir ,thedefining sequenceof the alignment correspondence.

Alignment correspondences are named for their connection to aligned schemaligned scheme is a zero-dimensional scheme aligned to a jet in the sense that thcorresponding to the scheme in the local ring at the point of support in the amvariety can be expressed in terms of the curvilinear ideal corresponding to the jet and tmaximal ideal in the local ring. Aligned schemes can arise from the process of specializconfigurations of fat points, called the method of Horace [1,3]. Under a suitable cholocal coordinates, the ideal corresponding to the aligned scheme is a monomial idI .Thus the varietyU(I) parametrizes aligned schemes isomorphic to the given one acanonically isomorphic to a jet bundle.

The interior of an alignment correspondence can be thought of as a generalof a jet-bundle. The points of such a variety parametrize approximations of coordframes in the ambient variety. Like jet-bundles, interiors of alignment correspondhave simple geometry as described in Theorem 3.1. Their isomorphism class is deteby a sequence ofn monomial ideals, called the measuring sequence of the aligncorrespondence, together with the group of permutations of thexi ’s preserving the defininsequence.

The simple geometry of the interiors of alignment correspondences is the prmotivation for requiring the defining sequences to be composed of monomial idealmotivation for having defining sequences rather than just a single defining ideal mayobvious. Counterintuitive as it may be, one can often obtain an alignment corresponwith simpler structure by adding ideals to the defining sequence. In fact, one of theeffective ways of understanding a given alignment correspondence correspondinsingle ideal is often through studying an alignment correspondence correspondinsequence of ideals that includes that single ideal.

Given an alignment correspondence corresponding to a sequence ofr ideals, manyrelations that hold among the ideals translate into relations among ther projections, evenon the boundary. For example, the varietyXr = C((x, y), (x, y2), . . . , (x, yr)) studiedin [5] corresponds to a sequence ofr nested ideals. Ther projections at a point othis variety correspond to a sequence ofr nested schemes. Incidence correspondebetween Hilbert schemes based on inclusions have also been studied by Cheand Nakajima [10]. Nakajima uses these correspondences to create operatorson the cohomology of punctual Hilbert schemes of a surface. Similarly, alignmecorrespondences corresponding to a pair of ideals also give rise to operatorscohomology rings of Hilbert schemes, but it is difficult to relate these maps to Nakajoperators.

For any pair of alignment correspondences, there is a third alignment with desequence given by concatenating the defining sequences of the pair of alig

H. Russell / Journal of Algebra 278 (2004) 553–570 555

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correspondences. The measuring sequence for this new alignment correspondenceby intersecting the ideals in the measuring sequences of the pair. A natural quto ask is whether there is a universal alignment correspondence for a given measequence in the sense that if we add more ideals to the defining sequencechanging the measuring sequence, we get an isomorphic alignment correspondeninitial expectation was that the answer would be positive for all measuring sequeThis is because relations among ideals in the defining sequence translate into relatioamong the corresponding projections at boundary points of an alignment corresponWe construct universal alignment correspondences for certain measuring sequeTheorem 4.3. However, in Theorem 4.2, we show that there is no universal aligncorrespondence for most of the remaining measuring sequences.

The motivation for this work is two-fold. On one hand we are motivated by applicato enumerative problems. This approach to studying Hilbert schemes has also beeby Danielle and Le Barz in [6], for example. An application of our work to counnumbers of curves in linear series on algebraic surface with a given type of singucan be found in [11]. Similar enumerative problems are studied in [9] using the sH(D) corresponding to an Enriques diagramD. The spaceH(D) is itself an alignmencorrespondence corresponding to a single ideal.

We are also motivated by the question of how families of schemes can degenSome classic work has been done on this subject by Briançon [2], Iarrobino [8]Yameogo [13]. An application of the theory developed here to the study of degeneratioof schemes can be found in [12].

2. Preliminaries

Let I1, . . . , Ir be a sequence of monomial ideals of finite colength inR andG(I1, . . . , Ir ) the group of automorphisms ofR stabilizing this sequence. The fiberU(I1, . . . , Ir ) over X can be identified with the quotient Aut(R)/G(I1, . . . , Ir ) via anyisomorphismϕ :R → OX,p . The group Aut(R) acts on the fiber ofC(I1, . . . , Ir ) overXthrough this isomorphism. IfG(I1, . . . , Ir ) is contained inG(J1, . . . , Js), this identificationinduces a map fromU(I1, . . . , Ir ) to U(J1, . . . , Js). This map need not extend to a mfrom C(I1, . . . , Ir ) to C(J1, . . . , Js). We will say that such maps as well as their extensiand restrictions arenatural.

Theorem 2.1. Given a sequence of monomial idealsI1, . . . , Ir of finite colengthsd1, . . . , dr

respectively, the spaceU(I1, . . . , Ir ), is a locally closed subset of the spaceH =Hilbd1(X) × · · · × Hilbdr (X).

Proof. From the product of Hilbert Chow morphisms

ϕ1 × · · · × ϕr :H → Symd1(X) × · · · × Symdr (X),

we see that there is a closed subvarietyY in H consisting of sequences of schemes wa single, common point of support. The spaceY is a fiber bundle overX containing


s

p.

by

ealsite

U(I1, . . . , Ir ). A fiber of U(I1, . . . , Ir ) overX is an Aut(R) orbit of Y . Hence, coveringXby neighborhoods in which there is a continuous section ofU(I1, . . . , Ir ) and a continuouaction of Aut(R) restricting to the usual action on the fibers overX, the spaceU(I1, . . . , Ir )

is locally closed inY and therefore inH . �

3. Measuring sequences

Definition. Let I1, . . . , Ir be a sequence of monomial ideals of finite colength inR. Foreach integeri between 1 andn, let Ai be the ideal generated by images ofxi underautomorphisms ofR fixing eachxj for i �= j and stabilizing eachIk . We will say thatthe sequenceA1, . . . ,An is themeasuring sequenceof the sequence of idealsI1, . . . , Ir .

Example 3.1. In characteristic 3, the measuring sequence of the ideal(x3, y3) is (x, y),(x, y). In all other characteristics, the measuring sequence is(x, y3), (x3, y).

Proposition 3.1. Each idealAi in a measuring sequence is a monomial ideal.

Proof. Since theIk ’s are monomial ideals, the subgroup of elements ofG(I1, . . . , Ir )

fixing xj for i �= j is stable under conjugation by the automorphisms scaling thexk ’s.ThusAi contains all the terms in the images ofxi under automorphisms in this subgrouSince these terms are sufficient to generateAi , we see thatAi is a monomial ideal. �

In characteristic 0, for theith idealAi in the measuring sequence ofI1, . . . , Ir we have

Ai = {f ∈ R: (Ij : xi) ⊂ (Ij : f ) ∀1 � j � r

}.

This is because the ring homomorphism sendingxi to xi +f and fixing the otherxj ’s takeselements ofIj to Ij if and only if (Ij : xi) ⊂ (Ij : f ).

By the following proposition, we see that theideals in a sequence can be constructedthe ideals in their measuring sequence.

Proposition 3.2. Let A1, . . . ,An be the measuring sequence of a sequence of idI1, . . . , Ir . Given a vectorα = (α1, . . . , αn) with non-negative integers as coordinates wr

α =m∑

i=0

pivi ,

wherep is the characteristic ofK, m is minimal and the coordinates of eachvi are non-negative integers that are less thanp if p is positive. Ifp is positive, letF be the Frobeniusmap. Let

Aα = Aα11 . . .Aαn

n , xα = xα11 . . . xαn

n , and A(α) = Av0F(Av1

). . .Fm

(Avm

).

If p is zero, thenAα andA(α) are the same. ThenIk is the sum of the idealsA(α) for themonomial generatorsxα of Ik .


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ains tof

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tialences

Proof. Suppose, by way of contradiction, that there is an exponent vectorα and amonomialf , such thatxα ∈ Ik , f ∈ A(α), andf /∈ Ik . Thenf can be expressed as tproductf0 . . . f

pm

m , where eachfi is a monomial inAvi . Chooseα so that eachfi = xwi f ′i

with f ′i ∈ Avi−wi of minimal degree. Leta be the largest integer such thatva �= wa . Let

j be an integer such that thej th coordinateb of va − wa is greater than 0. Then theare monomialsh1, . . . , hb ∈ Aj such thatf ′

a = h1 . . .hbh with h ∈ Ava−wa−bej and thereis an elementg ∈ G(I1, . . . , Ir ) with g(xj ) = xj + h1 and g(xi) = xi for i �= j . The

monomialxα(h1/xj )pb

has non-zero coefficient in the expansion ofg(xα) and hence isin the monomial idealIk . Let β be the exponent vector corresponding to this monomThenf is in A(β). This contradicts the minimality of the degrees off ′

a because if we chosβ instead ofα, f ′

a is reduced by a factor ofh1. Therefore ifxα is a monomial generatoof Ik , thenA(α) is contained inIk . Sincexα is contained inA(α), we see thatIk can beexpressed as the sum of the idealsA(α) for xα a monomial generator ofIk . �Corollary 3.1. There is a surjective morphism of varieties

ϕ :U(A1, . . . ,An) → U(Ij )

given by sending a pointa = (a1, . . . , an) to the sum of ideals of the forma(α) for a setof xα ’s generatingIj , wherea(α) is the ideal obtained by replacing eachAi by ai in theformula forA(α) in Proposition3.2.

Proof. Any map from the interior of an alignment correspondence that takes each pcorresponding to a sequence of ideals to the point corresponding to a fixed polynothe images of these ideals under powers of the Frobenius map is a surjective morphisthe interior of an alignment correspondence corresponding to a single ideal. It remshow that this single ideal isIj . The fact that thexα ’s generateIj ensures that the sum othe ideals of the formA(α) for each generatorxα of Ij containsIj . By Proposition 3.2the sum of these ideals is contained isIj . Thus the image of the map isU(Ij ). �Corollary 3.2. The groupG(A1, . . . ,An) consists of all the automorphisms ofR sendingeachxi to an element ofAi .

Proof. The groupG(A1, . . . ,An) is contained in the set of automorphisms ofR sendingeachxi to an element ofAi because elements ofAi must be sent to elements ofAi . Sincea measuring sequence is its own measuring sequence, it follows from Proposition 3.2, thall such automorphisms are contained inG(A1, . . . ,An). �Proposition 3.3. Given a measuring sequence,A1, . . . ,Ar , let< be the partial ordering onxi ’s defined byxi � xj if xi ∈ Aj . There is a bijection between completions of this parordering in which no new equivalences of variables are introduced and nested sequ

m2 � B1 � · · · � Bm = m

of distinct monomial ideals such thatG(A1, . . . ,An) is contained inG(B1, . . . ,Bm), andm is maximal.


goftains

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of

f

Proof. Given a set ofBi ’s, we can complete< to the (possibly non-strict) total orderinon thexi ’s such thatxi � xj if xj ∈ Bk impliesxi ∈ Bk . Conversely, there is a unique setBi ’s from which such a total ordering is derived. The fact that the total ordering conthe partial ordering coming from theAi ’s ensures thatG(A1, . . . ,An) is contained inG(B1, . . . ,Bm). The fact that the< is completed to a total ordering makesm maximal. �Lemma 3.1. Let A1, . . . ,An be the measuring sequence ofI1, . . . , Ir , and letB1, . . . ,Bm

be a nested sequence of distinct monomial ideals containingm2 with m maximal and suchthat G(A1, . . . ,An) is contained inG(B1, . . . ,Bm). Each elementg ∈ G(I1, . . . , Ir ) canbe expressed as a productg1g2 whereg1 andg2 are elements ofG(I1, . . . , Ir ) such thatg1 is linear andg2 acts trivially onm/m2.

Proof. Let g1 be the linear automorphism ofR having the same restriction tom/m2 asg.Then writingg as a product ofg1 andg2, the automorphismg2 acts trivially onm/m2. Ifthe linear automorphismg1 lies inG(I1, . . . , Ir ), theng2 must also. So, it remains to shothat g1 lies in G(I1, . . . , Ir ). Let α(t) ∈ G(I1, . . . , Ir ) be the element that scales eachxi

by t . Then the limit ast goes to zero ofα(t)gα(t−1) is g1. Since eachα(t)gα(t−1) is inG(I1, . . . , Ir ), their limit g1 must be as well. �Lemma 3.2. Letm2 = B0 � B1 � · · · � Bm = m be a sequence of nested monomial ideLet Wi be the span of the monomials generatingBi/Bi−1 as aK-vector space. Then ang ∈ G(I1, . . . , Ir ) stabilizing each of theWi ’s can be decomposed into a productg1g2whereg1 and g2 are linear automorphisms inG(I1, . . . , Ir ) such thatg1 stabilizes theWi ’s andg2 acts trivially on eachBi/Bi−1.

Proof. Using the same technique as the previous proof, we will obtaing1 fromg, by takinglimits of conjugations. Letαi(t) be the automorphism ofR scaling the variables inBi by t

and fixing the other variables. Theng1 can be expressed by

g1 = limt1→0

. . . limtm→0

α1(t1) . . .αm(tm)gαm

(t−1m

). . .α1

(t−11

)

with the innermost limits taken first. Thus sinceg1 can be expressed as the limitautomorphisms inG(I1, . . . , Ir ) it must itself be an element ofG(I1, . . . , Ir ). �Theorem 3.1. LetA1, . . . ,An be the measuring sequence ofI1, . . . , Ir , and letB1, . . . ,Bm

be a nested sequence of monomial ideals containingm2 such thatG(A1, . . . ,An) iscontained inG(B1, . . . ,Bm) andm is maximal. LetB denote the varietyC(B1, . . . ,Bm).Let G(B) denote the groupG(B1, . . . ,Bm) and C(I1, . . . , Ir ,B) denote the varietyC(I1, . . . , Ir ,B1, . . . ,Bm).

(1) The varietyB is a generalized flag bundle of the tangent bundle ofX.(2) The spaceU(A1, . . . ,An) is a locally trivial bundle overB.(3) The spaceU(I1, . . . , Ir ) is the quotient ofU(A1, . . . ,An) by the finite subgroup o

G(I1, . . . , Ir ) permuting thexi ’s.(4) The compactificationC(I1, . . . , Ir ,B) of U(A1, . . . ,An) is a fiber bundle overB.


t

,

seg

ce

nearn

n

nt

ver,ble byls

(5) The dimension ofC(I1, . . . , Ir ) is the sum of the colengths of theAi ’s plus a constandepending onB. For n � 3, or B a complete flag bundle, this constant is zero.

Example 3.2. The ideal(x2, y3) has measuring sequence(x, y3), (x2, y). By Theorem 3.1C((x2, y3), (x, y2), (x2, y)) is a fiber bundle over bothC((x, y2)) and C((x2, y)) withrespect to the natural projections.

Proof of Theorem 3.1(1). The spaceU(B1, . . . ,Bm) is a generalized flag bundle becauthe Bi ’s correspond to a nested sequence of subspaces ofm/m2. Since generalized flavarieties are compact,B andU(B1, . . . ,Bm) are one and the same.�Proof of Theorem 3.1(2). Let W1 ⊕ · · · ⊕ Wm be a decomposition of the vector spam/m2 into direct summands spanned by monomials such thatBi/m

2 = W1 ⊕· · ·⊕Wi . LetN be an integer large enough so that eachAi containsmN . For i from 1 tom, let Ji be theideal generated overmN by the monomials spanningWi . ThenG(J1, . . . , Jm) is containedin G(A1, . . . ,An) and is expressible as the semi-direct product of the group of liautomorphisms stabilizing theWi ’s with the group of automorphisms acting trivially om/mN . By Corollary 3.2, a set of coset representatives forG(A1, . . . ,An)/G(J1, . . . , Jm)

is given by the set of all automorphismsϕ of R with ϕ(xi) = xi + li + fi for fi alinear combination of monomials inAi of degree strictly between 1 andN and li inthe span of variables inequivalent toxi . Thus U(J1, . . . , Jm) is a trivial bundle overU(A1, . . . ,An). A set of coset representatives forG(B)/G(J1, . . . , Jm) is given by theset of automorphismsϕ of R with ϕ(xi) = xi + li + fi wherefi is a linear combinationof monomials of degree strictly between 1 andN and li is a linear combination ofxj ’sstrictly less thanxi . ThusU(J1, . . . , Jm) is also a trivial bundle overB. Therefore, sincethe map fromU(J1, . . . , Jm) to B factors throughU(A1, . . . ,An), U(A1, . . . ,An) is anaffine bundle overB as well. �Lemma 3.3. Let W1 ⊕ · · · ⊕ Wm be a decomposition of the vector spacem/m2 intonontrivial direct summands such thatm is maximal and eachW1 ⊕ · · · ⊕ Wi is stabilizedby some power of each element ofG(I1, . . . , Ir ). If K contains a unit of infinite order, thethe groupG of linear automorphisms stabilizing theWi ’s is contained inG(I1, . . . , Ir ).

Proof. Let M be the set of monic monomials inR. Let theG-topologyon M be thetopology such that the closed sets are those with span fixed by the groupG. To prove thelemma, it is enough to show that the intersection ofM andIk is closed in theG-topologyfor eachk. Let G′ be the intersection ofG(I1, . . . , Ir ) andG. Let theG′-topologyon M

be the topology such that the closed sets are those with span fixed by the groupG′. Thenthe intersection ofM andIk is closed in theG′-topology for eachk because each elemeof G′ stabilizes eachIk . Thus, it is sufficient to show thatG′-topology and theG-topologyare the same.

By maximality of m, theWi ’s are uniquely determined up to permutation. Moreothey are spanned by monomials since they are invariant under scaling any variasome power of a unit ofK that is not a root of unity. LetMi be the set of monic monomiain M that are products of variables inWi . We will call such monomialsi-monomials. Then


t

logies

ill

f

gree

tr itnt

sfnte

nures

e

lsat

M is the product of theMi ’s and theG′-topology and theG-topology are both productopologies with respect to this decomposition ofM. Moreover, in both topologiesMi isthe union of the disconnected piecesMi(d) where eachMi(d) is the set ofi-monomials ofdegreed . Thus, it remains to show that these two topologies restrict to the same topoon eachMi(d).

The restriction of theG-topology toMi(d) has an alternate description, which we wcall theexponent topology. It can be described as follows. If the characteristic ofK isa positive integerp, say that a monomialf hasexponent type(a0, . . . , am) if f can bewritten as the productf0f

p

1 . . . fpm

m where thefi ’s are pth power free monomials odegreeai . If the characteristic ofK is 0, then say thatf has exponent type(a0) wherea0 is the degree off . Let < be the partial ordering on exponent types of the same desuch that(a0, . . . , am) � (m0, . . . ,mb) exactly when

k∑i=0

aipi �

k∑i=0

mipi

for all k, takingal (respectivelyml) to be 0 ifl > m (respectivelyl > b). Then the exponentopology is the topology onMi(d) such that a set is closed if and only if whenevecontains a giveni-monomial, it contains all otheri-monomials of lesser or equal exponetype.

Suppose, by way of contradiction, that there are anl and ad such that the two topologieon Ml(d) are different, withd chosen to be minimal. LetB(a1, . . . , at ) denote the set ol-monomials of exponent type(a1, . . . , at ) or less. If(a0, . . . , at ) is not a possible exponetype, we will use the convention thatB(a0, . . . , at) is empty. LetB(f ) denote the closurof a monomialf with respect to theG′-topology. Then there is anl-monomialf of degreed and exponent type(a0, . . . , at ) such thatB(f ) is properly contained inB(a0, . . . , at ).Choosef to have minimal exponent type. Note that we cannot havea0 = 0 because thethepth root off would give a monomial of smaller degree having two different closin the two different topologies. Relabeling thexi ’s as needed, assume further thatf can beexpressed by

f = xe11 . . . x

eb

b hp

with the degree ofh and then theek ’s maximal, in ascending order. Letα be the sequenc

a0 − p, a1 − p + 1, . . . , ac−1 − p + 1, ac + 1, ac+1, . . . ,

wherec is the smallest integer withac �= n(p − 1). ThenB(α) is the set of monomials inMl(d) of exponent type smaller than(a0, . . . , at) not beginning witha0. It is empty if Khas characteristic 0 ora0 is less thanp. Assume that we chosef , among those monomiameeting the previous conditions, so thatZ ⊂ Ml(1) is a set of maximum order such thwe have the containment

xe1 . . . x

eb−1Zebhp ⊂ B(f ) ∪ B(α).
1 b−1


t

n

rs

f

in

n

es

as ah

t

nt

Note thatZ is nonempty since it containsxb. Giveng ∈ G′ let ik andx ′k be such thatx ′

k

differs fromg(xk) by a linear combination of thex ′i ’s for i < k and such that the coefficien

of xij in x ′k is 0 if j < k and non-zero ifj = k. Thus we can takex ′

1 to beg(x1), xi1 tobe any element ofMi(1) with non-zero coefficient ing(x1), x ′

2 to be a linear combinatioof g(x1) andg(x2) having noxi1 term, etc. By maximality ofe1, expandingg(x

e11 . . . x

eb

b )

out in terms of thex ′k ’s we see that modulo the span ofB(a0 − p,1), the largest powe

of x ′1 occurring ise1 and hence that modulo the span ofB(a0 − p,1), we can expres

g(xe11 . . . x

eb

b ) as the product of(x ′1)

e1 and a polynomial in thex ′k ’s for k � 2. Continuing

in this way, we see thatg(xe11 . . . x

eb−1b−1 ) is a multiple of(x ′

1)e1 . . . (x ′

b)eb modulo the span o

B(a0 − p,1). Sinceg is invertible, there must be anh′ of exponent type(a1, . . . , at ) suchthat the coefficient ofh′ in g(h) is non-zero. Any monomial with a non-zero coefficient

xe1i1

. . . xeb−1ib−1

g(Z)eb (h′)p

is in B(f ) ∪ B(α). This is because moduloB(α) there is no cancellation of terms whewe multiply g(x

e11 . . . x

eb−1b−1 Zeb ) with g(hp). By maximality of the dimension ofZ, the

monomial span ofg(Z) must be of the same dimension asZ. Therefore, the span ofZ isfixed by some power of any element ofG′ since there are only finitely many vector spacgenerated by elements ofMl(1).

We can conclude thatZ = Ml(1) as follows. For each element ofG(I1, . . . , Ir ), thereis a power stabilizing theBi ’s. By Lemmas 3.1 and 3.2, this power can be expressedproduct of an elementg1 in G (and hence inG′) and an element acting trivially on eacBi/Bi−1. Thus any subspace ofm/m2 is stabilized by some power of any element ofG′ isa direct sum ofWi ’s. ThusZ is forced to be theMl(1). It also follows thatek = p − 1 fork < b.

By minimality ofd , the exponent topology is the same as theG′-topology onMl(d −eb)

and

B(x

e11 . . . x

eb−1b−1 hp

) = B(a0 − eb, a1, . . . , at ).

Thus, the closure of the terms inxe11 . . . x

eb−1b−1 Wb

l hp in the G′-topology is the seB(a0, . . . , at ). Therefore the elements ofB(a0, . . . , at ) − B(f ) are contained inB(α). Ifa0 � p or p = 0, we have reached a contradiction since thenB(α) is empty.

It remains to show thatB(α) is contained inB(f ). By minimality of (a0, . . . , at ), itsuffices to show that there is a single element of exponent type(α) in B(f ). Given anyl-monomialf0 of exponent type(a0), by minimality of d , the setf0B(a1, . . . , at )

p iscontained inB(f ). Thus we need only show thatB(f0) contains a monomial of exponetype(a0 − p,1). We will use

f0 = (x1 . . . xib−1)p−1x

eb

b ,

where thexk ’s are as before.Let Z′ be the subset ofMl(1) with maximum number of elements such that

(x1 . . . xib−1)p−1(Z′)eb


weder

fws

3,ed

or

o

gt

e

is contained inB(f0) ∪ B(a0 − 2p,2). Assume that within all the possible choicescould have made for thexi ’s up to this point, we chose the ones maximizing the orof Z′. Suppose that there is an elementg ∈ G′ such that the monomial span ofg(Z′) hasdimension greater than that ofZ′. As before letik andx ′

k be such thatx ′k differs fromg(xk)

by a linear combination of thex ′i ’s for i < k and such that the coefficient ofxij in x ′

k is 0if j < k and non-zero ifj = k. Expandingg((x1 . . . xb−1)

p−1(Z′)eb ), if there is a term ofexponent type(a0−p,1), then we are done. Otherwise, we haveg((x1 . . . xb−1)

p−1(Z′)eb )

is a multiple of(x ′1 . . . x ′

b−1)p−1g(Z′)eb modulo the span ofB(a0 − 2p,2). Then since

(x ′

1 . . . x ′b−1

)p−1g(Z′)eb

is contained in the span ofB(f0) ∪ B(a0 − 2p,2), by maximality of the dimension ofZ′,any element ofG′ preserves the span ofZ′. ThereforeZ′ must be equal toWl . Sincethere is a term of(x ′

1 . . . x ′ib−1

)p−1(Wl)eb of exponent type(a0 − p,1), there is a term o

exponent type(a0 − p,1) in B(f0). Hence we have arrived at a contradiction. It follothat the groupG is contained inG(I1, . . . , Ir ). �Lemma 3.4. The decomposition ofm/m2 asW1 ⊕ · · ·⊕ Wm such that eachWi is spannedby monomials andBi/m

2 = W1 ⊕ · · · ⊕ Wi is a decomposition ofm/m2 into nontrivialsummands withm maximal such that some power of each element inG(I1, . . . , Ir )

stabilizes eachBi/m2.

Proof. Two variables are in the sameWi if and only if they are equivalent. By Lemma 3.the same can be said for two variables in aWi coming from a decomposition as describin that lemma. �Proof of Theorem 3.1(3). If the theorem holds for a given field, it also holds fany subfield, because we can use the same coset representatives forG(A1, . . . ,An) inG(I1, . . . , Ir ), namely the permutations of thexi ’s fixing theIj ’s modulo those fixing theAj ’s. Thus, we can assume thatK contains a unit of infinite order sinceK is contained inthe function fieldK(t).

Let W1 ⊕ · · · ⊕ Wm be a decomposition ofm/m2 such that eachWi is spannedby monomials andBi/m

2 = W1 ⊕ · · · ⊕ Wi . The Wi ’s are uniquely determined up tpermutation. Thus any elementg ∈ G(I1, . . . , Ir ), is expressible as a productg1g2g3g4whereg1 is a permutation of thexi ’s sending eachBi to g(Bi), g2 is an element of thegroupG of linear automorphisms stabilizing theWi ’s, g3 is a linear automorphism actintrivially on eachBi/Bi−1 andg4 acts trivially onm/m2. We will show thatg1 is an elemenof G(I1, . . . , Ir ) andg2, g3, andg4 are elements ofG(A1, . . . ,An).

We will first show thatg1 is an element ofG(I1, . . . , Ir ). Since eachIk is a monomialideal, it is enough to show that for any monomialf ∈ Ik , the imageg1(f ) is also amonomial inIk . Thus, it suffices to show that there is someg′ ∈ G(I1, . . . , Ir ) such that thecoefficient ofg1(f ) in g′(f ) is non-zero since all non-zero terms ofg′(f ) must be inIk .Recall that by Lemmas 3.3 and 3.4, the groupG is contained inG(I1, . . . , Ir ). So supposeby way of contradiction that there is a monomialf (not necessarily inIk) of smallest degresuch thatg1(f ) has coefficient 0 in every element ofGg(f ). Write f = f1f2 wheref1


ts

,

ot

e

m

f

rnot

.

e

t

is ani-monomial for somei andf2 is a product ofj -monomials forj < i. Let σ be theelement ofSm corresponding to the action ofg1 on theWi ’s. Then the terms of elemenof Gg(f2) are products ofσ(j)-monomials forj < i. Thus giveng′ ∈ Gg, there is nocancellation of terms when multiplying the part ofg′(f1) spanned byσ(i) monomials withg′(f2). Hence, by minimality of the degree off , eitherf1 or f2 must be a unit. Chooseiso thatf2 is a unit. Sinceg is invertible, there must be a non-zero term ing(f ) that is aσ(i)-monomial of the same exponent type asf . Thus by Lemma 3.3, allσ(i)-monomialsof this exponent type occur in some element ofGg(f ), includingg1(f ). Thus we havearrived at a contradiction and it follows thatg1 is an element ofG(I1, . . . , Ir ).

Next we will show thatg2, g3, andg4 are in G(A1, . . . ,An). By Lemmas 3.1, 3.3and 3.2, we see that each of these automorphisms is inG(I1, . . . , Ir ). The fact thatg2 is inG(A1, . . . ,An) follows from Lemmas 3.3, 3.2 and Corollary 3.2.

Let g′3 be a linear element ofg3G(A1, . . . ,An) such that the number of variables n

fixed by g′3 is minimal. Let i be minimal such that not all the variables inWi are fixed

by g′3. Let xj be a variable inWi such thatg′

3(xj ) = xj + f for f non-zero. One can sethatf must be inAj as follows. Fork �= i, let αk(t) be the automorphism ofR scaling thevariables inBk by t and fixing the other variables. Letαi be the automorphism ofR scalingthe variables inBk exceptxj by t and fixing the other variables. Then the automorphis

limti→0

. . . limtn→0

α1(ti) . . .αn(tn)g′3αn

(t−1n

). . .αi

(t−11

),

with the innermost limits taken first, is an automorphismh sendingxj to xj +f and fixingthe other variables. Since the limit of elements ofG(I1, . . . , Ir ) is itself an element othis group if it exists, we see thath is in G(I1, . . . , Ir ). This implies thatf is in Aj . ByCorollary 3.2, we then haveh in G(A1, . . . ,An). Composingg′

3 with h−1, we get anotheelement ofg3G(A1, . . . ,An) contradicting the assumption that the number of variablesfixed byg′

3 is minimal. This proves thatg3 is in G(A1, . . . ,An).It remains to show thatg4 is an element ofG(A1, . . . ,An). Let g′

4 ∈ g4G(A1, . . . ,An)

with

g′4(xk) = xk + fk

and the smallest degree non-zero piecehk of each non-zerofk of maximal degreeSuppose, by way of contradiction, thatg′

4 is not the identity map. Suppose thathk is in Ak.Then composingg′

4 with the element ofG(A1, . . . ,An) sendingxk to xk − hk and fixingthe otherxi ’s we get a new automorphism contradicting the maximality of the degrehk .Thus eitherxk is sent to itself orhk is not in Ak. Choosek so that the degree ofhk isminimal. Sincehk is not in Ak , there must be a monomialf ∈ Ij for somej such thatthe automorphismg′ sendingxk to xk + hk and fixing the otherxi ’s does not sendf to anelement ofIj . However, each term ing′(f ) is the lowest degree term ing′

4(f ) having thesame power ofxk. But, sinceIj is a monomial ideal, every term ofg′

4(f ) is in Ij , whichmeans thatg′(f ) is also inIk . So, we have arrived at a contradiction and it follows thag4is an element ofG(A1, . . . ,An).


f

s

f

at

f

r,as

neer is

gengthsing

Thus the coset representatives ofG(A1, . . . ,An) in G(I1, . . . , Ir ) are permutations othexi ’s fixing theIj ’s modulo those fixing theAj ’s. ThereforeU(I1, . . . , Ir ) is the quotientof U(A1, . . . ,An) by the group of these automorphisms.�Proof of Theorem 3.1(4). Given any pointp ∈ X, there is an isomorphismϕP :R →OX,p. The group Aut(R) acts transitively on the fiber ofB over X. This action extendto C(I1, . . . , Ir ,B), showing that its fiber overX is a fiber bundle over the fiber ofBoverX. Therefore, sinceC(I1, . . . , Ir ,B) is a fiber bundle overX, it is also a fiber bundleoverB. �Lemma 3.5. Let< be the partial ordering on thexi ’s such thatxi � xj if xj ∈ Bk impliesxi ∈ Bk for eachBk . Let I be the set of pairs(i, α) such thatxα is in the complement oAi and if xα is linear, thenxα < xi . A set of right coset representatives forG(B) overG(A1, . . . ,An) is given by the setS of automorphismsg of the form

g(xi) = xi +∑

(i,v)∈Iai,vx

v

for eachi.

Proof. Let us first show that ifg ∈ S andh ∈ G(A1, . . . ,An) are automorphisms such thg ◦ h ∈ S, thenh is the identity map. Suppose thath is not the identity map. Letk be suchthat the lowest graded piecehk of h(xi)−xi is minimal. The ordering< on the variables oR can be extended to a partial ordering on monomials satisfying the relationsf1f3 � f2f4

if f1 � f2 andf3 � f4. Then for each monomialf , the terms ofh(f ) − f are all eitherof higher degree thanf or of the same degree and strictly greater thanf . The strictnesscomes from the consequence of Lemma 3.3 thatG is contained inG(A1, . . . ,An). Hencethe minimal terms ofhk remain terms ofg ◦ h(xk). Thus by Lemma 3.2,g ◦ h /∈ S. Thuswe have arrived at a contradiction and it follows thath is the identity map. Moreoveby the proof of Theorem 3.1(1), the spaceS is an affine space of the same dimensionG(B)/G(A1, . . . ,An) and so must be a full set of coset representatives.�Proof of Theorem 3.1(5). The dimension ofC(I1, . . . , Ir ) is the same as the dimensioof its interior. SinceU(I1, . . . , Ir ) is a fiber bundle overB, its dimension is the sum of thdimension ofB and the dimension of the fiber. By Lemma 3.5, the dimension of the fibthe cardinality ofI. Let A′

i denote theith ideal in the measuring sequence ofB1, . . . ,Bm.Then for eachi, the monomialsxα with (i, xα) ∈ I form a set of monomials generatinA′

i/Ai as a vector space. Therefore, the dimension of the fiber is the sum of the colAi ’s minus the sum of the colengths of theA′

i ’s. Thus we see that the constant dependon B is the sum of the colengths of theA′

i ’s minus the dimension ofB. In the case thatBis the complete flag bundle on the tangent bundle ofX or the dimension ofX is at mostthree, the dimension ofB is equal to the sum of the colengths of theA′

i ’s. �


eirith thentrast,

address

tedet

espon-der-im-nce fornswer tospon-

gnmentpproachundle

nt

ectedignmentmanian.nment

4. Universality

In the previous section we classified alignment correspondences according to thmeasuring sequences. We found that the interiors of alignment correspondences wsame measuring sequence are isomorphic up to quotienting by a finite group. In cothe alignment correspondences themselves can vary quite a bit. In this section, wethe question of how much these alignment correspondences vary.

Definition. Given two alignment correspondencesY1 andY2, we will say that the alignmencorrespondence corresponding to the concatenation of their defining sequences is obtainby superimposingY1 andY2. We will say thatY1 is auniversalalignment correspondencwith its measuring sequence if whenever we superimposeY1 with another alignmencorrespondence the natural projection toY1 is an isomorphism.

For most practical purposes, understanding the geometry of the alignment corrdence obtained by superimposing two alignment correspondences is as good as unstanding the geometries of each of the two alignment correspondences that we superposed. Thus, it is natural to ask whether there is a universal alignment correspondea given measuring sequence. For most measuring sequences, we determine the athis question. In Theorem 4.2, we prove that there is no universal alignment corredence for certain measuring sequences. In Theorem 4.3, we construct universal alicorrespondences for several of the remaining measuring sequences. Our basic ais to focus our attention on a class of alignment correspondences with nice fiber bstructure as defined below.

Definition. If an alignment correspondenceY is a fiber bundle over another alignmecorrespondenceB such thatB is its own interior and the interior ofY is a trivial bundleoverB, we will say thatY is directedoverB.

By Theorem 3.1(4), every alignment correspondence is dominated by a diralignment correspondence with the same measuring sequence. Fibers of directed alcorrespondences can be understood by their embedding in an appropriate GrassUnderstanding these fibers is the heart of the problem of understanding directed aligcorrespondences.

Example 4.1. Consider the fiberF of C((x, y4), (x, y2)) over the spaceC((x, y2)). ByLemma 3.5, theG((x, y2)) orbit of (x, y4) is theA2 of ideals of the form

(x + ay2 + by3, y4).

As a subspace of the quotient(x, y2)/(x, y2)2, a basis for such an ideal is given by

{xy + ay3, x + ay2 + by3}.


ith

ment

ls

ofeeneous

ates,esr

Expanding

(xy + ay3) ∧ (

x + ay2 + by3)

with respect to the basis

{y3 ∧ xy, y3 ∧ y2, y3 ∧ x, xy ∧ y2, xy ∧ x, y2 ∧ x

}

of ∧2V4, we get a map fromF to P5x0,...,x5

given by

(x + ay2 + by3, y4) → (−b, a2, a, a,1,0

).

The closure of the image is cut out by

x2 − x3 = x5 = x1x4 − x22 = 0.

Thus,F is a cone over a conic. The boundary parameterizes theP1 of ideals of the form

(αxy + βy2, x2, xy2, y3).

The cone point corresponds to the ideal

(x2, xy, y3).

In the following three theorems, letC(I1, . . . , Ir ) be an alignment correspondence wmeasuring sequenceA1, . . . ,An that is directed overB.

The first theorem will serve as a basic tool for understanding fibers of directed aligncorrespondences.

Theorem 4.1. Let xi have weightei and letg be as in Lemma3.5. Considering theai,α ’sas coordinates for the fiberF of U(A1, . . . ,An) overB , giveai,α weightei − α, makingeachg(xi) homogeneous. Then there are homogeneous coordinate functions embeddingF

in projective space in such a way that the closureF is the fiber ofC(A1, . . . ,An) overB.Moreover, if the weights of the coordinates onF are independent, the normalization ofF

is a toric variety.

Proof. There is a natural embedding ofF in a product ofn Grassmanians. LetVi be thequotient of the union of the ideals in theG(B) orbit of Ai by the intersection of the ideain theG(B) orbit ofAi . Then the projection ofC(A1, . . . ,An) to C(Ai) gives rise to a mapfrom F to the Grassmanian of subspaces ofVi of the appropriate dimension. The pointF with coordinatesai,α is sent to the point corresponding tog(Ai) viewed as a subspacof Vi . These Grassmanians can then be embedded in projective space via homogPlücker coordinates. Sinceg preserves homogeneity, with respect to these coordinthe coordinate functions will be homogeneous. Mapping the product of projective spacinto a single projective space via the Segre embedding, the coordinate functions foF are


ghtsbe

in

the

ing

ring

e

ane will

in thealent

g to

case,fow. Inesne cann by

products of homogeneous coordinate functions and hence homogeneous. If the weiof the coordinate functions are independent, then the coordinate functions can onlymonomials. Hence the normalization ofF is a toric variety. The action of scaling thexi ’sgives the action of an open dense torus.�Definition. We will say thatxi is equivalentto xj if xi ∈ Aj andxj ∈ Ai . We will saythat two monomials areequivalentif they are of the same degree in the variableseach equivalence class. We will say that coordinatesai,v and aj,w are equivalentif xi

is equivalent toxj andxv is equivalent toxw.

Theorem 4.2. Suppose that there are two inequivalent coordinates for the fiberF ofU(A1, . . . ,An) over B. Then there is no universal alignment correspondence withmeasuring sequenceA1, . . . ,An.

Proof. We will prove the theorem by constructing an infinite sequence of ideals{Im}∞m=2with the given measuring sequence such that the alignment correspondences correspondto finite subsequences do not stabilize.

Recall from Proposition 3.3 that there is a partial ordering of thexi ’s corresponding tothe sequenceA1, . . . ,An andB corresponds to some completion of this partial ordeto a total ordering≺. Without loss of generality, assume that the indices of thexk ’s aresuch that≺ contains the total ordering on thexk ’s inherited from the total ordering on thindices. If there are inequivalent coordinates forF for one ordering≺ definingB, thenthere are inequivalent coordinates for all orderings. Thus we are free to choose≺. We firstshow that we can choose≺ so that there are two coordinates with indices given byentry of the first column of Table 1. Based on these two inequivalent coordinates wconstruct the sequence of ideals{Im}∞m=2 from the data in Table 1.

Suppose that there is a pair of inequivalent coordinatesai,ej andak,el . Since there maybe several such pairs, we will chose a pair to minimize the larger ofi − j and k − l.This difference must be 1 and the indices for the pair of coordinates must be asfirst or second row of the first column of Table 1. If there is no such pair of inequivcoordinates, then there is a coordinateai,ej +ek with i minimal andj andk maximal. Let≺ be chosen so thati = 1 andej + ek is either equal to 2en or en + en−1. In the formercase, ifn �= 2, one can chose≺ so there is an inequivalent coordinate correspondinthe second entry in the first column of the third, fourth, or fifth row. Ifn = 2, then therewill be a second coordinate corresponding to the entry in the sixth row. In the latterif characteristic ofK is not 2, then one can chooseB so that the indices of a pair oinequivalent coordinates are given by the entries in the first column of the seventh rcharacteristic 2, for a suitable choice of≺, there will be a pair of inequivalent coordinatcorresponding to the entries in the first column of one of the last three rows. Thus, oalways chooseB so that there is a pair of inequivalent coordinates with indices giveone of the rows of the first column of Table 1. Without loss of generality, assume thati < j

in the first row of Table 1. In the last two rowsN is an integer larger thanm that is 1 lessthan a power of 2.

Given two inequivalent coordinates forF with pair of indicesS given by one of the rowsin the first column of Table 1, we will construct the idealsJ1, J2, andIm from the data in


hathird

s ine not

elsor they

ee in

Table 1Generators forIm

S Im/J2 J1/Im

i, ei−1 xmi

xj−1, . . . , xixm−1i−1 xj−1 xm

i−1xj−1

j, ej−1 xmi−1xj

i, ei−1 xm+1i , . . . , xm−1

i−1 x2i , xm+1

i−1 xmi−1xi

i + 1, ei xmi−1xi+1

1,2en xm1 xn−1, . . . , x1x

2m−2n xn−1 x2m

n xn−1

1, en−1 + en x1x2m−1n

1,2en xm1 , . . . , x1x2m−2

n x2mn

2,2en x2x2m−2n

1,2en xm1 xi−1, . . . , x1xi−1x2m−2

n xi−1x2mn

i, ei−1 xix2mn

1,2en xm1 , . . . , x1x2m−2

n x2mn

1,3en x1x2m−3n

1, en−1 + en xm1 , . . . , x1x2m−2

n−1 x2mn−1

n, en−1 x2m−1n−1 xn

1, en−1 + en xm1 xi−1xN−1

n−1 xN−1n . . . , xi−1xN

n−1xNn xi−1xN

n−1xNn

i, ei−1 x1xixN−1n−1 xN−1

n

1, en−1 + en xm1 xN−m

n−1 xN−mn , . . . , x1xN−1

n−1 xN−1n xN

n−1xNn

2, en−1 + en x2xN−1n−1 xN−1

n

this row. We will constructJ1 andJ2 so thatIm is sandwiched in between them and so tJ1 is linearly generated overJ2 by monomials equivalent to those in the second and tcolumns of the row. LetJ2 be the ideal generated by monomials that appear as termtheG(B) image of the monomials in the second and third columns of this row, but arequivalent to any of them. LetIm be the ideal generated overJ2 by the monomials of thesecond column and equivalent monomials. ThenJ2 is the intersection of the ideals in thG(B) orbit of Im and these monomials are a basis forIm/J2. Moreover, these monomiatogether with the monomials equivalent to the one in the third column are a basis funionJ1 of the ideals in theG(B) orbit of Im overJ2. Let L be the lattice generated bthe vectorsei − v for i, v ∈ S. LetA denote the set of automorphismsg of R with

g(xi) = xi +∑

(i,α)∈Sai,αxα

for eachi from 1 ton. Let M be the subspace ofJ1/J2 generated by the monomials in thsecond and third columns. Then the differences of the weights of these monomials liL.

The closure of theA orbit of Im in the Grassmanian of hyper-planes inJ1/J2 is a coneover a rational normal curve of degreem. Embedding theA-orbit of (I2, . . . , Im) in the


nce

of

in

ring

t

tost andppet

eof

butf

it

e withre stillces

appropriate product of Grassmanians, the boundary of the image is a chain of(m − 1)

P1’s. Therefore, there is no finite sequence of ideals withA-orbit dominating theA-orbitof (I2, . . . , Im) for all values ofm. Thus, there is no universal alignment correspondefor the given measuring sequence.�Theorem 4.3. If the measuring sequenceA1, . . . ,An does not satisfy the hypothesesTheorem4.2 then the coordinates for the fiberF of U(A1, . . . ,An,B) over B are allcoordinates in an equivalence class of a coordinateai,v satisfying one of the following:

(1) v = ej ,(2) v = ej + ek with xj andxk equivalent,(3) v = ej + ek with xj andxk equivalent andj �= k,(4) v = ej + ek with xk in a unique equivalence class andj �= k.

The last two cases happen only in characteristic2. Let m be the number of elementsthe equivalence class ofxi andl the number in the equivalence class ofxj for any coordi-nateai,v . Then ifm or l = 1 in the first or fourth case,l = 1 in the second, orl = 2 in thethird, thenC(A1, . . . ,An,B) is a universal alignment correspondence for the measusequence(A1, . . . ,An).

Proof. The fact that these are the only cases follows from the proof of Theorem 4.2. LeG be the group of linear automorphism ofR fixing the subspaceV1 of m/m2 spannedby the variables equivalent toxi and the subspaceV2 spanned by variables equivalentxj and fixing all the variables outside of these two equivalence classes. In the firlast case,F can be identified with the unipotent radicalUm,l of the parabolic subgrouPm,l of GL(V1 ⊕ V2) fixing V1. The groupG can be identified with the Levi subgrouof Pm,l acting by conjugation. In the second caseF can be identified with the spacof m-tuples of quadratic forms inl variables. An element ofG viewed as an elemen(g1, g2) ∈ GL(V1) × GL(V2) acts by the composition of the action ofg1 by change ofvariables and the action ofg2 by matrix multiplication. The third case is similar to thsecond, except that the space ofm-tuples of quadratic forms is taken modulo the imagethe Frobenius on the space of linear forms.

If m and l are as in the hypotheses of the theorem thenG is the group of all lineartransformations of the affine spaceF . In the third case, this is less straightforward,using the fact that the characteristic is 2,GL(V2) acts onF by the cofactor matrix othe matrix giving the action onV2. The fiber ofC(A1, . . . ,An,B) overB is a projectivespace in which the only fixed subvariety underG is the boundary. Thus it follows thatis the universalG-equivariant compactification ofF and hence thatC(A1, . . . ,An,B) isuniversal. �

Universal alignment correspondences are relatively easy to understand. Thosfibers that are toric varieties by Theorem 4.1 are more complicated, but there atechniques for understanding them [11,12]. The remaining alignment correspondenseem quite difficult to understand.


4.3

rphic

hether

for

undero com-

oupsndingrs.

ndras help.ank

vent.

(1998)

otes in

es221.

0,

Amer.

Example 4.2. It is natural to hope that for all measuring sequences in Theoremthe alignment correspondenceC(A1, . . . ,An,B) is universal. However, the idealI =(x, y,m3) in K[[x, y, z,w]] has measuring sequence(x, y,m2), (x, y,m2),m,m. But,C((x, y,m2), I ) is not a universal alignment correspondence since it is not isomoto C((x, y,m2), I, I2).

The two simplest measuring sequences for which we have not yet determined wthere is a universal alignment correspondence are the measuring sequences(x,m3), m, m

and(x, y,m2), (x, y,m2), (z,w,m2), (z,w,m2). Fibers of alignment correspondencessequences of ideals with the first measuring sequence have fibers overB corresponding tocompactifications of the space of quadratic form in two variables that are equivariantthe action of scaling the two variables. In the second case, the fibers correspond tpactifications of the space of 2× 2 matrices equivariant under right and left multiplicationby GL2(K). A deeper understanding of equivariant compactification of algebraic grsuch asGLn in the setting of alignment correspondences may be key to understasome of the other alignment correspondences that do not have toric varieties as fibe

Acknowledgments

I thank Karen Chandler, Joe Harris, Tony Iarrobino, Ezra Miller, Keith Pardue, DipePrasad, Mike Roth, Jason Starr, Ravi Vakil, and Joachim Yameogo for their generouI would particularly like to thank Steve Kleiman for many helpful comments. I also ththe referee for suggestions which helped to improve the exposition of this paper.

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Math., vol. 1647, Springer-Verlag, Berlin, 1996.[7] A. Grothendieck, Techniques de construction et théorèmes d’existance en géométrie algébrique. IV, in: L

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