Carlsson's rank conjecture and a conjecture on square-zero upper triangular matrices

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Contents lists available atScienceDirect

Journal

of

Pure

and

Applied

Algebra

www.elsevier.com/locate/jpaa

Carlsson’s

rank

conjecture

and

a

conjecture

on

square-zero

upper

triangular

matrices

Berrin Şentürk∗, Özgün Ünlü1

Department of Mathematics, Bilkent University, Ankara, 06800, Turkey

a r t i c l e i n f o a b s t r a c t

Article history:

Received26May2017

Receivedinrevisedform20July 2018

Availableonline13September2018 CommunicatedbyS.Iyengar MSC: 55M35;13D22;13D02 Keywords: Rankconjecture Square-zeromatrices Projectivevariety Borelorbit

Letk beanalgebraicallyclosedfieldandA thepolynomialalgebrainr variables withcoefficientsink.Incasethecharacteristicofk is2,Carlsson[9] conjectured thatforanyDG-A-moduleM ofdimensionN asafreeA-module,ifthehomology ofM isnontrivialandfinitedimensionalasak-vectorspace,then2r_{≤ N.}_Here_we

statea strongerconjectureaboutvarietiesofsquare-zerouppertriangularN× N matriceswithentriesinA.UsingstratiﬁcationsofthesevarietiesviaBorelorbits, weshow that the stronger conjectureholds when N < 8 or r < 3 without any restrictiononthecharacteristicofk.Asaconsequence,weobtainanewprooffor manyoftheknowncasesofCarlsson’sconjectureandgivenewresultswhenN > 4 andr = 2.

1. Introduction

Awell-knownconjectureinalgebraictopologystatesthatif(Z/pZ)r_acts_freely_and_cellularly_on_a_ﬁnite

CW-complex homotopy equivalentto Sn1× . . . × Snm_, _then _{r is}_less_than _or _equal_to _m. _This _conjecture

is known to be true in several cases: In the equidimensional case n1 = . . . = nm =: n, Carlsson [7],

Browder [6], andBenson–Carlson [5] proved theconjecture undertheassumption thattheinducedaction on homology istrivial. Without thehomology assumption,theequidimensional conjecturewas provedby Conner[11] for m= 2,Adem–Browder[1] forp= 2 orn= 1,3,7,andYalçın [25] forp= 2,n= 1.Inthe non-equidimensionalcase,theconjectureisprovedbySmith[23] form= 1,Heller[13] form= 2,Carlsson [10] forp= 2 andm= 3,Refai[20] forp= 2 andm= 4,andOkutan–Yalçın[19] forproductsinwhichthe average ofthedimensionsissuﬃciently largecomparedto thediﬀerencesbetween them.Thegeneralcase

m≥ 5 isstillopen.

* Correspondingauthor.

E-mail addresses:berrin@fen.bilkent.edu.tr(B. Şentürk),unluo@fen.bilkent.edu.tr(Ö. Ünlü).

1

ThesecondauthorispartiallysupportedbyTÜBA-GEBİP/2013-22.

https://doi.org/10.1016/j.jpaa.2018.09.007

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LetG= (Z/pZ)r_and_{k be}_an_{algebraically}_closed_ﬁeld_of_{characteristic}_p._Assume_that_{G acts}_freely_and

cellularlyonaﬁniteCW-complexX homotopyequivalenttoaproductofm spheres.Onecanconsider the cellularchaincomplexC∗(X;k) asaﬁnitechaincomplexoffreekG-moduleswithhomologyH∗(X;k) that

hasdimension 2m as ak-vectorspace. Hence,astrongerand purely algebraicconjecture canbe statedas follows:If C_∗ isaﬁnitechaincomplexoffreekG-moduleswith nonzerohomologythen dimkH∗(C∗)≥ 2r.

However,Iyengar–Walkerin[15] disprovedthis algebraicconjecture whenp= 2,butthealgebraic version forp= 2 remainsopen.

LetR beagraded ring.Apair(M,∂) isadiﬀerential graded R-moduleifM isagradedR-moduleand

∂ isanR-linearendomorphismofM thathasdegree−1 andsatisﬁes∂2_{= 0.}_Moreover,_a_DG-R-module_is free if theunderlyingR-moduleisfree.

LetA= k[y1,. . . ,yr] bethepolynomialalgebrainr variables,wherek isaﬁeldandeachyihasdegree−1.

Usingafunctorfrom thecategory ofchaincomplexesofkG-modulesto thecategoryofdiﬀerentialgraded

A-modules, Carlsson showed in [8], [9] that the above algebraic conjecture is equivalent to the following conjecturewhenthecharacteristic ofk is2:

Conjecture1.Letk beanalgebraicallyclosed field,A thepolynomialalgebra inr variableswithcoefficients ink,and N apositiveinteger.If(M,∂) isafree DG-A-moduleof rankN whosehomology isnonzeroand finite dimensionalasa k-vectorspace, then N≥ 2r_.

When the characteristic of k is 2, Conjecture 1 was proved by Carlsson [10] for r ≤ 3 and Refai [20] forN ≤ 8. Avramov,Buchweitz,and Iyengarin[4] dealtwith regularringsand inparticular theyproved Conjecture 1 for r ≤ 3 without any restriction on the characteristic of k. See also Proposition 1.1 and Corollary1.2 in[2],Theorem5.3 in[24] forresultsincharacteristicnotequalto2.

Inthis paperwe consider theconjecture from theviewpointof algebraicgeometry. We showthat Con-jecture1isimplied bythefollowing inSection2:

Conjecture 2. Let k be an algebraically closed ﬁeld, r a positive integer, and N = 2n an even positive integer.Assumethatthereexistsanonconstantmorphismψ fromtheprojectivevarietyPr_k−1totheweighted quasi-projectivevarietyofrankn square-zerouppertriangularN×N matrices(xij) withdeg(xij)= di−dj+1

forsomeN -tupleof nonincreasingintegers (d1,d2,. . . ,dN). ThenN ≥ 2r.

WewillgiveamoreprecisestatementofConjecture2inSection3afterdiscussingnecessary deﬁnitions andnotation.Weproposethefollowing,whichisstrongerthanConjecture2:

Conjecture3. Letk, r, N , n andψ be as in Conjecture 2. Assume 1≤ R,C ≤ N and thevalue of xij at

every pointintheimage ofψ is 0 wheneveri≥ N − R+ 1 orj≤ C. ThenN ≥ 2r−1(R+C).

Note that inConjecture 3 wehave 2r−1(R+C) ≥ 2r _because₁ _{≤ R and} ₁_{≤ C.} _The _main _result _of _the

paperis aproofofConjecture 3whenN < 8 orr < 3,seeTheorem 2andTheorem 6. AsConjecture 3is thestrongestconjecturementionedabove,weobtainproofsofalltheconjecturesinthisintroductionunder the sameconditions,including themain resultof Carlsson in[9]. Also note thatfor r = 2,taking N > 4

gives novel resultsnot covered inthe literature. However, when the characteristic of theﬁeld k is not 2, Iyengar–Walker[15] gaveacounterexampletoConjecture1foreachr≥ 8.HencebySection2,wecansay thatConjectures 2and3are alsofalsewhen r≥ 8 andthecharacteristic of theﬁeldk isnot2.Allthese conjecturesare stillopen incasethecharacteristic ofk is2.InSection4,we concludewithexamplesand problems.

Wethankthereferee forgivingusextensivefeedback,ashorterproofofTheorem1andencouragingus toextendourresultstoﬁeldswithcharacteristicsotherthan2.WearealsogratefultoMatthewGelvinfor commentsandsuggestions.

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2. SomenotesonConjectures 1, 2,and3

To show thatConjecture 3is the strongest conjecturein Section1, it is enoughto provethe following theorem.

Theorem 1.Conjecture2impliesConjecture 1.

Proof. Let k,r,andA beas inConjecture1.Let(M,∂) beafreeDG-A-moduleofrankN which satisﬁes thehypothesisinConjecture1.Withoutlossofgenerality,wecanassumethatN isthesmallestrankofall suchDG-A-modules.

Suppose the image of the diﬀerential ∂ is not contained in (y1,. . . ,yr)M . Then, there exists a basis

b1,. . . ,bN of M and there are some i and j so that ∂(bi) = cbj +

l=jglbl for some non-zero c ∈ k

and some gl ∈ A. Replacing bj with ∂(bi) gives a new basis b1,...,bN such that ∂(bi) = bj. Now form

the acyclic sub-DG-A-module (E,∂) of (M,∂) spanned by b_i, b_j. The map (M,∂) → (M,∂)/(E,∂) is a surjective quasi-isomorphismandM/E is freeofrankN− 2.This isacontradiction.Hence,theimage of thediﬀerential∂ iscontainedin(y1,. . . ,yr)M .

Now pick any basis c1,. . . ,cN of M such that deg(c1) ≤ · · · ≤ deg(cN). Let m be such that

deg(cN−m+1) = · · · = deg(cN) and deg(ci) < deg(cN−m+1) for all i < N − m + 1. For each i, we have

∂(ci) = jgijcj, for some homogeneous polynomials gij ∈ A. Since the image of ∂ is contained in

(y1,. . . ,yr)M , no gij is anon-zero constant. Thus, whenever gij is non-zero, we have deg(gij)≤ −1 and

hence deg(cj) = deg(ci)− 1 − deg(gij)≥ deg(ci). It follows that the diﬀerential on M restricts to one on

thesubmodule

L = AcN−m+1⊕ · · · ⊕ AcN.

Moreprecisely,foralli∈ {N − m+ 1,. . . ,N} wehave∂(ci) =N_j=N_−m+1gijcj whereeachnonzerogij isa

linearpolynomial.Hence,relativeto thebasisc_N−m+1,. . . ,cN,thediﬀerential∂ onL isgivenbyamatrix

intheform y1X1+· · · + yrXr where eachXi is anm× m matrix withentriesink. Since∂2= 0 wehave

X2

i = 0 and XiXj+ XjXi= 0 foralli,j.Incasethecharacteristic oftheﬁeldk is 2,by aclassicalresult

aboutcommutingsetofmatrices,forexampleseeTheorem7 onpage207 in[14],there existsaninvertible

m×m matrixT withcoeﬃcientsink suchthatT−1XiT isuppertriangularforalli∈ {1,. . . ,r}.Incasethe

characteristicoftheﬁeldk isnot2,foreverypolynomialQ innoncommutativer variables,thesquareofthe matrixQ(X1,. . . Xr)(XiXj−XjXi) iszero.Therefore,byatheoremofMcCoyasstatedin[22] (seealso[12],

[17]),againthere exists amatrixT as abovewhich simultaneouslyconjugatesallXi’s to uppertriangular

matrices. Inotherwords,thereisak-linearchangeofbasisinwhicheachXi isuppertriangular.Itfollows

that,relativetothisnewbasisc_N−m+1 ,. . . ,c_NofL onehas∂(c_i)∈ ⊕N_j=i+1Ac_j foralli∈ {N −m+1,. . . N}.

Note that M/L is a free DG-A-module whose diﬀerential has an image in (y1,. . . ,yr)(M/L) and so, by

inductive argumentsonrank,we mayassume thatM/L admits abasiswhichmakesits diﬀerentialupper triangular. The unionof any liftof this basisto M with thebasis c_N_−m+1,. . . ,c_N gives abasisB for M

where ∂ is representedby anuppertriangular matrixΨ.Moreover, Propositions8 and9 in [10] work for any characteristic. Hence N isdivisible by2 and for any γ in kr_{− {0} the}_evaluation _of _Ψ _at _{γ gives} _a

matrixofrankN/2.

LetS = k[x1,. . . ,xr] bethepolynomialalgebrawithdeg(xi)= 1. For1≤ i≤ r,replaceyi withxi inΨ

toobtainΨ.NotethatΨ canbeconsideredasa nonconstantmorphismfromtheprojectivevarietyPr−1_k to theweightedquasi-projectivevarietyofrankN/2 square-zeroupper triangularN× N matrices(xij) with

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3. Varietiesofsquare-zero matrices

Weassumethatk is analgebraicallyclosed ﬁeld,n apositiveinteger,N = 2n,andd= (d1,d2,. . . ,dN)

anN -tupleofnonincreasingintegers.

3.1. Statementsof conjectures

Wegiveherethenotationfortheaffine andprojectivevarieties usedto provetheconjectures discussed above.Firstwefix anaffine varietyUN,aringR(UN),andasubvarietyVN asfollows:

• UN istheaﬃnevarietyofstrictlyuppertriangularN× N matricesoverk.

• R(UN)= k[ xij |1≤ i< j≤ N ] isthecoordinateringofUN.

• VN isthesubvarietyofsquare zeromatricesinUN.

Deﬁne anaction oftheunit groupk∗ onUN by λ· (xij)= (λdi−dj+1xij) forλ∈ k∗. Usingthis actionwe

settwo morenotation.

• U (d) istheweightedprojectivespace givenbythequotient ofUN− {0} bytheactionofk∗.

• R(U (d)) isthe homogeneouscoordinateringofU (d).Inotherwords, R(U (d)) isR(UN) consideredas

agradedringwithdeg(xij)= di− dj+ 1.

Note thatthepolynomialpij = j−1

m=i+1

ximxmj inR(U (d)) is homogeneousof degreedi− dj+ 2 whenever

1≤ i< j ≤ N.Similarly,the n× n-minors of(xij) are homogeneous polynomials inR(U (d)).Hence,we

deﬁnetwo subvarietiesofU (d) asfollows:

• V (d) istheprojectivevarietyofsquarezeromatricesinU (d).

• L(d) isthesubvarietyofmatricesofranklessthann in V (d).

Wecanusethis terminologytorestateConjecture2:

Conjecture4.Letk bean algebraicallyclosedﬁeld, r a positiveinteger,andd anN -tupleof nonincreasing integers. If thereexists anonconstant morphismψ fromthe projectivevariety Pr_k−1 to thequasi-projective varietyV (d)− L(d), thenN ≥ 2r.

LetU beanopen subsetofV (d).Wesayψ :P_kr−1 → U isanonconstantmorphism ifψ isrepresentedby (ψij) sothatthefollowingconditionsaresatisﬁed:

(I) there exists a positive integer m so that each ψij is a homogeneous polynomial in the variables

x1,x2,. . . ,xrinS ofdegreem(di− dj+ 1) for 1≤ i< j≤ N,

(II) foreveryγ∈ Pr_k−1 there existsi,j suchthatψij(γ)= 0.

Inparticular,ifψ :Pr_k−1 → U isanonconstantmorphism, ψ canbeconsideredasafunctionfromPr_k−1 to

U representedbyanonconstantpolynomialmapψ from Ar

k totheconeoverU suchthatψ( Ark− {0}) does

notcontain the zeromatrix inVN. Each indeterminate xij canbe viewed as homogeneous polynomialin

R(U (d)).Hencefor1≤ R,C ≤ N wedeﬁneanimportantsubvarietyofV (d):

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Now werestatetheConjecture3asfollows:

Conjecture5.Letk beanalgebraicallyclosed ﬁeld, r apositiveinteger,andd an N -tupleof nonincreasing integers. If there existsa nonconstantmorphismψ from theprojective varietyPr−1_k tothequasi-projective variety V (d)RC− L(d), thenN ≥ 2r−1(R+C).

Hence,these varietiesare themain interestinthis paper.

3.2. Actionof aBorelsubgroupon VN

HereweintroduceanactionofaBorelsubgroupinthegroupofinvertibleN×N matricesonthevarieties discussed intheprevioussubsection.Firstweset anotationfortheBorelsubgroup.

• BN isthegroupofinvertible uppertriangularN× N matriceswith coeﬃcientsink.

ThegroupBN actsonVN byconjugation.

• VN/BN denotes theset oforbitsoftheactionof BN onVN.

• BX denotestheBN-orbitthatcontainsX∈ VN.

A partial permutation matrix is a matrixhaving at most onenonzero entry, which is 1, in eachrow and column. A result of Rothbach (Theorem 1 in [21]) implies that each BN-orbit of VN contains a unique

partial permutationmatrix.Henceweintroducethefollowingnotation:

• PM(N ) denotes the set of nonzero N × N strictly upper triangular square-zero partial permutation matrices.

Thereisaone-to-onecorrespondencebetweenPM(N ) andVN/BN sendingP toBP.Wecanidentifythese

partial permutationmatriceswithasubsetofthesymmetricgroupSym(N ):

• P(N ) isthesetofinvolutionsinSym(N );i.e., thesetofnon-identitypermutationswhosesquareisthe identity().

ForP ∈ PM(N) andσ∈ P(N),

• σP denotes thepermutationinP(N ) thatsendsi toj ifPij= 1;

• Pσ denotes thepartial permutationmatrixinPM(N ) that satisﬁes (Pσ)ij = 1 ifand onlyifσ(i)= j

andi< j.

ClearlytheassignmentsP → σP andσ→ Pσaremutualinversesandsodeﬁneaone-to-onecorrespondence

betweenP(N ) and PM(N ).

3.3. Apartial orderon theset of orbits

There are important partial orders on VN/BN, P(N ), PM(N ), all of which are equivalent under the

one-to-one correspondence mentioned above (cf. [21]). We begin with VN/BN. For Borel orbits B,B ∈

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• B≤ B meanstheclosureofB,consideredasasubspaceofVN, containsB.

Second, we deﬁne a partial order on PM(N ). To do this, we consider ranks of certain minors of partial permutationmatrices.Ingeneral,foranN× N matrixX,

• rij(X) denotestherankofthelowerleft((N− i+ 1)× j) submatrixofX,where 1≤ i< j≤ N.

ForpartialpermutationmatricesP,P ∈ PM(N),

• P ≤ P meansrij(P)≤ rij(P ) for alli,j.

Third,wedeﬁne apartial orderonP(N ). Forpositiveintegersi< j,letσ(i,j) denotetheproductof the permutationsσ and(i,j) andσ(i,j)_the_conjugate_of_{σ by}_(i,_j)._For_σ,_σ_{∈ P(N),}

• σ≤ σ ifσ canbe obtainedfromσ byasequence ofmovesof thefollowing form: – Type Ireplacesσ withσ(i,j) ifσ(i)= j andi= j.

– Type IIreplacesσ withσ(i,i)ifσ(i)= i< i < σ(i). – Type IIIreplacesσ withσ(j,j)ifσ(j)< σ(j)< j< j.

– Type IVreplacesσ withσ(j,j) _if_σ(j₎_{< j}_{< j = σ(j).}

– Type Vreplacesσ withσ(i,j) _if_i_{< σ(i)}_{< σ(j)}_{< j.}

Theideaofdescribingorder viathesemovescomes from[16].Although weusediﬀerent namesformoves, the set of possible moves are same. We represent a permutation (i1,j1)(i2,j2). . . (is,js) in P(N ) by the

matrix i1 i2 . . . is j1 j2 . . . js .

Forexample,wedrawtheHassediagram ofP(4) inwhicheachedgeislabelledbythetypeofthemove itrepresents(seeFig.1).

WhenN ≥ 6,theHassediagram forP(N ) istoo largetodraw here.Weare actuallyonlyinterestedin asmallpartofthis diagram,whichwediscussinSection3.6.

Onecanconsider Fig.1as astratiﬁcation ofV4.In thenextsection, weuse thestratiﬁcation of VN to

stratifyV (d).

3.4. Stratiﬁcation ofV (d)

Ford= (d1,d2,. . . ,dN) anN -tupleof nonincreasingintegers,λ∈ k∗,andX = (xij)∈ VN,wehave

λ· X = λ · (xij) = (λdi−dj+1xij) = DλIλXDλ−1

where Dλ denotes the diagonal matrixwith entries λd1,λd2,. . . ,λdN and Iλ is the scalar matrixwith all

diagonal entries λ. LetPX ∈ PM(N) be the uniquepartial permutation matrix inthe Borelorbit of X.

Considerb∈ BN suchthat

PX = b−1Xb.

Let Iλ,X be the diagonalmatrix whose jth entry isλ if (PX)ij = 1 for somei and 1 otherwise.Then we

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Fig. 1. Hasse diagram of P(4).

IλPX = I_λ,X−1 PXIλ,X.

Hence,wehave

λ· X = Dλb Iλ,X−1 b−1X b Iλ,Xb−1D−1λ = Z−1XZ

where Z = bIλ,Xb−1D−1λ is in BN. Thus, for any X ∈ V (d) there exists a well-deﬁned Borel orbit in

VN/BN thatcontainsarepresentativeofX inVN.Hencewecansetthefollowingnotation.ForX ∈ V (d),

• BX denotestheBorelorbitinVN/BN thatcontainsarepresentative ofX inVN.

Letψ :Pr_k−1→ V (d)− L(d) beanonconstantmorphism.Thereisaliftofthismorphismtoamorphism from Ar

k − {0} to the cone over V (d)− L(d) that canbe extended to a morphism ψ : Ark → VN. Since

Ar

k is anirreducible aﬃne variety,there exists aunique maximal Borelorbitamong the Borelorbitsthat

intersects the image of ψ nontrivially. Note that this maximal Borel orbit is independent of the lift and extension we selected becauseit is also maximal inthe set {BX|X ∈ V (d)}. Hence we mayassociate a

permutationto thenonconstantmorphismψ:

• σψ isthepermutationthatcorrespondstotheuniquemaximalBorelorbitBX whereX isintheimage

ofψ.

Note that every point in the image of a morphism ψ as above must have rank n. Hence σψ must be a

product ofn distincttranspositions.InSection3.6,wewill restrictourattentiontosuchpermutations.

3.5. Operationson polynomialmapsfrom Ar k toVN

Another way to see that BX is well-deﬁned for X ∈ V (d) is to consider the fact that a minor of a

representative ofX iszeroifandonlyifthecorresponding minorofanotherrepresentative iszero.Weuse this factseveral timestoproveourmain result.Henceweintroducethefollowingnotation. ForX ∈ VN,

• mi1i2... ik

j1j2... jk(X) denotesthedeterminantofthek×k submatrixobtainedbytakingthei1th,i2th,. . . , ikth

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Firstnotethatmi1i2... ik

j1j2... jk canbeconsideredasamorphismfromVN tok,andhencecanbecomposedwith

the morphism ψ mentioned inthe previous subsection.Here we discuss several other morphisms thatwe cancomposewithsuchmorphisms.Foru∈ k,

• Ri,j(u) isthefunctionthattakes asquarematrixM andmultipliestheith rowofM byu andaddsit

tothejth rowofM whilemultiplyingthejth columnofM byu andaddingittotheith columnofM .

Note that Ri,j(u)(M ) is a conjugate of M . In fact, they are inthe same Borelorbit when M ∈ VN and

i > j. Hence,for i > j, we can consider Ri,j(u) as an operation thattakes a morphism from Ask to VN

and transformsitto amorphismfrom As+1_k to VN by consideringu as anewindeterminate andapplying

Ri,j(u) tothemorphism.Forv∈ k∗,

• Di(v) denotesthefunctionthattakesasquarematrixM andmultipliestheith rowofM byv andthe

ith columnof M by1/v.

Letq beapolynomialins indeterminates.WedeﬁneDi(q) asanoperationthattakesarationalmapfrom

thequasi-aﬃnevarietyAs_k− Z toVN and transformsitintoarationalmap fromAsk− Z ∪ V (q) toVN by

applyingDi(q),usingthefollowingnotation:

• V (q1,q2,. . . ,qk) isthevarietydeterminedbytheequationsq1= q2= . . . qk = 0.

Weusetheabovenotationalsoforvarietiesinprojectivespacesdeterminedbythehomogeneouspolynomials

q1,q2,. . . ,qk.

3.6. Rank oforbitsand proofof ﬁrstmainresult

Each σ∈ P(N) isaproduct ofdisjoint transpositions.Hencefor σ∈ P(N), wedeﬁne therank of σ to

be the numberof transpositionsinσ. Note thatunderthe one-to-one correspondence between P(N ) and PM(N ),therankofapermutationisequalto therankofthecorresponding partialpermutationmatrix.

• RP(N ) denotesthepermutationsinP(N ) ofrankn.

NotethatallmovesotherthantypeIpreservetherankofσ.Indeed,theonlywayofobtainingσ ofsmaller rankbyapplying ourmovesis bydeletingatransposition, whichistheeﬀectofmoveof typeI.Alsonote thatitisimpossibletohaveamoveoftypeIIor amoveoftypeIVbetweentwopermutationsinRP(N ).

Forexample,wedrawtheHassediagramforRP(6) whereeachdottedlinedenotesamoveoftypeIIIand solidlinedenotesamoveoftypeV(seeFig.2).SuchHassediagrams,withparticularattentionpaidtothe maximalelements,willleadtotheproofofourﬁrstmainresult.

Theorem2.Conjecture 5holdsforN < 8.

Proof. TakeN < 8,d= (d1,d2,. . . ,dN) an N -tupleofnonincreasingintegers,andψ :Prk−1 → V (d)− L(d)

anonconstantmorphism.Thenσ = σψisinRP(N ).ByconsideringFigs.1and2,wenotethatthereexists

aunique maximal σ ∈ RP(N) suchthat σ can be obtained from σ bya sequence ofmoves of type III. Since movesof type III do not change the number of leading zero rows and ending zero columns of the corresponding partial permutation matrices, theBorel orbitcorresponding to σ is containedinV (d)_RC if and onlyif theBorel orbitcorresponding to σ iscontained inV (d)_RC for allR,C.Hence itis enoughto

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Fig. 2. Hasse diagram of RP(6).

consider thecaseswhereσ is lessthanorequaltoamaximalelement inRP(N ) forN = 2,4,6.Wecover these casesbyproving inthefollowingeightstatements:

(i) Ifσ = (1,2) thenr < 2.

Assumetothecontrarythatσ = (1,2) andr≥ 2.Ifwealsowriteψ foritsrestrictiontoP1

k⊆ Pr−1k ,weget

amapoftheform

ψ(x : y) = 0 p12 0 0 ,

where p12 isahomogeneouspolynomialink[x,y].Since k isalgebraically closed,there exists γ∈ P1_k such thatp12(γ)= 0.Thismeansψ(γ) isinL(d),whichisacontradiction.

(ii) Ifσ≤ (1,2)(3,4) thenr < 3.

Supposetothecontrarythatσ≤ (1,2)(3,4) andr≥ 3.Whenwerestrictψ toP2

k,wegetamapoftheform

ψ(x : y : z) = ⎡ ⎢ ⎢ ⎢ ⎣ 0 p12 p13 p14 0 0 0 p24 0 0 0 p34 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎦.

Note thatthere existsγ inP2

k suchthat

p12(γ) = 0 and p13(γ) = 0.

Againthis meansψ(γ)∈ L(d).Hencethiscaseisprovedbycontradiction aswell. (iii) If σ≤ (1,4)(2,3) thenr < 2.

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Supposewehave ψ(x : y) = ⎡ ⎢ ⎢ ⎢ ⎣ 0 0 p13 p14 0 0 p23 p24 0 0 0 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎦. Letmi1i2... ik

j1j2... jk beas inSection3.5andusethesamenotationtodenoteitscompositionwithψ.Thenthere

existsγ in P1

k suchthat

m12₃₄(γ) = (p13p24− p23p14)(γ) = 0.

Thisagaingivesacontradiction.

(iv) Ifσ≤ (1,2)(3,4)(5,6) thenr < 3.

Supposeotherwise.Wehave

ψ(x : y : z) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 p12 p13 p14 p15 p16 0 0 0 p24 p25 p26 0 0 0 p34 p35 p36 0 0 0 0 0 p46 0 0 0 0 0 p56 0 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .

Ifp12andp13 arenotrelatively primehomogeneouspolynomials thenthereexists γ∈ P2k suchthat

p12(γ) = 0, p13(γ) = 0, and m123456(γ) = 0.

Moreover,ifp46(γ)= 0 andp56(γ)= 0,then therankofψ(γ) isatmost2,whichleadsto acontradiction. Hencewehavep46(γ)= 0 orp56(γ)= 0. Let

c4(γ) := ⎡ ⎢ ⎣ p14(γ) p24(γ) p34(γ) ⎤ ⎥ ⎦ and c5(γ) := ⎡ ⎢ ⎣ p15(γ) p25(γ) p35(γ) ⎤ ⎥ ⎦ .

Sinceψ2= 0, c4(γ)p46(γ)+ c5(γ)p56(γ)= 0. Bythefactthatp46(γ)= 0 orp56(γ)= 0,c4(γ) andc5(γ) are

linearlydependent.Thustherankofψ(γ) isatmost2,whichisacontradiction.

Therefore wemay assumep12 and p13 are relatively prime. Sinceψ2 = 0,wehave p12p24+ p13p34 = 0

and p12p25+ p13p35 = 0. This impliesthatp12 dividesp34 and p35, and similarly p13 dividesp24 and p25. Thenthere exists γ inP2

k suchthat

p12(γ) = 0 and p13(γ) = 0.

Thismeansp12, p13,p24,p25,p34,andp35 allvanishatγ.Henceψ(γ)∈ L(d),whichisacontradiction.

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These casesaresymmetric,soitisenoughtoprove (v). Consider ψ(x : y : z) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 p12 p13 p14 p15 p16 0 0 0 0 p25 p26 0 0 0 0 p35 p36 0 0 0 0 p45 p46 0 0 0 0 0 0 0 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .

Wemodifyψ bytheoperationsinSection3.5.FirstapplyR6,5(u) toψ foranewvariableu.Ifp46+up45= 0,

apply D5(1/(p46+ up45)) andthenR5,6(−p45) to obtainamatrixoftheform

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 p12 p13 p14 ∗ ∗ 0 0 0 0 m24 56 p26+ up25 0 0 0 0 m34 56 p36+ up35 0 0 0 0 0 p46+ up45 0 0 0 0 0 0 0 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .

Hencebyselectingacorrectvalueforu wewouldbedoneifV (m24

56,m3456) V (p45).Wemayassume V (m24₅₆, m34₅₆)⊆ V (p45).

Similarly,wemayalsoassume

V (m23₅₆, m34₅₆)⊆ V (p35) and V (m23₅₆, m24₅₆)⊆ V (p25). Therefore,

V (m2356, m3456, m2456)⊆ V (p25, p35, p45) =∅.

Thus,{m23₅₆,m34₅₆,m2456} isaregularsequenceink[x,y,z].Ifp45andp46arenotrelativelyprime,thereexists

γ suchthatm23

56(γ)= 0,andp45(γ)= p46(γ)= 0.Hence,wemayassumep45 andp46are relativelyprime,

whichleadsacontradictionaswehave

p45m23₅₆+ p25m34₅₆− p35m24₅₆= 0. (vii) Ifσ≤ (1,6)(2,3)(4,5) then r < 2.

To provethiscase,consider

ψ(x : y) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 p13 p14 p15 p16 0 0 p23 p24 p25 p26 0 0 0 0 p35 p36 0 0 0 0 p45 p46 0 0 0 0 0 0 0 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .

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V (m124₃₅₆)⊆ V (p13, p23) and V (m123456)⊆ V (p14, p24).

Hence{m124

356,m123456} mustbearegularsequenceink[x,y].Howeverthisisimpossiblebecausethedeterminant

of ⎡ ⎢ ⎣ gcd(p13, p14) p15 p16 gcd(p23, p24) p25 p26 0 gcd(p35, p45) gcd(p36, p46) ⎤ ⎥ ⎦ dividesbothm124 356 andm123456. (viii) Ifσ≤ (1,6)(2,5)(3,4) thenr < 2.

Itisenoughtoconsider aroot ofm123

456to provethiscase. 2

Note thatintheaboveproofthe last twocasesproveConjecture 5whenN ≤ 6 andr≤ 2. Intherest of the paper we will generalize these ideas to provethe conjecture for r ≤ 2. To do this we examine the dimensionsofthesevarieties.

3.7. Orbit dimensionsandproof of secondmain result

Wenowintroducenotationfordimensionsofthesevarieties.Inthissubsection,forσ∈ P(N),iftherank ofσ iss,then weobtaintwo sequencesofnumbersi1,. . . ,isandj1,. . . ,jssatisfyingthefollowing:

σ = (i1, j1)(i2, j2) . . . (is, js)

withi1< i2<· · · < isandia< jaforall1≤ a≤ s.In[18], Melnikovgivesaformulaforthedimensionof

aBorelorbitBσ forσ inP(N ) asfollows:

• ft(σ):= #{jp| p< t,jp < jt} + #{jp| p< t,jp< it} for 2≤ t≤ s, • dim(Bσ)= N s+ s t=1 (it− jt)− s t=2 ft(σ).

WedeﬁneanewsubsetofRP(N ):

• DP(N ) istheset ofallσ inRP(N ) suchthatdim(Bσ)= dim(Bσ)− 1 wheneverσ isapermutation

obtainedbyapplyingasinglemoveoftypeI toσ.

For instance, the following is the Hasse diagram of DP(8) (Fig. 3). Note that in the Hasse diagram of

DP(8) allmovesare oftypeV.This isgenerallythecase,whichwe willprovebelow.Beforewedoso,we willproveaneasierresultthatwillintroducethenotationandargumentstylethatwillbenecessary.

Fixσ∈ DP(N).Weusetheourconventionforσ atthebeginningofthissubsectionwhichimpliesi1= 1.

Forq∈ {1,. . . ,n},letσ betheresultofapplying themoveoftypeIthatdeletes theqthtransposition ofσ,so that σ = (1, j1) . . . (iq−1, jq−1)(iq+1, jq+1) . . . (in, jn) = ⎛ ⎜ ⎝ 1 i2 . . . iq−1 iq iq+1 . . . in j1 j2 . . . jq−1 jq jq+1 . . . jn ⎞ ⎟ ⎠ .

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Fig. 3. Hasse diagram of DP(8).

Then byMelnikov’sformulawehave

dim(Bσ) = N n + n t=1 (it− jt)− n t=2 ft(σ) , and dim(Bσ) = N (n− 1) + n−1 t=1 (it− jt)− n−1 t=2 ft(σ).

To simplifyourcalculation,wewriteft(σ)= ft1(σ)+ ft2(σ) for2≤ t≤ n,where

f_t1(σ) = #{jp| p < t, jp < jt} and ft2(σ) = #{jp| p < t, jp< it},

and weusethenotation:

ft,ql (σ) = ⎧ ⎪ ⎨ ⎪ ⎩ f_tl(σ) if t≤ q − 1 0 if t = q fl t−1(σ) if t≥ q + 1 forl = 1,2.

Lemma 3.If N= 2 and thetransposition(1,N ) appears inσ,thenσ /∈ DP(N).

Proof. If (1,N ) appears in σ and σ ∈ DP(N), let q = 2 and σ be the result of deleting the second transpositionfrom σ.Sincethei’sareincreasingandia< jaforall1≤ a≤ n,wehavei2= 2,so

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σ= ⎛ ⎜ ⎝ 1 2 i3 . . . in N j2 j3 . . . jn ⎞ ⎟ ⎠ . Thus3≤ j2≤ N − 1.Letj2= N− b forsome1≤ b≤ N − 3.Then

σ = ⎛ ⎜ ⎝ 1 2 . . . in N N− b . . . jn ⎞ ⎟ ⎠ ,

andany numberbetweenN− b andN hastoappearas aj orani thatisbiggerthanj2.Therefore, _n t=2 f_t1(σ)− f_t,21 (σ) + _n t=2 f_t2(σ)− f_t,22 (σ) = b− 1.

Hence,dim(Bσ)− dim(Bσ)= N + 2− N + b− (b− 1)= 3,soσ /∈ DP(N). 2

Nowweprovethemainpropositionofthissubsection.

Proposition 4.If σ∈ DP(N), then jp< jt forallp< t, andthereforewe cannot applya moveof typeIII

toσ.Conversely,if σ∈ RP(N) and wecannot applyamoveof type ofIII toσ,thenσ∈ DP(N).

Proof. Assumethatσ∈ DP(N).Wewill provethefollowingstatementbyinductiononk:

jn−k< . . . < jn−1< jn and∀ (p < n − k), jp< jn−k. (∗)

Supposek = 0. To prove(∗), weneed to show that∀p< n,jp < jn. Letσ be obtained bydeletingnth

transpositionofσ. σ= ⎛ ⎜ ⎝ 1 i2 . . . in−1 in j1 j2 . . . jn−1 jn ⎞ ⎟ ⎠ . Sinceσ∈ DP(N),wehave 1 = dim(Bσ)− dim(Bσ) = N + (in− jn)− f_n1(σ) + f_n2(σ) . (1)

Since thetotalnumberofpossible j exceptjn isn− 1,and anynumberbetween in and jn hasto appear

as j,we havefn2(σ) = n− 1 − (jn− in− 1).Bythe equation(1),fn1(σ)= n− 1,so that∀p< n,jp < jn

istrue.Thereforejn= N .

Nowassumethestatement(∗) istruefork. Thenwecanvisualiseσ asfollows:

σ = ⎛ ⎜ ⎝ 1 < i2 < . . . < in−k−1 < in−k < in−k+1 < . . . < in j1 j2 . . . jn−k−1 jn−k < jn−k+1 < . . . < jn ⎞ ⎟ ⎠ Weneedtoprove(∗) fork + 1,thatis,

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By the second part of the inductive hypothesis for k, we have j_n−k−1< j_n−k so the ﬁrst part of (∗) is already true, and we onlyneed to show that thesecond partholds. In other words, itis enough to show thatf_n1_−k−1(σ)= n− k − 2.Letσ be obtainedbydeleting(n− k − 1)thtranspositionofσ.Thenwehave

n t=n−k f_t1(σ)− f_t,n1 _−k−1(σ) = k + 1. Letw := #{ip| in−k−1< ip< jn−k−1}.Then f_n−k−12 (σ) = n− (k + 2) − (jn−k−1− in−k−1− 1 − w), and n t=n−k f_t2(σ)− f_t,n2 _−k−1(σ) = k + 1− w.

Bythefactthatdim(Bσ)− dim(Bσ)= 1,wehavef_n−k−11 (σ)= n− k − 2.Thustheﬁrstclaimisproved.

Conversely, givenσ∈ RP(N),suppose thatσ isthe resultofapplying themoveoftypeI thatdeletes theq-th transpositionfrom σ.Notethatf1

q(σ)= q− 1 and n t=q+1ft1(σ)− ft,q1 (σ)= n− q.Hence, n t=2 f_t1(σ)− f_t,q1 (σ) = n− 1. Then, dim(Bσ)− dim(Bσ) = N + (iq− jq)− (n − 1) − n t=2 f_t2(σ)− f_t,q2 (σ) .

Wealsohavethediﬀerencef2

t(σ)− ft2(σ)= 0 whent∈ {1,. . . ,q− 1}.Therefore, n t=2 f_t2(σ)− f_t,q2 (σ) = n t=q #{jp| p < t, jp< it} − n t=q+1 #{jp| p < t, p = q, jp< it} = #{jp| jp< iq} + #{it| jq< it}.

Let F = #{jp | jp < iq} and G = #{it | jq < it}. Note that numbers between iq and jq must appear

as jl for l < q or as is where s > q. Let a = #{jp | iq < jp < jq} and b = #{it | iq < it < jq}. Then

a+ b = jq− iq− 1. Let A= #{jp | iq < jp} and B = #{it| it < jq}.Wehave A− a+ B− b− 1 = n.

Therefore, A+ B = n+ jq− iq. σ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ B i1, . . . , iq. . . b G . . . , in j1, . . . F a _{. . . j} q, . . . , jn A ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

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Sinceσ∈ RP(N),A+ B + F + G= 2n.ThenF + G= n− jq+ iq.Consequently,dim(Bσ)− dim(Bσ)= 1,

soσ∈ DP(N). 2

Lemma5.Forevery X inV (d)− L(d) we have

rij(X)≥ j − i + 1 − n.

Proof. TherankofX isn,sor1N(X)= n.Theresultfollows fromtheinequality

rij(X) + (i− 1) + (N − j) ≥ r1N(X). 2

Wenowdeﬁneourlast setofpermutations:

• MP(r,N ) isthesetofminimalpermutationsinP(N ) thatappearasapermutationintheformσψfor

somed andnonconstantmorphismψ :Pr_k−1→ V (d)− L(d).

Wenowstateandproveoursecond mainresult.

Theorem6.Conjecture 5holdsforr≤ 2.

Proof. We haveMP(1,N )= {(1,n+ 1)(2,n+ 2). . . (n,N )} (see Example 1). This means Conjecture 5

holdsforr = 1,becauseN− n+ (n+ 1)− 1= N ≤ N.HenceitisenoughtoproveConjecture5forr = 2.

SupposethatConjecture5doesnotholdforr = 2.ThenthereexistsanN -tupleofnonincreasingintegers

d= (d1,d2,. . . ,dN),twopositiveintegersR,C,andanonconstantmorphismψ :P1k→ V (d)RC− L(d) such

thatN < 2(R+C), or equivalently n<R+C.Write σψ = (i1,j1)(i2,j2). . . (in,jn) with 1= i1 < i2 < · · · < in andia< jaforall1≤ a≤ n.

Firstassumethatσ∈ DP(N).ByProposition4,wehavej1< j2<· · · < jn = N .Therefore,C = j1− 1

andR= N− in.Moreover,foreverya wehaveja>C andia< N− R+ 1.SetI :={i1,. . . ,in}.Notethat

{1,. . . ,C}⊆ I since∀a,ja>C.So,ia= a ifa≤ C.Similarly,ja−n= a ifa≥ N − R+ 1.Set

i := n− R + 1 and i := C

and

j := N− R + 1 and j := C + n.

Wehave

i− i = R + C − n − 1 = j − j.

Inparticular,i≤ i andj≤ j,sinceweassumedthatn<R+C.

Fori≤ i andj ≤ j, letA_j,ji,i denote thesubmatrix of thepartial permutationmatrixPσ obtained by

consideringtherowsfromi toi andcolumnsfromj toj.NotethatA1,i_1,N hasC many1’sandA1,N_j,N hasR many1’s.HenceA1,i_j,N musthave(R+C − n) many1’s.However,there isno1 in A1,i_j,N−1; otherwisethere wouldexist a such thatia< i = n− R+ 1 and ja≥ j,whichleadsto acontradiction byconsideringthe

number of 1’s in the region determined by the union of A1,ia−1

1,N and A

1,N

ja+1,N. Similarly, there is no 1 in

A1,i

j+1,N.Hencethe(R+C − n)× (R+C − n)-submatrixA i,i

j,j contains(R+C − n) many1’s.Thus,A i,i j,j is

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1 C j j ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ 0 . . . 0 .. . ... i = n− R + 1 Ai,i j,j 0 . . . 0 i =C 0 . . . 0 j = N− R + 1 .. . ... 0 . . . 0 N

This inparticular meansthatforeveryX intheimage ofψ wehave

ri,j−1(X) = rn−R+1,N−R(X) = 0

and

r_i+1,j(X) = r_C+1,C+n(X) = 0.

Sincek isalgebraicallyclosed,thereexistsarootoftheminormi,i+1...,i

j,j+1...,j.Thus,thereexistsX intheimage

of ψ suchthat

r_i,j(X)≤ i − i, whichmeans

r_n−R+1,C+n(X)≤ R + C − n − 1. Lemma5impliesthatforeveryX intheimage ofψ wehave

rn−R+1,C+n(X)≥ C + n − (n − R + 1) + 1 − n = R + C − n.

This isacontradiction,sowearedonewiththecaseσ∈ DP(N).

Now assumethat σ /∈ DP(N). We recursivelydeﬁne perturbations of ψ sothat we canagain use the square submatrix Ai,i_j,j to get acontradiction similar to that of the previous case. Set ψ0 = ψ, n0 = 0,

Z0=∅.Wehavearationalmapψ0_:_A2+n0

k − Z0→ VN.Nowgiven

ψs:A2+ns

k − Zs→ VN

we deﬁneψs+1_:_A2+ns+1

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Assumeσs∈ DP(N)./ ByProposition4,thereexists amoveoftypeIIIthatwecanapplyto σs.Hence, wemaydeﬁne l_s:= min{ l| l < l < σs(l) < σs(l)} and ls:= σs(min{ σs(l)| ls < l < σs(l) < σs(ls)}) . Incasel_s< i,wedeﬁne ns+1:= ns+ ls− ls+ 1.

Note thatls> ls and so ns+1 ≥ ns+ 2. Hencethe aﬃnevariety Zs canbe considered as asubvarietyof

A2+ns+1

k byconsideringA

2+ns

k ⊂ A

2+ns+1

k .Herewewrite(x,y,u1,u2,. . . uns+1) todenoteapointinA

2+ns+1

k .

HenceA2+ns

k correspondstothepointswhere uns+1= . . . = uns+1= 0.

Representψs_by_a_matrix_(ψs

ij) whose(i,j)-entryψsi,jisarationalfunctionink(x,y,u1,u2,. . . uns).Letpi

denotetheentryψs

i,σ(ls)whichisapolynomial.Alsoletp be¯ thegreatestcommondivisorofpls,pls+1,. . . ,pls.

Setp_i := pi/¯p fori∈ {ls,ls+ 1,. . . ,ls}.Wedeﬁne

Zs+1= Zs∪ V ⎛ ⎝ uns+1, ls i:=l_s uns+i−ls+1p i ⎞ ⎠ .

We obtain ψs+1 from ψs by ﬁrst applying Dls(uns+1), then Ri,ls(ui) where l

s < i ≤ ls, then Di(li:=ls suns+i−l s+1p

i) for ls < i ≤ ls, and ﬁnally applying Rls,i(−p

i) for ls < i ≤ ls. Notice that pi

alsodependsons,so wewritep_s,i insteadofp_i whens isnotclear.

We canrepeat this process until it is nolonger possible to ﬁnd amove of typeIII with l_s< i. At the end of this partof the process we obtain a rational map ψt _for _some _t. _Then _we _can _continue _with _the

symmetric(withrespectto thediagonalofthematrixrunningfromthelowerleftentry totheupperright entry)operationsassumingψsisdeﬁnedfors≥ t.Wedeﬁne ψs+1as follows:

l_s:= max{ l| l< l < σs(l) < σs(l)}

and

ls:= σs(max{ σs(l)| ls < l < σs(l) < σs(ls)}) .

We repeat the symmetric operations as long as we have σs(ls) > j. We deﬁne ns+1 := ns+ σs(ls)−

σs(ls)+ 1.Let pj denote theentryψlss,j whichisapolynomial.Also letq be¯ thegreatestcommon divisor

ofpσs(ls),pσ(ls)+1,. . . ,pσs(ls).Similarly,wewritep

s,j insteadofpj whens isnotclear.

Attheendofthis processwe obtainarationalmapψ¯t_from _the_quasi_aﬃne_variety_{U =}_A2+nt¯

k − Z¯tto

VN where wehaveri,j−1(X)= rn−R+1,N−R(X)= 0 and ri+1,j(X)= rC+1,C+n(X)= 0 foreveryX in the

imageofthisrationalmap.Denote thecompositionofm_j,...ji,...,i withψt¯bym.Noticethatm isapolynomial ink[x,y,u1,. . . un¯t].Deﬁneanotherpolynomialp inthesamepolynomialalgebraas follows:

p = ⎛ ⎝t−1 s=0 uns+1 ls i=ls uns+i−ls+1p s,i ⎞ ⎠ ⎛ ⎝t−1¯ s=t uns+1 σ(ls) j=σ(ls) uns+σ(ls)−j+1p s,j ⎞ ⎠ .

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Notice p_s,l s,p

s,ls+1,. . . ,p

s,ls are relatively prime for 1≤ s ≤ t and similarly p

s,σ(ls),p s,σ(ls)+1,. . . ,p s,σ(l_s)

are also relativelyprimefor t + 1≤ s ≤ ¯t. Moreover, thepolynomialm has anirreducible factorink[x,y]

or for some s, the polynomial m has an irreducible factor in the form f uns + g, where f and g are in

k[x,y,u1,. . . uns−1] so thatf is neitheran associate of ps,ls norp

s,σ(ls). Hence,there exists asolution to

theequations p= 1 and m= 0, whichis againacontradictionbyLemma5. 2 4. Examplesandproblems

Thelast inequalityinConjecture5isequivalentto

r≤ ! log₂ N R + C " + 1.

Onemightaskhow strictthisupperboundforr is.

In allthefollowing examples,we deﬁneψ from P_kr−1 to V (d)RC− L(d) for diﬀerentvalues ofr, N and R+C where r = # log₂ $ N R+C %&

+ 1. It follows that we do nothave abetter upper bound for r in these cases.

Example 1.Forr = 1,N = 2n,d= (0,. . . ,0) andR+C = N,deﬁne

ψ(x) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 M 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ where M = ⎡ ⎢ ⎣ x 0 . .. 0 x ⎤ ⎥ ⎦ .

Note thatσψ= (1,n+ 1)(2,n+ 2). . . (n,N ).Thisexampleshowsthat

MP(1, N ) ={(1, n + 1)(2, n + 2) . . . (n, N)}.

Example 2.Forr = 2 andN = 4,d= (0,0,0,0) andR+C = 2,deﬁne

ψ(x, y) = ⎡ ⎢ ⎢ ⎢ ⎣ 0 x y 0 0 0 0 y 0 0 0 −x 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎦ Inthis example,σψ= (1,2)(3,4).Hence,

MP(2, 4) ={(1, 2)(3, 4)}.

Example 3.Forr = 2,N = 6,d= (0,−1,−1,−1,−1,−1),andR+C = 3,set:

ψ(x, y) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 x2 xy y2 0 0 0 0 0 0 y 0 0 0 0 0 −x y 0 0 0 0 0 −x 0 0 0 0 0 0 0 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .

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Here,wehaveσψ = (1,2)(3,5)(4,6).ConsideringtheHasse diagram forRP(6) inFig.2andsymmetryit

isclearthat

MP(2, 6) ={ (1, 2)(3, 5)(4, 6) , (1, 3)(2, 4)(5, 6) }.

Theaboveexamplecanbegeneralized:

Example4.Forr = 2,N = 2n,d= (0,−n+ 2,. . . ,−n+ 2),andR+C = n,set:

ψ(x, y) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 xn−1 xn−2y . . . yn−1 0 . . . 0 0 0 . . . 0 y ... 0 −x y −x . .. 0 . .. . .. y −x 0 .. . 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .

Wecanusetheaboveexamplesto obtainnewones bythechessboardconstruction:

Construction1.Let(l1,l2,. . . ,lm) beanm-tupleof positiveintegersand V (d)(l1,l2,...,lm) thesubvarietyof

V (d) suchthatxij = 0 whenl1+ l2+ . . . + l(s−1)+ 1≤ i< j≤ l1+ l2+ . . . + lsforsome1≤ s≤ m. For

example,thefollowingmatrixψ(x,y) isinV (d)(1,3,2) where d is6-tupleof nonincreasingintegers:

ψ(x, y) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 x2 _xy _y2 ₀ ₀ 0 0 0 0 y 0 0 0 0 0 −x y 0 0 0 0 0 −x 0 0 0 0 0 0 0 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . Takeψ1∈ V (d1)(l1

1,l12,...,l1m)where d1 isN1-tuplenonpositiveintegersandψ2∈ V (d2)(l21,l22,...,l2m)where d2is

N2-tupleofnonincreasing integers.Wearrange a(N1+ N2)× (N1+ N2) matrixina2m× 2m-chessboard

asfollows:Theij-squarecontainsalε_'i i+1

2

(_{× l}_#εj j+1

2

&matrixsuchthatεk= 1 ifk isoddorεk = 2 ifk iseven

integer.Nowwe colortheij squareblack ifεi= εj andwhiteifεi = εj.Fill intheij squarewithzerosif

it isa blacksquare and otherwise ﬁllitin with(xij) where i≤ i ≤ i andj ≤ j ≤ j part of ψεi where

i,i,j,j are deﬁnedby s = 's+1 2 ( −1 m=1 lεs m+ 1 and s = 's+1 2 ( m=1 lεs m.

Forinstance,usingchessboardconstructionwecanobtainanexample:

Example5. Forr = 2,N = 4+ 6= 10,d= (0,0,−1,−1,−1,−1,−1,−1,−1,−1), andR+C = 3+ 2= 5 weobtainanexamplebyapplying thechessboardconstructiononthemorphismsinExamples2and3.

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0

Z

0

x2

0

xy

0

y2

Z

0

Z

0

Z

0

x

0

y

Z

0

Z

0

Z

0

y

0

Z

0

Z

0

Z

0

−x

0

y

Z

0

Z

0

Z

0

−x

Z

0

Z

0

Z

0

y

Z

0

Z

0

Z

0

−x

0

Z

0

Z

0

Z

0

Z

0

Z

0

Z

0

Z

0

Z

0

Wealsohaveotherwell-knownconstructionsliketheKoszulcomplexconstruction[4] givingusexamples as below.

Example 6.Forr = 3,N = 8 andR+C = 2, deﬁne

ψ(x, y, z) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 x y z 0 0 0 0 0 0 0 0 y −z 0 0 0 0 0 0 −x 0 z 0 0 0 0 0 0 x −y 0 0 0 0 0 0 0 0 z 0 0 0 0 0 0 0 y 0 0 0 0 0 0 0 x 0 0 0 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . Inthis example,σψ= (1,2)(3,5)(4,6)(7,8).

Weendwithafewquestionsforfutureresearch.NoticethatallexamplesdiscussedaboveareinDP(N ).

Henceonecanask:

Question 1.IsMP(r,N )⊆ DP(N)?

Foralltheseexamples ₂rN−1 isaninteger.Forinstance,wecanﬁndanexampleseeExample7forr = 3,

N = 12 but wedonotknowtheanswerofthefollowingquestion:

Question 2. Is there any example for r = 3 and N = 10? More precisely, canwe say that MP(3,10) is nonempty?

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Example7.Forr = 3,N = 12,d= (0,0,−1,. . . ,−1) andR+C = 3,consider ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 x y2 _yz _z2 ₀ ₀ ₀ ₀ ₀ ₀ ₀ 0 0 0 0 0 y2 _yz ₀ _z2 ₀ ₀ ₀ 0 0 0 0 0 −x 0 −z 0 0 0 0 0 0 0 0 0 0 −x y 0 z 0 0 0 0 0 0 0 0 0 0 −x −y 0 0 0 0 0 0 0 0 0 0 0 0 −z 0 0 0 0 0 0 0 0 0 0 0 y −z 0 0 0 0 0 0 0 0 0 0 x 0 0 0 0 0 0 0 0 0 0 0 0 y 0 0 0 0 0 0 0 0 0 0 0 −x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Note thatwe canobtain anexample for r = 3 andN = 4s for everys ≥ 2 byusing the examplesfor

r = 3,N = 8 andtheexampleforr = 3,N = 12 and applyingthechessboardconstructionas manytimes asnecessary. Iftheanswerto question2isnegative,thenonecanask thefollowingquestion:

Question3.Dothereexist anyperiodicityresultsaboutnonemptinessof MP(r,N )?

Anotherobservationwemakeabouttheseexamplesisthattherealwaysexistsasequenceofpermutations

σ1 < σ2 <· · · < σr such that the image of the morphism contains apoint from each Borel orbit

corre-spondingtothetheseσi’sandeachpairofconsecutiveσi’sconsist ofdistincttranspositions.Forexample,

puttingx= 1 andy = 0 toψ inExample2,wegetapointintheBorelorbitcorrespondingtopermutation

σ2= (1,2)(3,4), andputting x= 0 and y = 1,we getσ1= (1,3)(2,4).Henceonecouldask thefollowing question:

Question 4.Given σ in MP(r,N ) does there always exists amorphism ψ :Pr_k−1 → V (d)− L(d) with a sequence permutationsσ1 < σ2<· · · < σr and pointsX1,X2,. . . ,Xr intheimage ofψ such thatσψ = σ

andXi isintheBorelorbitofσi foralli andσi andσi+1 hasnocommontranspositions?

Iftheanswerisaﬃrmativetothisquestionthenonecansaythattheinequalities

n(n + 1) 2 ≤ dim(Bσi)≤ n 2 and dim(Bσi) + )_n 2 * ≤ dim(Bσi+1)

holdand theygivetheinequalityN≥ 2r.

Note thatAlldayand Puppe [3] haverelated results:Ifk, A,r, N ,and M areas inConjecture1,then theyproveN ≥ 2r.Moreover,Avramov,BuchweitzandIyengar[4] veriﬁedthatN ≥ 2r inamoregeneral case.

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