A blowup algebra of hyperplane arrangements

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by  Mehdi Garrousian

A BLOWUP ALGEBRA OF HYPERPLANE ARRANGEMENTS
MEHDIGARROUSIAN,ARONSIMISANDS¸TEFANO.TOHA˘NEANU
ABSTRACT. Itisshown thattheOrlik-Teraoalgebraisgraded isomorphictothespecial fiberoftheideal I
generatedbythe(n−1)-foldproductsofthemembersofacentralarrangementofsizen. Thismomentum
7
iscarriedovertotheReesalgebra(blowup)ofI anditisshownthatthisalgebraisoffiber-typeandCohen-
1
0 Macaulay.ItfollowsbyaresultofSimis-VasconcelosthatthespecialfiberofIisCohen-Macaulay,thusgiving
2 anotherproofofaresultofProudfoot-SpeyerabouttheCohen-MacauleynessoftheOrlik-Teraoalgebra.

n
a
J
2 INTRODUCTION
1
Thecentral themeofthispaperistostudytheidealtheoretic aspects oftheblowupofaprojective space

] along a certain scheme of codimension 2. To be more precise, let A = {ker(ℓ ),...,ker(ℓ )} be an
C 1 n
arrangement ofhyperplanes inPk−1 andconsidertheclosureofthegraphofthefollowingrationalmap
A
Pk−1 99K Pn−1, x 7→ (1/ℓ (x) : ··· : 1/ℓ (x)).
. 1 n
h
t Rewriting the coordinates of the map as forms of the same positive degree in the source Pk−1 =
a
m Proj(k[x1,...,xk]), we are led to consider the corresponding graded k[x1,...,xk]-algebra, namely, the
Reesalgebra oftheidealofk[x ,...,x ]generated bythe(n−1)-foldproducts ofℓ ,...,ℓ .
[ 1 k 1 n
It is our view that bringing into the related combinatorics a limited universe of gadgets and numerical

1
invariants from commutative algebra may be of help, specially regarding the typical operations with ideals
v
0 and algebras. This point of view favors at the outset a second look at the celebrated Orlik-Terao algebra
7 k[1/ℓ ,...,1/ℓ ] which is regarded as a commutative counterpart to the combinatorial Orlik-Solomon al-
1 n
4
gebra. The fact that the former, as observed by some authors, has a model as a finitely generated graded
3
k-subalgebra of a finitely generated purely transcendental extension of the field k, makes it possible to
0
. recoveritasthehomogeneous coordinate ringoftheimageofacertainrational map.
1
This is our departing step to naturally introduce other commutative algebras into the picture. As shown
0
7 inTheorem 2.4,theOrlik-Terao algebra nowbecomes isomorphic, asgraded k-algebra, tothespecial fiber
1 algebra(alsocalledfiberconealgebraorcentralalgebra)oftheidealI generatedbythe(n−1)-foldproducts
:
v ofthemembersofthearrangementA. ThisalgebraisinturndefinedasaresiduealgebraoftheReesalgebra
i
X ofI,soitisonlynatural tolookatthisandrelated constructions. Oneoftheseconstructions takesustothe
symmetricalgebraofI,andhencetothesyzygiesofI. SinceI turnsouttobeaperfectidealofcodimension
r
a 2,itssyzygiesarerathersimpleandallowustofurther understand thesealgebras.
Asasecondresultalongthislineofapproach, weshowthatapresentation idealoftheReesalgebraofI
canbe generated bythe syzygetic relations andthe Orlik-Terao ideal (see Theorem 4.2). Thisproperty has
beencoinedfibertypeproperty intherecentliterature ofcommutativealgebra.
2010MathematicsSubjectClassification. Primary13A30,14N20;Secondary13C14,13D02,13D05.
Keywordsandphrases. Reesalgebra,specialfiberalgebra,Orlik-Teraoalgebra,Cohen-Macaulay.
The second author has been partially supported by a CNPq grant (302298/2014-2) and by a CAPES-PVNS Fellowship
(5742201241/2016); heiscurrentlyholding aoneyear SeniorVisitingProfessorship (2016/08929-7) at theInstitutefor Mathe-
maticalandComputerSciences,UniversityofSa˜oPaulo,Sa˜oCarlos,Brazil.
Garrousian’s Address: Department of Mathematics, University of Western Ontario, London, ON N6A 5B7, Canada, Email:
[email protected].
Simis’Address:DepartamentodeMatema´tica,CentrodeCieˆnciasExatasedaNatureza,UniversidadeFederaldePernambuco,
50740-560,Recife,Pernambuco,Brazil,Email:[email protected].
Tohaneanu’s Address: Department of Mathematics, University of Idaho, Moscow, Idaho 83844-1103, USA, Email: to-
[email protected],Phone:[phone],Fax:[phone].
1
2 MEHDIGARROUSIAN,ARONSIMISANDS¸TEFANO.TOHA˘NEANU
The third main result of this work, as an eventual outcome of these methods, is a proof of the Cohen-
Macaulayproperty oftheReesalgebraofI (seeTheorem4.9).
The typical argument in the proofs is induction on the size or rank of the arrangements. Here we draw
heavilyontheoperationsofdeletionandcontractionofanarrangement. Inparticular,weintroduceavariant
of a multiarrangement that allows repeated linear forms to be tagged with arbitrarily different coefficients.
Then the main breakthrough consists in getting a precise relation between the various ideals or algebras
attached totheoriginalarrangement andthoseattached totheminors.
One of the important facts about the Orlik-Terao algebra is that it is Cohen-Macaulay, as proven by
Proudfoot-Speyer [12]. Using a general result communicated by the second author and Vasconcelos, we
recoverthisresultasaconsequence oftheCohen-Macaulay property oftheReesalgebra.
Thestructure ofthispaperisasfollows. Thefirstsection isanaccount oftheneeded preliminaries from
commutative algebra. The second section expands on highlights of the settled literature about the Orlik-
Teraoidealaswellasatangential discussion ontheso-called non-linear invariants ofouridealssuchasthe
reduction number and analytic spread. The third section focuses on the ideal of (n−1)-fold products and
the associated algebraic constructions. Thelast section is devoted to the statements and proofs ofthe main
theoremswherewedrawvariousresultsfromtheprevious sectionstoestablish thearguments.
1. IDEAL THEORETIC NOTIONS AND BLOWUP ALGEBRAS
TheReesalgebraofanidealI inaringRistheR-algebra
R(I) := Ii.
i≥0
M
This is a standard R-graded algebra with R(I) = R, where multiplication is induced by the internal
0
multiplication rule IrIs ⊂ Ir+s. One can see that there is a graded isomorphism R[It] ≃ R(I), where
R[It] is the homogeneous R-subalgebra of the standard graded polynomial R[t] in one variable over R,
generated bytheelementsat,a ∈ I,ofdegree1.
Quitegenerally,fixingasetofgeneratorsofI determinesasurjectivehomomorphismofR-algebrasfrom
apolynomial ringoverRtoR[It]. Thekernel ofsuch amapiscalled apresentation ideal ofR[It]. Inthis
generality,evenifRisNoetherian(soI isfinitelygenerated)thenotionofapresentationidealisquiteloose.
In this work we deal with a special case in which R = k[x ,...,x ] is a standard graded polynomial
1 k
ring over a field k and I = hg ,...,g i is an ideal generated by forms g ,...,g of the same degree. Let
1 n 1 n
T = R[y ,...,y ] = k[x ,...,x ;y ,...,y ], a standard bigraded k-algebra with degx = (1,0) and
1 n 1 k 1 n i
degy = (0,1). Usingthegivengenerators toobtainanR-algebrahomomorphism
j
ϕ: T = R[y ,...,y ]−→ R[It], y 7→ g t,
1 n i i
yields apresentation ideal I which is bihomogeneous in the bigrading of T. Therefore, R[It] acquires the
corresponding bigrading.
Changingk-linearlyindependent setsofgenerators inthesamedegreeamountstoeffectinganinvertible
k-linearmap,sotheresultingeffectonthecorrespondingpresentationidealisprettymuchundercontrol. For
thisreason,wewillbyabusetalkaboutthepresentationidealofI byfixingaparticularsetofhomogeneous
generatorsofI ofthesamedegree. Occasionally, wemayneedtobringinafewsuperfluousgeneratorsinto
asetofminimalgenerators.
Sincethegivengenerators havethesamedegreetheyspanalinearsystemdefiningarationalmap
Φ :Pk−1 99K Pn−1, (1)
bytheassignment x 7→ (g (x) :··· :g (x)),whensomeg (x) 6= 0.
1 n i
TheidealI isoftencalledthebaseideal(toagreewiththebasescheme)ofΦ. AskingwhenΦisbirational
ontoitsimageisofinterest andwewillbrieflydealwithitaswell. Againnotethatchanging toanother set
of generators in the same degree will not change the linear system thereof, defining the same rational map
uptoacoordinate changeatthetarget.
ABLOWUPALGEBRAOFHYPERPLANEARRANGEMENTS 3
TheReesalgebra brings along other algebras ofinterest. Inthe present setup, one ofthem isthespecial
fiber F(I) := R[It]⊗ R/m ≃ ⊕ Is/mIs,where m = hx ,...,x i ⊂ R. TheKrull dimension ofthe
R s≥0 1 k
specialfiberℓ(I):= dimF(I)iscalledtheanalytic spreadofI.
The analytic spread is a significant notion in the theory of reductions of ideals. An ideal J ⊂ I is said
to be a reduction of I if Ir+1 = JIr for some r. Most notably, this is equivalent to the condition that the
natural inclusion R[Jt] ֒→ R[It] is a finite morphism. The smallest such r is the reduction number r (I)
J
withrespect toJ. Thereduction number ofI istheinfimumofallr (I)forallminimalreductions J ofI;
J
thisnumberisdenotedbyr(I).
Geometrically, the relevance of the special fiber lies in the following result, which we isolate for easy
reference:
Lemma1.1. LetΦbeasin(1)andI itsbaseideal. Thenthehomogeneous coordinate ringoftheimageof
Φisisomorphic tothespecialfiberF(I)asgradedk-algebras.
To see this, note that the Rees algebra defines a biprojective subvariety of Pk−1 × Pn−1, namely the
closure of the graph of Φ. Projecting down to the second coordinate recovers the image of Φ. Atthe level
of coordinate rings this projection corresponds to the inclusion k[I t] = k[g t,...,g t] ⊂ R[It], where
d 1 n
g ,...,g areformsofthedegree d,thisinclusion isasplitk[I t]-module homomorphism withmR[It]as
1 n d
directcomplement. Therefore, onehasanisomorphism ofk-gradedalgebras k[I ] ≃k[I t]≃ F(I).
d d
Asnotedbefore, thepresentation idealofR[It]
I = I ,
(a,b)
(a,b)∈N×N
M
isabihomogeneous idealinthestandardbigradingofT. TwobasicsubidealsofI arehI iandhI i.
(0,−) (−,1)
Thetwocomeinasfollows.
Considerthenatural surjections
T ϕ // R[It]⊗RR/m//44 F(I)
ψ
wherethekerneloftheleftmostmapisthepresentation idealI ofR[It]. Then,wehave
T T k[y ,...,y ]
1 n
F(I) ≃ ≃ ≃ .
kerψ hkerϕ| ,mi hI i
(0,−) (0,−)
Thus,hI iisthehomogeneous defining idealofthespecial fiber(or, asexplained inLemma1.1,ofthe
(0,−)
imageoftherational mapΦ).
AsforthesecondidealhI i,onecanseethatitcoincideswiththeidealofT generatedbythebiforms
(−,1)
s y +···+s y ∈ T, whenever (s ,...,s )is asyzygy of g ,...,g of certain degree in R. Thinking
1 1 n n 1 n 1 n
about the one-sided grading in the y’sthere isno essential harm in denoting this ideal simply byI . Thus,
1
T/I is apresentation ofthe symmetric algebra S(I) ofI. Itobtains anatural surjective mapof R-graded
1
algebras
S(I) ≃ T/I ։ T/I ≃ R(I).
1
Asamatterofcalculation, onecaneasilyshowthatI = I : I∞,thesaturation ofI withrespecttoI.
1 1
Theideal I issaid tobe of linear type provided I = I , i.e., when the above surjecton is injective. Itis
1
saidtobeoffibertypeifI = I +hI i= hI , I i.
1 (0,−) 1 (0,−)
Abasic homological obstruction foranidealtobeoflinear typeistheso-called G condition ofArtin-
∞
Nagata [2], also known as the F condition [7]. A weaker condition is the so-called G condition, for a
1 s
suitable integer s. All these conditions can be stated in terms of the Fitting ideals of the given ideal or,
equivalently, in terms of the various ideals of minors of a syzygy matrix of the ideal. In this work we will
haveachancetousecondition G ,wherek = dimR < ∞. Givenafreepresentation
k
Rm −ϕ→ Rn −→ I → 0
4 MEHDIGARROUSIAN,ARONSIMISANDS¸TEFANO.TOHA˘NEANU
ofanidealI ⊂ R,theG condition forI meansthat
k
ht(I (ϕ)) ≥ n−p+1, for p ≥ n−k+1, (2)
p
whereI (ϕ)denotestheidealgeneratedbythet-minorsofϕ. Notethatnothingisrequiredaboutthevalues
t
ofpstrictlysmallerthann−k+1sinceforsuchvaluesonehasn−p+1 > k = dimR,whichmakesthe
sameboundimpossible.
A useful method to obtain new generators of I from old generators (starting from generators of I ) is
1
via Sylvester forms (see [8, Proposition 2.1]), which has classical roots as the name indicates. It can be
defined in quite generality as follows: let R := k[x ,...,x ], and let T := R[y ,...,y ]as above. Given
1 k 1 n
F ,...,F ∈ I,letJ betheidealofRgenerated byallthecoefficients oftheF –theso-called R-content
1 s i
ideal. SupposeJ = ha ,...,a i,wherea areformsofthesamedegree. Thenwehavethematrixequation:
1 q i
F a
1 1
F a
 .2  = A· .2 ,
. .
. .
   
 Fs   aq 
   
whereAisans×q matrixwithentriesinT.   
If q ≥ s and if the syzygies on F′s are in mT, then the determinant of any s × s minor of A is an
i
element of I. These determinants arecalled Sylvester forms. Themainuse inthis workis toshow that the
Orlik-Teraoidealisgenerated bysuchforms(Proposition 3.5).
The last invariant we wish to comment on is the reduction number r(I). For convenience, we state the
followingresult:
Proposition1.2. Withtheabovenotation,supposethatthespecialfiberF(I)isCohen-Macaulay. Thenthe
reduction numberr(I)ofI coincides withtheCastelnuovo-Mumford regularity reg(F(I))ofF(I).
Proof. By [19, Proposition 1.85], when the special fiber is Cohen-Macaulay, one can read r(I) off the
Hilbertseries. Write
1+h s+h s2+···+h sr
1 2 r
HS(F(I),s) = ,
(1−s)d
withh 6= 0andd = ℓ(I),thedimensionofthefiber(analytic spread). Then,r(I) = r.
r
Since F(I) ≃ S/hI i, where S := k[y ,...,y ], we have that F(I) has a minimal graded S-free
(0,−) 1 n
resolution of length equal to m := hthI i, and reg(F(I)) = α − m, where α is the largest shift in
(0,−)
the minimal graded free resolution, occurring also at the end of this resolution. These last two statements
mentioned herecomefromtheCohen-Macaulayness ofF(I).
The additivity of Hilbert series under short exact sequences of modules, together with the fact that
sv
HS(Su(−v),s) = u givesthatr+m = α = m+reg(F(I)), sor(I) = reg(F(I)). (cid:3)
(1−s)n
2. HYPERPLANE ARRANGEMENTS
Let A = {H ,...,H } ⊂ Pk−1 be a central hyperplane arrangement of size n and rank k. Here
1 n
H = ker(ℓ ),i = 1,...,n, where each ℓ is a linear form in R := k[x ,...,x ] and hℓ ,...,ℓ i =
i i i 1 k 1 n
m := hx ,...,x i. Fromthealgebraic viewpoint, thereisanaturalemphasis onthelinearformsℓ andthe
1 k i
associated idealtheoretic notions.
Deletion and contraction are useful operations upon A. Fixing an index 1 ≤ i ≤ n, one introduces two
newminorarrangements:
A′ = A\{H }(deletion), A′′ = A′∩H := {H ∩H |1 ≤ j ≤ n,j 6= i}(contraction).
i i j i
Clearly, A′ isasubarrangement ofAof sizen−1and rank atmostk, whileA′′ isan arrangement of size
≤ n−1andrankk−1.
ABLOWUPALGEBRAOFHYPERPLANEARRANGEMENTS 5
Contraction comes with a natural multiplicity given by counting the number of hyperplanes of A′ that
givethesameintersection. Amodifiedversionofsuchanotionwillbethoroughly usedinthiswork.
Thefollowingnotionwillplayasubstantial roleinsomeinductivearguments throughout thepaper: ℓ is
i
calledacoloopiftherankofthedeletionA′withrespecttoℓ isk−1,ie. dropsbyone. Thissimplymeans
i
that H isalineratherthantheorigininAk. Otherwise,wesaythatℓ isanon-coloop.
j6=i j i
2.1.TThe Orlik-Terao algebra. One of our motivations is to clarify the connections between the Rees
algebra andtheOrlik-Teraoalgebrawhichisanimportantobjectinthetheoryofhyperplane arrangements.
Westatethedefinitionandreviewsomeofitsbasicproperties below.
Let A ⊂ Pk−1 be a hyperplane arrangement as above. Suppose c ℓ +··· + c ℓ = 0 is a linear
i1 i1 im im
dependency among m of the linear forms defining A, denoted D. Consider the following homogeneous
polynomial inS := k[y ,...,y ]:
1 n
m m
∂D := c y . (3)
ij ik
j=1 j6=k=1
X Y
Notethatdeg(∂D) = m−1.
TheOrlik-TeraoalgebraofAisthestandard gradedk-algebra
OT(A):= S/∂(A),
where∂(A)istheidealofSgeneratedby{∂D|D adependency ofA},with∂Dasin(3)–calledtheOrlik-
Terao ideal. This algebra was introduced in [10] as a commutative analog of the classical combinatorial
Orlik-Solomonalgebra, inordertoansweraquestion ofAomoto.
Thefollowingremarkstatesafewimportant properties ofthisalgebra.
Remark2.1.
i. Recalling that a circuit is a minimally dependent set, one has that ∂(A) is generated by ∂C, where C
runs over the circuits of A ([11]). In addition, these generators form an universal Gro¨bner basis for
∂(A)([12]).
ii. OT(A)isCohen-Macaulay ([12]).
iii. OT(A) ≃ k[1/ℓ ,...,1/ℓ ], a k-dimensional k-subalgebra of the field of fractions k(x ,...,x )
1 n 1 k
([15,17]). Thecorresponding projectivevarietyiscalledthereciprocalplaneanditisdenotedbyL−1.
A
iv. Although the Orlik-Terao algebra is sensitive to the linear forms defining A, its Hilbert series only
depends ontheunderlying combinatorics ([17]). Let
π(A,s) = µ (F)(−s)r(F).
A
F∈L(A)
X
be the Poincare´ polynomial where µ denotes the Mo¨bius function, r is the rank function and F runs
A
overtheflatsofA. Then,wehave
s
HS(OT(A),s) = π(A, ).
1−s
See[10]fordetailsand[17]and[3]forproofsoftheabovestatement.
2.2. Idealsofproductsfromarrangements. LetA = {ℓ ,...,ℓ }denoteacentral arrangement inR :=
1 n
k[x ,...,x ],n ≥ k. Denoting[n] := {1,...,n},ifS ⊂ [n],thenwesetℓ := ℓ ,ℓ := 1. Alsoset
1 k S i∈S i ∅
Sc := [n]\S.
LetS := {S ,...,S },whereS ⊆ [n]aresubsetsofthesamesizee. WeareQinterestedinstudyingthe
1 m j
Reesalgebras ofidealsoftheform
IS := hℓS1,...,ℓSmi ⊂ R. (4)
Example2.2. (i)(TheBoolean case) Letn = k and ℓi = xi,i = 1,...,k. Thenthe ideal IS ismonomial
for any S. In the simplest case where e = n−1, it is the ideal of the partial derivatives of the monomial
x ···x –alsothebaseidealoftheclassicalMo¨biusinvolution. Fore= 2theidealbecomestheedgeideal
1 k
6 MEHDIGARROUSIAN,ARONSIMISANDS¸TEFANO.TOHA˘NEANU
ofasimplegraphwithkvertices. Ingeneral,itgivesasubidealoftheidealofpathsofagivenlengthonthe
completegraphand,assuch,ithasaknowncombinatorial nature.
(ii)((n−1)-fold products) Hereone takes S := [n]\{1},...,S := [n]\{n}. Wewilldesignate the
1 n
correspondingidealbyI (A). Thiscasewillbethemainconcernofthepaperandwillbefullyexamined
n−1
inthefollowingsections.
(iii)(a-foldproducts)Thisisanaturalextensionof(ii),whereI (A)istheidealgenerated byalldistinct
a
a-products of the linear forms defining A. The commutative algebraic properties of these ideals connect
strongly toproperties ofthe linear code built onthedefining linear forms (see[1]). Inaddition, thedimen-
sionsofthevectorspacesgenerated bya-foldproducts giveanewinterpretation totheTuttepolynomial of
thematroidofA(see[3]).
Wecannaturally introduce thereciprocal planealgebra
1 1
L−1 := k ,..., (5)
S
(cid:20)ℓS1c ℓSmc (cid:21)
asageneralized versionofthenotionmentionedinRemark2.1(iii).
Proposition 2.3. Intheabovesetupthereisagradedisomorphism ofk-algebras
1 1
k[ℓ ,...,ℓ ] ≃ k ,..., .
S1 Sm (cid:20)ℓS1c ℓSmc (cid:21)
Proof. Considerbothalgebrasashomogeneousk-subalgebras ofthehomogeneoustotalquotientringofthe
standard polynomial ring R,generated indegrees eand−(d−e), respectively. Thenmultiplication bythe
totalproduct ℓ givestherequired isomorphism:
[d]
k 1 ,..., 1 −·ℓ→[d] k[ℓ ,...,ℓ ]. (cid:3)
(cid:20)ℓS1c ℓSmc (cid:21) S1 Sm
Aneatconsequence isthefollowingresult:
Theorem 2.4. Let A denote a central arrangement of size n, let S := {S ,...,S } be a collection of
1 m
subsets of[n]ofthesamesizeandletIS beasin(4). Thenthereciprocal plane algebraL−S1 isisomorphic
to the special fiber of the ideal IS as graded k-algebras. In particular, the Orlik-Terao algebra OT(A) is
gradedisomorphic tothespecialfiberF(I)oftheidealI = I (A)of(n−1)-fold productsofA.
n−1
Proof. Itfollowsimmediately fromProposition 2.3andLemma1.1. (cid:3)
Remark2.5. InthecaseoftheOrlik-Teraoalgebra,theaboveresultgivesananswertothethirdquestionat
theendof[14]. Namely,letk ≥ 3andconsidertherationalmapΦasin(1). ThenTheorem2.4saysthatthe
projectionofthegraphofΦontothesecondfactorcoincideswiththereciprocalplaneL−1(seeRemark2.1
A
(iii)). Inaddition, theidealI := I (A)hasasimilarprimarydecomposition asobtained in[14,Lemmas
n−1
3.1and3.2],forarbitrary k ≥ 3. By[1,Proposition 2.2],onegets
I = I(Y)µA(Y),
Y∈\L2(A)
Theorem2.4contributesadditionalinformationoncertainnumericalinvariantsandpropertiesinthestrict
realmofcommutativealgebraandalgebraic geometry.
Corollary 2.6. Let I := I (A) denote the ideal generated by the (n−1)-fold products coming from a
n−1
centralarrangement ofsizenandrankk. Onehas:
(a) ThespecialfiberF(I)ofI isCohen-Macaulay.
(b) Theanalyticspreadisℓ(I) = k.
(c) ThemapΦisbirational ontoitsimage.
(d) Thereductionnumberisr(I) = k−1.
ABLOWUPALGEBRAOFHYPERPLANEARRANGEMENTS 7
Proof. (a)ItfollowsfromfromTheorem2.4viaRemark2.1(ii).
(b)Itfollowsbythesametokenfrom2.1(iii).
(c) This follows from and [5, Theorem 3.2] since the ideal I is linearly presented (see proof of Lemma
3.1),andℓ(I) = k,maximumpossible.
(d)FollowsfromPart(a),Proposition 1.2,and[14,Theorem3.7].
It may be interesting to remark that, because of this value, in particular the Orlik-Terao algebra is the
homogeneous coordinate ring of a variety of minimal degree if and only if k = 2, in which case it is the
homogeneous coordinate ringoftherationalnormalcurve. (cid:3)
3. IDEALS OF(n−1)-FOLD PRODUCTS AND THEIR BLOWUP ALGEBRAS
AsmentionedinExample2.2,aspecialcaseoftheidealIS,extendingthecaseoftheidealgeneratedby
the(n−1)-foldproducts, isobtained byfixinga ∈ {1,...,n}andconsidering thecollection ofallsubsets
of[n]ofcardinality a. Thenthecorresponding idealis
I (A):= hℓ ···ℓ |1 ≤ i < ··· < i ≤ ni⊂ R
a i1 ia 1 a
and is called the ideal generated by the a-fold products of linear forms of A. The projective schemes
definedbytheseidealsareknownasgeneralizedstarconfigurationschemes. Unfortunately, onlyfewthings
are known about these ideals: if d is the minimum distance of the linear code built from the linear forms
definingAandif1 ≤ a ≤ d,thenI (A)= ma (cf. [18,Theorem3.1]);andthecasewhena = nistrivial.
a
In the case where a = n − 1, some immediate properties are known already, yet the more difficult
questions in regard to the blowup and related algebras have not been studied before. These facets, to be
throughly examinedinthesubsequent sections, isourmainendeavor inthiswork.
Henceforth, wewillbe working withthe following data: Ais anarrangement with n ≥ k and for every
1 ≤ i≤ n,weconsider the(n−1)-foldproducts ofthenlinearformsdefiningthehyperplanes ofA
f := ℓ ···ℓˆ ···ℓ ∈ R,
i 1 i n
andwrite
I := I (A):= hf ,...,f i.
n−1 1 n
LetT = k[x ,...,x ,y ,...,y ] = R[y ,...,y ]asbeforeanddenotebyI(A,n−1) ⊂ T thepresenta-
1 k 1 n 1 n
tionidealoftheReesalgebra R[It]corresponding tothegenerators f ,...,f .
1 n
3.1. Thesymmetric algebra. LetI (A,n−1) ⊂ T stand forthesubideal ofI(A,n−1)presenting the
1
symmetricalgebraS(I)ofI = I (A).
n−1
Lemma3.1. Withtheabovenotation, onehas:
(a) TheidealI = I (A)isaperfectidealofcodimension 2.
n−1
(b) I (A,n−1) = hℓ y −ℓ y |1 ≤ i≤ n−1i.
1 i i i+1 i+1
(c) I (A,n−1)is anideal of codimension k;in particular, it isacomplete intersection ifand only if
1
n= k.
Proof. (a) This is well-known, but we give the argument for completeness. Clearly, I has codimension 2.
Thefollowingreduced Koszullikerelations aresyzygies ofI: ℓ y −ℓ y ,1 ≤ i ≤ n−1. Theyalone
i i i+1 i+1
formthefollowingmatrixofsyzygies ofI:
ℓ
1
−ℓ ℓ
2 2
ϕ =  −ℓ ... .
3
 ... ℓ 
 n−1 
 −ℓ 
 n 
 
8 MEHDIGARROUSIAN,ARONSIMISANDS¸TEFANO.TOHA˘NEANU
Since the rank of this matrix is n−1, it is indeed a full syzygy matrix of I; in particular, I has linear
resolution
0 −→ R(−n)n−1 −ϕ→ R(−(n−1))n −→ I −→ 0.
(b)Thisisanexpression ofthedetailsof(a).
(c) Clearly, I (A,n − 1) ⊂ mT, hence its codimension is at most k. Assuming, as we may, that
1
{ℓ ,...,ℓ } is k-linearly independent, we contend that the elements s := {ℓ y −ℓ y ,1 ≤ i ≤ k},
1 k i i i+1 i+1
form a regular sequence. To see this, we first apply a k-linear automorphism of R to assume that ℓ = x ,
i i
for 1 ≤ i ≤ k – this will not affect the basic ideal theoretic invariants associated to I. Then note
that in the set of generators of I (A,n − 1) the elements of s can be replaced by the following ones:
1
{x y −ℓ y ,1 ≤ i≤ k}. Clearly,thisisaregularsequence –forexample,becausehx y ,1 ≤ i≤ ki
i i k+1 k+1 i i
istheinitialidealoftheidealgenerated bythissequence, intherevlexorder. (cid:3)
There are two basic ideals that play a distinguished role at the outset. In order to capture both in one
single blow, we consider the Jacobian matrix of the generators of I (A,n − 1) given in Lemma 3.1 (b).
1
Its transpose turns out to be the stack of two matrices, the first is the Jacobian matrix with respect to the
variablesy ,...,y –whichcoincideswiththesyzygymatrixφofI asdescribedintheproofofLemma3.1
1 n
(a) – while the second is the Jacobian matrix B = B(φ) with respect to the variables x ,...,x – the so-
1 k
called Jacobian dual matrix of [16]. The offspring are the respective ideals of maximal minors of these
stacked matrices, the first retrieves I, while thesecond gives anideal I (B) ⊂ S = k[y ,...,y ]that will
k 1 n
play a significant role below (see also Proposition 4.1) as a first crude approximation to the Orlik-Terao
ideal.
Proposition3.2. LetS(I) ≃ T/I (A,n−1)standforthesymmetricalgebraoftheidealI of(n−1)-fold
1
products. Then:
(i) depth(S(I)) ≤ k+1.
(ii) AsanidealinT,everyminimalprimeofS(I)iseithermT,theReesidealI(A,n−1)orelsehas
the form (ℓ ,...,ℓ ,y ,...,y ), where 2 ≤ s ≤ k −1,t ≥ 1, {i ,...,i }∩{j ,...,j } = ∅,
i1 is j1 jt 1 s 1 t
andℓ ,...,ℓ arek-linearlyindependent.
i1 is
(iii) Theprimarycomponents relativetotheminimalprimesm = (x)T andI(A,n−1)areradical;in
addition, withtheexception ofmT,everyminimalprimeofS(I)containstheidealI (B).
k
Proof. (i) Since I(A,n−1) is a prime ideal which is asaturation of I (A,n−1) then itis an associated
1
primeofS(I). Therefore, depth(S(I)) ≤ dimR(I) = k+1.
(ii)SinceI(A,n−1)isasaturation ofI (A,n−1)byI,onehas I(A,n−1)It ⊂ I (A,n−1), for
1 1
some t ≥ 1. This implies that any (minimal) prime of S(I) in T contains either I or I(A,n−1). By the
proof of (i), I(A,n−1) is an associated prime of S(I), hence it must be a minimal prime thereof since a
minimalprimeofS(I)properly contained initwouldhavetocontainI,whichisabsurd.
Now,supposeP ⊂ T isaminimalprimeofS(I)containingI. OneknowsbyLemma3.1thatm = (x)T
isaminimalprimeofS(I). Therefore,weassumethatmT 6⊂ P. SinceanyminimalprimeofI isacomplete
intersection of two distinct linear forms of A then P contains at least two, and at most k − 1, linearly
independent linear forms of A. On the other hand, since I (A,n−1) ⊂ P, looking at the generators of
1
I (A,n−1)asinLemma3.1(b),byadominoeffectprinciplewefinallyreachthedesiredformatforP as
1
stated.
(iii)Withthenotation priortothestatementoftheproposition, weclaimthefollowingequality:
I (A,n−1) : I (B)∞ = mT
1 k
Itsufficestoshow forthefirstquotient asmT isaprimeideal. Theinclusion mI (B) ⊂ I (A,n−1)isa
k 1
consequence oftheCramerrule. Thereverseinclusion isobvious becauseI (A,n−1) ⊂ mT impliesthat
1
I (A,n−1) : I (B) ⊂ mT : I (B) = mT,asmT isaprimeideal.
1 k k
Notethat,asaverycrudeconsequence, onehasI (B) ⊂ I(A,n−1).
k
ABLOWUPALGEBRAOFHYPERPLANEARRANGEMENTS 9
Now,letP(mT)denotetheprimarycomponent ofmT inI (A,n−1). Then
1
mT = I (A,n−1) : I (B)∞ ⊂ P(mT) :I (B)∞ = P(mT).
1 k k
The same argument goes through for the primary component of I(A,n− 1) using the ideal I instead of
I (B).
k
Toseethelaststatementoftheitem,letP denotetheprimarycomponentofoneoftheremainingminimal
primes P of S(I). Since P : I (B)∞ is P-primary and m 6⊂ P, then by the same token we get that
k
I (B)⊂ P. (cid:3)
k
Remark3.3. (a)Itwillbeshowninthelastsection thattheestimatein(i)isactuallyanequality.
As a consequence, every associated prime of S(I) viewed in T has codimension at most n − 1. This
will give a much better grip on the minimal primes of the form hℓ ,...,ℓ ,y ,...,y i. Namely, one
i1 is j1 jt
musthaveinaddition thats+t ≤ n−1and, moreover, due tothedominoeffect principle, one musthave
s = k−1,hencet ≤ n−k.
(b)Weconjecture thatS(I)isreduced.
Theproperty(R )ofSerre’siseasilyverifiedduetotheformatoftheJacobianmatrixasexplainedbefore
0
theaboveproposition. Theproblemis,ofcourse,theproperty(S ),theknownobstruction fortheexistence
1
ofembedded associated primes. Thecasewheren = k+1,iseasilydetermined. Heretheminimalprimes
areseentobem,hx ,...,x ,y iandtheReesidealhI (A,k),∂i,where∂ istherelationcorresponding
1 k−1 k 1
totheuniquecircuit. Acalculation willshowthatthethreeprimesintersect inI (A,k). Asaside,thisfact
1
aloneimpliesthatthemaximalregularsequenceintheproofofLemma3.1(c)generatesaradicalideal. For
n ≥ k +2 the calculation becomes sort of formidable, but we will prove later on that the Rees ideal is of
fibertype.
(c)TheweakerquestionastowhethertheminimalcomponentofS(I)isradical seemspliable.
Iftheconjectural statement inRemark3.3(b)istruethen, foranylinearformℓ = ℓ thefollowingbasic
i
formulaholds
I (A,n−1) : ℓ = I(A,n−1)∩ P ,
1
!
ℓ∈/P
\
whereP denotes aminimalprimeotherthatmT andI(A,n−1),asdescribed inproposition 3.2(i). Thus
onewouldrecoversectorsoftheOrlik-Teraogenerators insidethiscolonideal.
Fortunately, this latter virtual consequence holds true and has a direct simple proof. Forconvenience of
lateruse,westateitexplicitly. Let∂(A| )denotethesubidealof∂(A)generatedbyallpolynomialrelations
ℓ
∂ corresponding tominimaldependencies (circuits)involving thelinearformℓ ∈ A.
Lemma3.4. ∂(A| ) ⊂ I (A,n−1) :ℓ.
ℓ 1
Proof. Say,ℓ = ℓ . LetD : a ℓ +a ℓ +···+a ℓ = 0beaminimaldependency involving ℓ ,forsome
1 1 1 2 2 s s 1
3 ≤ s ≤n. Inparticular, a 6= 0,i = 1,...,s. Thecorresponding generator of∂(A| )is
i ℓ1
∂D := a y y ···y +a y y ···y +···+a y y ···y .
1 2 3 s 2 1 3 s s 1 2 s−1
Thefollowingcalculation isstraightforward.
10 MEHDIGARROUSIAN,ARONSIMISANDS¸TEFANO.TOHA˘NEANU
ℓ ∂D = a ℓ y y ···y +(ℓ y −ℓ y )(a y ···y +···+a y ···y )
1 1 1 2 3 s 1 1 2 2 2 3 s s 2 s−1
+ ℓ y (a y ···y +···+a y ···y )
2 2 2 3 s s 2 s−1
= (a ℓ +a ℓ )y y ···y +(ℓ y −ℓ y )(a y ···y +···+a y ···y )
1 1 2 2 2 3 s 1 1 2 2 2 3 s s 2 s−1
+ ℓ y (a y y ···y +···+a y y ···y )
2 2 3 2 4 s s 2 3 s−1
= (−a ℓ −···−a ℓ )y y ···y +(ℓ y −ℓ y )(a y ···y +···+a y ···y )
3 3 s s 2 3 s 1 1 2 2 2 3 s s 2 s−1
+ ℓ y2(a y ···y +···+a y ···y )
2 2 3 4 s s 3 s−1
= (ℓ y −ℓ y )(a y ···y +···+a y ···y )+y (ℓ y −ℓ y )a y ···y
1 1 2 2 2 3 s s 2 s−1 2 2 2 3 3 3 4 s
+ ···+y (ℓ y −ℓ y )a y ···y
2 2 2 s s s 3 s−1
= a y ···y (ℓ y −ℓ y )+a y y ···y (ℓ y −ℓ y )+···+a y ···y (ℓ y −ℓ y ).
2 3 s 1 1 2 2 3 2 4 s 1 1 3 3 s 2 s−1 1 1 s s
Hencetheresult. (cid:3)
3.2. Sylvester forms. The Orlik-Terao ideal ∂(A) has an internal structure of classical flavor, in terms of
Sylvesterforms.
Proposition 3.5. Thegenerators ∂(A)of the Orlik-Terao ideal are Sylvester formsobtained from thegen-
eratorsofthepresentation idealI (A,n−1)ofthesymmetricalgebraofI.
1
Proof. Let D be a dependency c ℓ +···+c ℓ = 0 with all coefficients c 6= 0. Let f = n ℓ .
i1 i1 im im ij i=1 i
EvaluatingtheOrlik-Teraoelement∂D ontheproductswehave
Q
m fm−1 m fm−1 fm−2
∂D(f ,...,f ) = c = c ℓ = (c ℓ +···+c ℓ ) = 0.
1 n ijΠm ℓ ijΠm ℓ ij ℓ ···ℓ i1 i1 im im
j=1 j6=k=1 ik j=1 k=1 ik i1 im
X X
Therefore,∂D ∈ I(A,n−1),andsince∂D ∈ S := k[y ,...,y ],then∂D ∈ hI(A,n−1) i.
1 n (0,−)
Forthesecondpart,suppose thattheminimalgenerators ofI (A,n−1)are
1
∆ := ℓ y −ℓ y ,∆ := ℓ y −ℓ y ,...,∆ := ℓ y −ℓ y .
1 1 1 2 2 2 2 2 3 3 n−1 n−1 n−1 n n
Withoutlossofgeneralitysupposeℓ = c ℓ +···+c ℓ issomearbitrarydependencyD. Wehave
j 1 1 j−1 j−1
y −y 0 ··· 0 0
∆ 1 2 ℓ
1 0 y −y ··· 0 0 1
∆  2 3  ℓ
2 . . . . . 2
 ..  = .. .. .. .. .. · .. .
.   .
 ∆j−1   −c01yj −c02yj −c03yj ······ −cyjj−−22yj yj−1−−yjc−j1−1yj   ℓj−1 
     
 
Thedeterminant ofthe(j −1)×(j −1)matrixweseeaboveis±∂D. (cid:3)
3.3. Alemmaondeletion. Inthisandthenextpartswebuildonthemaintoolofaninductiveprocedure.
LetA′ = A\{ℓ },anddenoten′ := |A′|= n−1. Wewouldliketoinvestigatetherelationshipbetween
1
theReesidealI(A′,n′−1)ofIn′−1(A′)andtheReesidealI(A,n−1)ofIn−1(A),bothdefinedinterms
ofthenaturally givengenerators.
Towit,wewilldenotethegenerators ofIn′−1(A′)as
f := ℓ ,...,f := ℓ .
12 [n]\{1,2} 1n [n]\{1,n}
Onecanmovebetweenthetwoidealsinasimplemanner,whichiseasytoverify:
In−1(A) :ℓ1 = In′−1(A′).