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Advances
in
Applied
Mathematics
www.elsevier.com/locate/yaama
CW-complex
Nagata
idealizations
✩Armando Capassoa,Pietro De Poib, Giovanna Ilardic,∗
a
ScuolaPolitecnicaedelleScienzediBase, UniversitàdegliStudidiNapoli “FedericoII”,corsoProtopisaniNicolangelo70,Napoli,C.A.P.80146,Italy bDipartimentodiScienzeMatematiche,InformaticheeFisiche,Universitàdegli StudidiUdine,viadelleScienze206, Udine,C.A.P.33100,Italy
cDipartimentodiMatematicaedApplicazioni“R.Caccioppoli”,Universitàdegli StudidiNapoli“FedericoII”,viaCintia21,Napoli,C.A.P.80126,Italy
a r t i c l e i n f o a b s t r a c t
Articlehistory:
Received23June2020
Receivedinrevisedform25June 2020
Accepted25June2020 Availableonlinexxxx
MSC:
primary13A30,05E40
secondary57Q05,13D40,13A02, 13E10
Keywords:
Lefschetzproperties ArtinianGorensteinalgebra Nagataidealization CW-complex
Weintroduce aconstructionwhichallowsusto identifythe elements of the skeletons of a CW-complex P (m) and the monomialsinm variables.Fromthis,weinferthatthereisa bijectionbetweenfiniteCW-subcomplexesofP (m),whichare quotientsoffinitesimplicialcomplexes,andcertainbigraded standardArtinianGorensteinalgebras,generalizingprevious constructionsofFaridiandourselves.
Weapplythisto ageneralization ofNagataidealizationfor levelalgebras.Thesealgebras arestandard gradedArtinian algebras whose Macaulay dual generator is given explicitly as a bigraded polynomial of bidegree (1,d). We consider thealgebra associated to polynomials of thesame bidegree (d1,d2).
©2020TheAuthor(s).PublishedbyElsevierInc.Thisisan openaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).
✩ P.D.P.&G.I.aremembersofINdAM- GNSAGAandP.D.P.issupportedbyPRIN2017“Advancesin
ModuliTheoryandBirationalClassification”.
* Correspondingauthor.
E-mailaddresses:[email protected](A. Capasso),[email protected](P. De Poi),
[email protected](G. Ilardi).
https://doi.org/10.1016/j.aam.2020.102079
0196-8858/©2020TheAuthor(s). PublishedbyElsevierInc. ThisisanopenaccessarticleundertheCC BYlicense(http://creativecommons.org/licenses/by/4.0/).
0. Introduction LetX = V (f )⊂ PN
K beahypersurface,wheretheunderlyingfieldK hascharacteristic
0;theHessian determinantoff (which wecall theHessian off ortheHessianofX)is thedeterminant oftheHessian matrixof f .
Hypersurface with vanishing Hessian were studied for the first time in 1851 by O. Hesse; hewrote two papers([10,11]) accordingto which these hypersurfacesshouldbe necessarilycones.In1876GordanandNoether([7])provedthatHesse’sclaimistruefor N ≤ 3, anditisfalseforN ≥ 4.TheyandFranchettaclassifiedallthecounterexamples toHesse’sclaiminP4(see [7,3,4,6]).In1900,Perazzoclassifiedcubichypersurfaceswith
vanishing Hessianfor N ≤ 6 ([14]). This work wasstudied and generalizedin [5], and theproblem isstillopeninspacesofhigherdimension.
Hessians of higher degreewere introduced in [13] and used to controlthe so called Strong LefschetzProperties (forshort,SLP).TheLefschetzpropertieshaveattracteda great attentioninthe last years.Thebasic papersof the algebraic theory ofLefschetz properties were the original ones of Stanley [15–17] and the book of Watanabe and others [8].
AnalgebraictoolthatoccursfrequentlyinthesepapersistheNagataIdealization:it is atool toconvertany moduleM overa(commutative)ring(withunit)R toanideal ofanotherringRM.Thestartingpointistheisomorphismbetweentheidealizationof anidealI = (g0,. . . ,gn) ofK[u1,. . . ,um] anditslevelalgebrasee [8,Definition2.72].In
thisway,thenewringisaStandardGradedArtinianGorensteinAlgebra (SGAGalgebra, forshort). Anexplicitformula fortheMacaulaygeneratorf is:
f = x0g0+· · · + xngn∈ K[x0, . . . , xn, u1, . . . , um](1,d).
A generalizationofthisconstructionistoconsider polynomialsoftheform: f = xd0g0+· · · + xdngn ∈ K[x0, . . . , xn, u1, . . . , um](d,d+1);
thesearecalledNagatapolynomialsofdegreed.TheLefschetzpropertiesfortherelevant associated algebrasA,thegeometryofNagatahypersurfacesof degreed,theinteraction betweenthecombinatoricsoff andthestructureofA werestudiedin [1],wherethegi’s
are squarefreemonomials,usingasimplicialcomplexassociatedto f .
In this paper we use the CW-complexes, to study Nagata polynomials of bidegree (d1,d2).We study theHilbertvectorand we giveacomplete descriptionofthe idealI
foreverycase,alsoifthegi’sarenotsquarefreemonomials.
The geometry of the Nagata hypersurface is very similar to the geometry of the hypersurfaceswithvanishing Hessian.
Moreprecisely,weintroduceanewConstruction3.10whichallowsustoidentifyeach (monic) monomial of degree d in m variables with an element of the (d− 1)-skeleton of aCW-complex thatwe call P (m).This CW-complex is constructedby generalizing
the construction introduced in[2] which associates to a(monic) square-free monomial inm variablesof degreed a unique(d− 1)-cellof thesimplexofdimensionm− 1,and vice versa. We consider an h-power uh
i as a product of h linear forms: ˜u1· · · ˜uh; this
correspondstoa(h− 1)-simplex,andweidentifyalltheδ-faces,withδ < h− 1,ofthis simplexto just oneδ-face, recursively,starting from δ = 0 to δ = h− 2: forδ = 0 we identify all the points to one, then if δ = 1 we obtain a bouquet of h-circles, and we identify allthese circles, and so on. Generalizing this construction to a general monic monomialandattachingthecorrespondingCW-complexesalongthecommonskeletons, weobtainP (m).
Thepaperisorganizedasfollows:inSection1werecallsomegeneralitiesaboutgraded ArtinianGorensteinAlgebrasandLefschetzProperties,withtheirconnectionswiththe vanishingsofhigherorderHessians.InSection2werecallwhattheNagataidealization is,whatweintendforahigherNagataidealizationandweshowitsconnectionwiththe Lefschetz Properties for bihomogeneous polynomials.Section3 isthe core ofthis arti-cle.After recalling generalities about bigradedalgebras and thetopological definitions thatweneed, we givetheconstruction oftheCW-complex P (m);then, weapply it to the Nagata polynomials (Definition 2.5) inTheorems 3.16 and 3.18, which give Theo-rem3.16aprecisedescriptionoftheArtinianGorensteinAlgebraassociatedtoaNagata polynomialand Theorem 3.18the generatorsof the annihilatorof the polynomial.We show that from these theorems a generalization of the principal results of [1] follows: Corollaries3.17and3.19.
WethinkthatthestudyoftheNagatahypersurfacescanbe—amongotherthings—a usefultoolfortheclassificationofthehypersurfaceswithvanishing HessianinPn.
Notations.Inthisthepaperwefixthefollowing notationsandassumptions: • K isafieldofcharacteristic 0.
• R := K[x0,. . . ,xn] willalwaysbetheringofpolynomialsinn+1 variablesx0,. . . ,xn.
• Q:= K[X0,. . . ,Xn] will bethe ringofdifferentialoperators ofR,where Xi= ∂ ∂xi
. • Thesubscript ofagradedK-algebra willindicatethepartof thatdegree;Rd isthe
K-vector space of the homogeneous polynomials of degreed, and Qδ the K-vector
spaceofthehomogeneousdifferentialoperators oforderδ. 1. GradedArtinianGorensteinalgebrasandLefschetzproperties 1.1. Graded Artinian Gorensteinalgebrasare Poincaré algebras
Definition 1.1. Let I be a homogeneous ideal of R such that A = R/I =
d
i=0
Ai is a
graded ArtinianK-algebra, where Ad = 0. The integerd isthe socle degree of A. The
Hilb(A)= (1,h1,. . . ,hd) is called Hilbert vectorof A. Since I1= 0, thenh1 = n+ 1 is
called codimensionof A.
Wealsorecallthefollowing definitions.
Definition1.2.AgradedArtinianK-algebraA=i=0d AiisaPoincaréalgebra if·: Ai× Ad−i→ Ad isaperfect pairing fori∈ {0,. . . ,d}.
Definition1.3.AgradedArtinianK-algebraA isGorenstein if(andonlyif)dimKAd= 1
and itisaPoincaréalgebra.
Remark 1.4. The Hilbert vector of a Poincaré algebra A is symmetric with respect to hd
2
,thatisHilb(A)= (1,h1,h2,. . . ,h2,h1,1).
1.2. GradedArtinian Gorenstein quotientalgebrasofQ
Foranyd≥ δ ≥ 0 thereexistsanaturalK-bilinearmapB : Rd× Qδ → Rd−δ defined
bydifferentiation
B(f, α) = α(f )
Definition1.5. LetI =f1,. . . ,f —wheref1,. . . ,f areformsinR—beafinite
dimen-sionalK-vector subspaceofR.Theannihilatorof I in Q is thefollowing homogeneous ideal
Ann(I) :={α ∈ Q | ∀f ∈ I, α(f) = 0}.
Inparticular,ifI isgeneratedbyahomogeneouselementf ,wewriteAnn(I)= Ann(f ). LetA= Q/Ann(f ),wheref ishomogeneous.ByconstructionA isastandardgraded ArtinianK-algebra;moreoverA isGorenstein.
Theorem 1.6 ([12], §60ff, [13], theorem 2.1 ).Let I be ahomogeneous ideal of Q such thatA= Q/I isastandardArtiniangradedK-algebra.ThenA isGorensteinifandonly if thereexistd≥ 1 and f ∈ Rd suchthat A∼= Q/Ann(f ).
Remark1.7.Usingthenotationas above,A iscalledtheSGAGK-algebraassociatedto f .Thesocledegreed ofA isthedegreeoff andthecodimensionisn+ 1,sinceI1= 0.
1.3. LefschetzpropertiesandtheHessiancriterion
LetA=
d
i=0
Definition1.8.Ifthere existsanL∈ A1 suchthat:
(1) Themultiplication map·L: Ai→ Ai+1 isofmaximalrankforalli,then A hasthe WeakLefschetz Property (WLP,forshort);
(2) The multiplicationmap ·Lk: A
i → Ai+k isofmaximal rankforalli andk, thenA
hastheStrongLefschetz Property (SLP,forshort);
Definition1.9.LetA betheSGAGK-algebra associatedto anelementf ∈ Rd, andlet Bk={αj ∈ Ak | j ∈ {1,. . . ,σk}} beanorderedK-basisofAk.Thek-th Hessianmatrix
off withrespectto Bk is
Hesskf = (αiαj(f ))σi,j=1k .
Thek-thHessian off withrespect toBk is
hesskf = det
Hesskf
.
Theorem 1.10([18] Theorem 4). Anelement L = a0X0+· · · + anXn ∈ A1 is a strong
LefschetzelementofA if andonlyifhesskf(a0,. . . ,an)= 0 forallk∈
0, . . . , d 2 .In particular,if forsomek onehas hesskf = 0,thenA does nothave SLP.
2. HigherorderNagataidealization 2.1. Nagataidealizations
Definition 2.1.Let A be a ring and let M be an A-module. The Nagata idealization A M of M istheringwithunderlyingset A× M andoperationsdefinedasfollows:
(r, m) + (s, n) = (r + s, m + n), (r, m)· (s, n) = (rs, sm + rn). 2.1.1. BigradedArtinian Gorenstein algebras
LetA=
d
i=0
Ai be aSGAGK-algebra,itis bigraded if:
Ad= A(d1,d2)∼= K, Ai= i
h=0
A(i,h−i) for i∈ {0, . . . , d − 1},
sinceA isaGorensteinring,andthepair(d1,d2) issaidthesoclebidegreeof A.Inthis
casewecallA anSBAG algebra.
Remark 2.2.By Definition 1.3, Ai ∼= A∨d−i = HomK(Ad−i,K) and since the duality
Wefix notationasinTheorem 2.4:
• S := R⊗KK[u1,. . . ,um]= K[x0,. . . ,xn,u1,. . . ,um] is thebigradedringof
polyno-mialsinm+ n+ 1 variablesx0,. . . ,xn,u1,. . . ,um;
• WehavechosenthenaturalbigradingofS:xihasbidegree(1,0) andujhasbidegree
(0,1);
• DefineS(d1,d2)tobetheK-vectorspaceofbihomogeneouspolynomialsf ofbidegree
(d1,d2);thatis, f canbewrittenas
p
i=0
aibi,where ai∈ Rd1 = K[x0,. . . ,xn]d1 and bi ∈ K[u1,. . . ,um]d2.
• T := Q⊗K K[U1,. . . ,Um] = K[X0,. . . ,Xn,U1,. . . ,Um] is the (bigraded) ring of
differential operators of S, where Xi = ∂ ∂xi and Uj = ∂ ∂uj ; Xi has bidegree (1,0)
andUj hasbidegree (0,1).
A homogeneousidealI ofS isabihomogeneousideal if:
I = ∞
i,j=0
I(i,j), where∀i, j ∈ N≥0, I(i,j)= I∩ S(i,j).
Letf ∈ S(d1,d2),thenI = Ann(f ) isabihomogeneousidealandusingTheorem1.6,A= T /(Ann(f )) isaSBAGK-algebraofsoclebidegree(d1,d2) (andcodimensionm+ n+ 1).
Remark2.3.Usingtheabovenotations,onehas:
∀i > d1, j > d2, I(i,j)= T(i,j).
Indeed,forallα∈ T(i,j)with i> d1,j > d2,α(f )= 0;asaconsequence:
∀k ∈ {0, . . . , d1+ d2}, Ak = 0≤i≤d1 0≤j≤d2 i+j=k A(i,j).
Moreover, the evaluation map α∈ T(i,j) → α(f)∈ A(d1−i,d2−j) provides the following short exactsequence:
0 I(i,j) T(i,j) A(d1−i,d2−j) 0. (1) Thefollowingtheorem,whichlinksNagataidealizationswithbihomogeneous polyno-mials, holds.
Theorem 2.4 ([8], Theorem 2.77). Let S := K[u1,. . . ,um] and S := R⊗KS be rings
differential operators, where Xi = ∂ ∂xi and Uj = ∂ ∂uj . Let g0,. . . ,gn be homogeneous elementsof S ofdegree d,letI betheT-submoduleofS generatedby{∂(gi)∈ R | ∂ ∈ T,i∈ {0,. . . ,n}} and letA := T/Ann(I). Define f = x0g0+· · · + xngn ∈ R, itis a bihomogeneous polynomial of bidegree(1,d), andlet A:= T /Ann(f ).Considering I as anA-module,A I ∼= A.
2.2. Lefschetzpropertiesforhigher Nagataidealizations Definition2.5.A bihomogeneouspolynomial
f = n i=0 xd1 i gi ∈ S(d1,d2)
is called aCW-Nagata polynomial of degree d1 ≥ 1 if gi ∈ K[u1,. . . ,um], i = 0,. . . ,n,
arelinearlyindependentmonomialsofdegreed2≥ 2.
Remark2.6.Oneneedsn≤
m + d2− 1
d2
otherwisethegi cannotbelinearly
indepen-dent.
Fromnowon,we assumethatn satisfiesthiscondition. Wewillneedthefollowingpropositions.
Proposition 2.7 ([4] Proposition 2.5).Let n+ 1 ≥ m ≥ 2,d2 > d1 ≥ 1 and s >
m + d1− 1
d1
; for any j ∈ {1,. . . ,s}, let fj ∈ S(d1,0), gj ∈ S(0,d2). Then the form f = f1g1+· · · + fsgs ofdegree d1+ d2 satisfies
hessd1 f = 0;
that is,A= T /Ann(f ) doesnothave theSLPcondition.
Proposition2.8([1] Proposition 2.7).Letn+ 1≥ m≥ 2,d1≥ d2.ThenL=
n
i=0 Xi isa
WeakLefschetzElement;that is, A= T /Ann(f ) has theWLPcondition. 3. CW-complexNagataidealizationofbidegree(d1,d2)
LetS andT beas intheprevioussubsection.
Definition3.1.A bihomogeneousCW-Nagatapolynomial
f = n i=0 xd1 i gi ∈ S(d1,d2)
iscalledasimplicialNagatapolynomialof degreed1ifthemonomialsgi aresquarefree.
Remark3.2.Oneneedsn≤
m d2
otherwisethegi cannot besquarefree.
3.1. CW-complexesandbihomogeneouspolynomials 3.1.1. Abstractfinite simplicialcomplexes
Definition 3.3. LetV = {u1,. . . ,um} be afinite set. Anabstract simplicial complex Δ with vertexset V isasubsetof 2V suchthat
(1) ∀u∈ V ⇒ {u}∈ Δ,
(2) ∀σ ∈ Δ,τ σ,τ = ∅⇒ τ ∈ Δ.
Theelements σ ofΔ arecalled faces orsimplices;aface withq + 1 verticesiscalled q-face orfaceofdimensionq andonewritesdim σ = q;themaximalfaces(withrespect to theinclusion) arecalled facets;ifall facetshavethesamedimensiond≥ 1 then one saysthatΔ isof puredimensiond.ThesetΔk offacesofdimensionat mostk iscalled k-skeleton of Δ.2V is calledsimplex (ofdimensionm− 1).
Remark3.4.
(1) (cfr. [1, Remark 3.4]) There is a natural bijection, introduced in [2], between the squarefreemonomials,ofdegreed,inthevariablesu1,. . . ,umandthe(d− 1)-faces
ofthesimplex2V,withvertexsetV ={u1,. . . ,um}.Infact,asquarefreemonomial g = ui1· · · uid corresponds to the subset {ui1,. . . ,uid} of 2
V. Vice versa, to any
subsetF ofV withd elementsoneassociatesthefreesquaremonomialmF =
ui∈F ui ofdegreed. (2) Letf = n i=0 xd1
i gi ∈ S(d1,d2)be asimplicialNagatapolynomial;byhypothesisthere isbijectionbetweenthemonomialsgi andtheindeterminatesxi.From this,wecan
associatetof asimplicialcomplexΔf withverticesu1,. . . ,umwherethefacetwhich
corresponds togi isidentifiedwith xdi1. 3.1.2. CW-complexes
Forthetopologicalbackground, wereferto [9]. Westartbyfixingsomenotations. Definition3.5.Letk∈ N≥1.Atopologicalspaceekhomeomorphictotheopen(unitary)
ball {(x1,. . . ,xk)∈ Rk | x12+· · · + x2k < 1} of dimensionk (withthe naturaltopology
induced by Rk+1) is called a k-cell. Its boundary, i.e. the (k− 1)-dimensional sphere will be denoted bySk−1 ={(x1,. . . ,xk)∈ Rk | x21+· · · + x2k = 1} and itsclosure, i.e.
theclosed (unitary) k-dimensionaldisk will be denotedby Dk := {(x1,. . . ,xk)∈ Rk | x21+· · · + x2k≤ 1}.
Wereallthefollowing
Definition3.6. ACW-complex isatopologicalspaceX constructedinthefollowingway: (1) There exists afixed and discrete set of points X0 ⊂ X,whose elements are called
0-cells;
(2) Inductively,thek-skeletonXkofX isconstructedfromXk−1byattachingk-cellsek α
(with index set Ak)viacontinuous maps ϕαk: Sαk−1 → Xk−1 (the attachingmaps).
This means thatXk is aquotient of Yk = Xk−1
α∈Ak
Dαk under theidentification x ∼ ϕα(x) for x ∈ ∂Dαk; the elements of the k-skeleton are the (closure of the)
attached k cells; (3) X =
k∈N≥0
Xk andasubsetC ofX isclosedifandonlyifC∩ Xk isclosed forany k (closed weaktopology).
Definition3.7.AsubsetZ of aCW-complexX isaCW-subcomplex ifitistheunionof cellsofX,suchthattheclosure ofeachcellisinZ.
Definition3.8.A CW-complexisfinite ifitconsists ofafinitenumberofcells. Wewillbeinterestedmainlyinfinite CW-complexes.
Example3.9 (Geometricrealization of an abstract simplicial complex).It isan obvious factthattoany simplicialcomplexΔ onecanassociateafinite CW-complexΔ via the geometricrealization ofΔ asasimplicialcomplex(asatopologicalspace)Δ.
Inwhatfollows wewill always identifyabstract simplicialcomplexes withtheir cor-respondingsimplicialcomplexes.
Construction 3.10.InRemark 3.4, we saw that to any degreed square-free monomial ui1· · · uid∈ K[u1,. . . ,um]donecanassociatethe(d−1)-face{ui1,. . . ,uid} oftheabstract
(m− 1)-dimensionalsimplexΔ(m):= 2{u1,...,um},andviceversa:ifwecall
ρd:={f ∈ K[u1, . . . , um]d| f = 0 is a square-free monic monomial} D(m)d:= Δ(m)d\ Δ(m)d−1,
wehaveabijection
ui1· · · uid → {ui1, . . . , uid} .
Alternatively, we can associate to ui1· · · uid the element of the (d − 1)-skeleton
{ui1, . . . , uid} ∈ Δ(m) d−1 ,so wehaveabijection σd: ρd→ Δ(m) d−1 ui1· · · uid→ {ui1, . . . , uid}
between the square-free monomials and the (d− 1)-faces of the (topological) simplex
Δ(m).
Using CW-complexes,we willextendthis constructionto thenon-square-free monic monomials.Weproceedasfollows.Letg := uj1
1 · · · ujmmbeagenericdegreed:= j1+· · ·+
jmmonomial.Considerthefollowingfiniteset:W :=
u11, . . . , uj1
1 , . . . , u1m, . . . , ujmm
,and ifΔ(d):= 2W istheabstractassociated (finite)simplex,we considerthecorresponding
(topological)simplex(whichisaCW-complex)Δ(d).
If jk ≤ 1 we do nothing, while ifjk ≥ 2, we recursivelyidentify, for varying from
0 to jk− 2, the -facesofthesubsimplex
2 u1 k,...,ujkk
⊂ Δ(d):start with = 0, andwe identifyallthejk pointstoonepoint—callituk.Then,for = 1,weobtainabouquetof
jk+1
2
circles,andweidentifythem injustonecircleS1passingthroughu
k,andsoon, upto thefacetsof2 u1 k,...,ujkk
,i.e.itsjk+ 1 (jk− 1)-faces,which,bytheconstruction,
havealltheirboundaryincommon,andweidentifyallofthem.
Make allthese identifications for all j1,. . . ,jm; in this way, we obtain afinite
CW-complexX = Xgofdimensiond− 1,with0-skeletonX0={ui| ji= 0}⊂ {u1,. . . ,um},
obtainedfrom the(d− 1)-dimensionalsimplexΔ(d), withtheaboveidentification. Inthisway,weobtainafiniteCW-complexX = Xgofdimensiond−1,with0-skeleton X0={ui| ji= 0}⊂ {u1,. . . ,um},obtainedfromthe(d− 1)-dimensionalsimplexΔ(d),
with the above identification. Under this identification each closure of a (jk − 1)-cell
u1 k, . . . , u jk k
becomesapoint ifjk = 1, acircleS1 ifjk = 2,atopological spacewith
fundamentalgroupZ3ifjk = 2 (i.e.itisnotatopologicalsurface),etc.Wewilldenote
these spacesinwhatfollowsbyjk−1
k ,i.e. jk−1
k correspondstou jk
k ,and viceversa:
Proposition 3.11.Everypowerin uj1
1 · · · ujmm (uptoa permutationofthevariables) cor-responds toajk−1
k ,andviceversa.
WecanseeXg asa(d− 1)-dimensionaljoin betweenthesespacesjkk−1 andthespan
of the 0-skeleton X0 i.e. the simplex S
X ⊂ Δ(m) associated to it; SX ∼= Δ( ), where = #X0≤ m.
Remark 3.12.This last observation suggests we consider an alternative construction: recallthatthecellulardecomposition ofthereal projective spaceis obtainedattaching asinglecellateachpassage;indeed,Pn
R isobtainedfrom PRn−1 byattachingonen-cell
withthequotientprojectionϕn−1: Sn−1→ PRn−1 as theattaching map. Then,toeachpowerujk
k weassociatearealprojectivespaceofdimensionjk−1 Pkjk−1
andimmersionsik−1: Pjk−1
k → P jk
k ;soPk0= uk∈ Pkjk−1.
Finally,tog = uj1
1 · · · ujmm weassociatethejoinbetweentheP jk−1
k andtheSXdefined
above;ifwecallthisjoinbyXg,wecanproceedinanequivalentway,bychangingjkk−1
withPjk−1
k .
It is clear how to glue two of these finite CW-complexes—say X = Xuj1
1 ···ujmm and
Y = Yuk1
1 ···ukmm , of degree d= j1+· · · + jm and d
= k
1+· · · + km—along Δ(m): we
simplyattachX andY viatheinclusionmapsSX⊂ Δ(m) andSY ⊂ Δ(m), whereSX
andSY arethesimplexes associatedto,respectively,X andY .
Finally, taking allthese finite CW-complexestogether, we obtain aCW-complex P inthefollowing way:
C :=
uj11 ···ujmm ∈K[u1,...,um]
Xuj1
1 ···ujmm P (m) := C/∼
where∼ istheequivalencerelationinducedbytheabovegluing.
Proposition3.13.Thereisbijectionbetweenthemonomialsofdegree d inK[u1,. . . ,um] andtheelements ofthe(d− 1)-skeleton of P (m).
Inotherwords,ifwedefine
ρd:={f ∈ K[u1, . . . , um]d| f = 0 is a monic monomial}
wehaveabijection,using theabovenotation
σd: ρd → P (m)d−1 uj1
1 · · · ujmm → Xuj11 ···ujmm .
Proposition3.14.Xuj1
1 ···ujmm ⊂ Xuk11 ···ukmm ifandonly ifu
j1 1 · · · ujmm divides u k1 1 · · · ukmm. Letf = n i=0 xd1
i gi ∈ S(d1,d2)beaCW-Nagatapolynomial;byhypothesisthereis bijec-tionbetweenthemonomialsgi andtheindeterminatesxi.Fromthis,wecanassociateto f afinite(d2−1)-dimensional,CW-subcomplexofP (m),Δf wherethe(d2−1)-skeleton
isgivenbytheXgi’sgluedtogetherwiththeaboveprocedure.EachXgicanbeidentified
withxd1
Thepreviousconstructiongeneralizestheanalogousonegivenin [1]. 3.2. TheHilbert functionofSBAG algebras
Thefirstmain resultofthispaperisthefollowinggeneraltheorem. Remark3.15.Inordertostateit,weobserve thatthecanonicalbasesof
S(d1,d2)= K[x0, . . . , xn]d1⊗ K[u1, . . . , um]d2 and
T(d1,d2)= K[X0, . . . , Xn]d1⊗ K[U1, . . . , Um]d2 given bymonomialsaredualbaseseachother, i.e.
Xk0 0 · · · XnknU 1 1 · · · Umm(x i0 0 · · · xinnu j1 1 · · · ujmm) = δ i0,...,in,j1,...,jm k0,...,kn,1,...,m where i0+· · · + in = k0+· · · + kn = d1, j1+· · · + jm = 1 +· · · + m = d2 and δi0,...,in,j1,...,jm
k0,...,kn,1,...,m istheKroneckerdelta.
This simpleobservation allows us to identify—given aCW-Nagata polynomialf = n
r=0 xd1
r gr ∈ S(d1,d2)— the dual differential operator Gr of the monomial gr—i.e. the monomial Gr ∈ K[U1,. . . ,Um]d2 such that Gr(gr) = 1 and Gr(g) = 0 for any other monomial g ∈ K[u1,. . . ,um]d2—with the sameelement of the (d2− 1)-skeleton of Δf associatedtogr.Inotherwords,weassociatetogr= u1j1· · · ujmm andtoGr= U1j1· · · Umjm
theCW-subcomplexofΔf ⊂ P (m),Xuj1 1 ···ujmm . Theorem 3.16. Let f = n r=0 xd1 r gr ∈ R(d1,d2), with gr = u j1 1 · · · ujmm, be a CW-Nagata polynomialof(positive)degreed1,wheren≤
m d2
,letΔf betheCW-complexassociated to f andletA= Q/Ann(f ). Then
A = d=d1+d2 h=0 Ah where Ah= A(h,0)⊕ · · · ⊕ A(p,q)⊕ · · · ⊕ A(0,h), p≤ d1, q≤ d2, Ad= A(d1,d2) and moreover,∀j ∈ {0,1,. . . ,d2},
dim A(i,j)= ai,j = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ fj i = 0 n r=0 fj,r i∈ {1, . . . , d1− 1}, fd2−j i = d1 where:
• fj isthenumberoftheelementsofthe(j− 1)-skeletonoftheCW-complexΔf (with theconvention that f0= 1);
• fj,r is the number of theelements of the (j− 1)-skeleton of the CW complex XGr
(with theconvention thatf0,r= 1,sothat dim A(i,0)= n+ 1).
Moreprecisely,abasisforA(i,j),∀j ∈ {0,1,. . . ,d2}, isgivenby
(1) If i = 0, {Ω1,. . . ,Ωfj}, where any Ωs := U
s1
1 · · · Umsm, with s1+· · · + sm = j, is associatedtotheelement Xus1
1 ···usmm ofthe(j− 1)-skeletonof Δf;
(2) Ifi= 1,. . . ,d1− 1, Ωi,s1,...,sm s s∈{0,...,n} sk≤,r k,k=1,...,m ksk=j where Ωi,s1,...,sm s := Xsi· U1s1· · · Umsm isassociated totheelementXus1
1 ···usmm ofthe(j− 1)-skeletonof Xgs;
(3) If i = d1, Xd1 0 Ω1(f ), . . . Xnd1Ωfd2−j(f ) , where Ω1, . . . Ωfd2−j
is the basis for A(0,d2−j) ofcase(1).
In the cases (1) and (2) thebasis are given by monomials, in the case (3), in general, not.
Proof. Wedividethe proofinto computing thedimensionof A(i,j) andfind abasisfor
it,asi varies:
i = 0: A(0,0)∼= K.
Then,bydefinition,ifj ∈ {1,. . . ,d2},A(0,j) isgenerated bythe(canonical
images of the)monomials Ωs∈ Qj = K[U1,. . . ,Um]j ∼= Q(0,j) thatdo not
annihilatef .Thismeansthat,ifwewrite
Ωs= U1s1· · · Umsm s1+· · · + sm= j,
there exists an rs ∈ {0,. . . ,n} such thatgrs = u
s1
1 · · · usmmgrs, where g
rs ∈
Rd2−j isa(nonzero)monomial; thismeansthatXus11 ···usmm isanelement of
the(j− 1)-skeletonoftheCW-complexΔf byProposition3.14.
We need to prove that these monomials are linearly independent over K: let {Ω1,. . . ,Ωfj} be a system of monomials of Q(0,j), where any Ωs =
Us1
−1)-skeleton of the CW-complex Δf; take a linear combination of them and apply ittof : 0 = fj s=1 csΩs(f ) = fj s=1 cs n r=0 xd1 r Ωs(gr) = n r=0 xd1 r fj s=1 csΩs(gr).
Bythelinearindependenceofthexd1 r ’s fj
s=1
csΩs(gr) = 0, ∀r ∈ {0, . . . , n}. (2)
By hypothesis, for any index s there exists an rs ∈ {0,. . . ,n} such that
Ωs(grs)= grs ∈ Rd2−j\ {0},then foranyindex s onehascs= 0, sincethe linearcombinationsin(2) areformedbylinearlyindependentmonomials(gr
isfixed ineachlinearcombination!).Inotherwords,dim A(0,j) = fj.
0 < i < d1: ObservethatXaXb(f )= 0 ifa= b.ThereforeA(i,j)isgeneratedbytheonly
(canonical images of) the monomials Ωi,s1,...,sm
s := XsiU s1
1 · · · Umsm ∈ Q(i,j),
with s1+· · · + sm= j, thatdo notannihilate f . Inparticular, abasis for A(i,0) is givenby X0i,. . . ,Xni and wecansuppose from now onthatj > 0.
Since
Ωi,s1,...,sm
s (f ) = xds1−i(U1s1· · · Umsm) (gs),
inordertoobtainthatthisisnotzero,wemusthavethatgs= us11· · · usmmgs,
where gs ∈ Rd2−j is anonzeromonomial. This meansXus11 ···usmm ⊂ Xgs by
Proposition3.14.
Asabove,wecanprovethatthese monomialsarelinearlyindependentover K:let Ωi,s1,...,sm s s∈{0,...,n} sk≤,r k,k=1,...,m ksk=j
be asystemofmonomialsofQ(i,j),whereanyΩsi,s1,...,sm = Xsi· U1s1· · · Umsm
is associatedto theelementXus1
1 ···usmm ofthe(j− 1) skeletonof Xgs ⊂ Δf,
i.e.Xus1
1 ···usmm ⊂ Xgs ⊂ Δf byProposition3.14.
Take alinearcombinationofthem andapplyittof :
0 = s∈{0,...,n} sk≤,r k,k=1,...,m ksk=j ci,s1,...,sm s Ωi,ss 1,...,sm(f ) = n s=0 xd1−i sk≤,r k,k=1,...,m ksk=j ci,s1,...,sm s gi,ss 1,...,sm (3)
where gi,s1,...,sm
s ∈ Rd2−j is the nonzero monomial such that gs = us1
1 · · · usmmgsi,s1,...,sm.From (3) wededuce,as intheprecedingcase,that
sk≤,r k,k=1,...,m ksk=j
ci,s1,...,sm
s gsi,s1,...,sm = 0 s = 0, . . . , n; (4)
asbefore, givenonechoiceofs1,. . . ,sm there existsans∈ {0,. . . ,n} such gi,s1,...,sm
s (f ) isanonzeromonomial,andthe(nonzero)gsi,s1,...,sm’sin(4) are
linearlyindependent sinceareobtainedbyafixed gs.
i = d1: By duality, see Remark2.2, A(d1,j) ∼= A∨(0,d2−j) so dim A(d1,j) = fd2−j. To findabasisforA(d1,j),weconsidertheexactsequence(1) givenbyevaluation atf ,whichinthiscasereads
0→ I(0,d2−j)→ Q(0,d2−j)→ A(d1,j)→ 0, (5) thena basisfor A(d1,j) is obtainedinthefollowing way: if{Ω1,. . . Ωfd2−j}
isthebasisforA(0,d2−j)∼= Q(0,d2−j)/I(0,d2−j)ofthecasei= 0,thenabasis forA(d1,j)is Xd1 0 Ω1, . . . , Xnd1Ωfd2−j(f ) .
AsacorollaryofTheorem3.16weseethatwecandeducethegeneralcaseofthe sim-plicialNagatapolynomial,whichisaslightimprovementofthefirstpartof[1,Theorem 3.5]. Corollary 3.17. Let f = n r=0 xd1 r gr ∈ R(d1,d2), with gr = xr1· · · xrd2, be a simplicial
Nagata polynomial of (positive) degree d1, where n ≤
m d2
, let Δf be the simplicial complex associatedtof andletA= Q/Ann(f ). Then
A = d=d1+d2 h=0 Ah where Ah= A(h,0)⊕ · · · ⊕ A(p,q)⊕ · · · ⊕ A(0,h), p≤ d1, q≤ d2, Ad= A(d1,d2) andmoreover,∀j ∈ {0,1,. . . ,d2},
dim A(i,j)= ai,j =
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ fj i = 0 n r=0 fj,r i∈ {1, . . . , d1− 1}, fd2−j i = d1
where:
• fj isthenumber of(j− 1)-cells oftheΔf (withtheconvention that f0= 1);
• fj,r isthenumberof(j−1)-subcellsofΔgr,i.e.the(d2−1)-celloftheΔf associated
togr (with theconventionthat f0,r= 1,sothat dim A(i,0)= n+ 1).
Moreprecisely,a basisforA(i,j),∀j ∈ {0,1,. . . ,d2},isgiven by
(1) Ifi= 0,{Ω1,. . . ,Ωfj},whereanyΩs:= Us1· · · Usj isassociatedtothe(j−1)-subcell
{us1,. . . ,usj} of Δf; (2) If i = 1,. . . ,d1− 1, Ωi,s1,...,sj s s∈{0,...,n} s1,...,sj∈ r1,...,rd2 where Ω i,s1,...,sj s := XsiUs1· · · Usj
isassociatedtothe(j− 1)-subcell {us1,. . . ,usj} ofΔgs(⊂ Δf);
(3) If i = d1,
Xd1
0 Ω1(f ), . . . Xnd1Ωfd2−j(f )
, where {Ω1,. . . Ωfd2−j} is the basis for
A(0,d2−j) of case(1).
In thecases (1) and(2) the bases are given by monomials, in the case (3), in general, not. Theorem3.18.Letf = n r=0 xd1 r gr∈ S(d1,d2),withgr= x r1 1 · · · xrmm suchthatr1+· · ·+rm= d2,beaCW-NagatapolynomialwhoseassociatedCW-complexisΔf,asinthepreceding theorem.
Then I := Ann(f ) isgeneratedby:
(1) XiXj and Xkd1+1,fori,j,k∈ {0,. . . ,n},i< j;
(2) U1,. . . ,Um d2+1,i.e. allthe(monic) monomialsofdegree d2+ 1;
(3) The monomials Us1
1 · · · Umsm such that s1+· · · + sm = j, where Xus11 ···usmm is a
(minimal) element of the (j− 1)-skeleton of P (m) not contained in Δf (for j ∈ {1,. . . ,d2});
(4) The monomialsXrUi, where ui does not divide gr (i.e. {ui} is not an element of the0-skeletonof Xgr);
(5) The monomials XsU1r1· · · Umrm such that r1 +· · · + rm = j, where ur11· · · urmm is minimalamongthosethatdonotdividegs(i.e.the(minimal)elementofthe(j −1)-skeletonof P (m),Xur1
1 ···urmm ,isnotcontainedin Xgs),forj ∈ {1,. . . ,d2};
(6) ThebinomialsXd1 r U
ρ1
1 · · · Umρm−Xsd1U1σ1· · · Umσmwithρ1+· · ·+ρm= σ1+· · ·+σm= j suchthatgr,s= GCD(gr,gs) andgr= uρ11· · · umρmgr,s,gs= u1σ1· · · uσmmgr,s(i.e.Xgr,s
istheelement ofthe(d2− j − 1)-skeletonofΔf whichrepresentstheintersectionof Xgr andXgs:Xgr,s= Xgr∩ Xgs).
By Theorem 3.16, (1) a basis for A(0,j), ∀j ∈ {1,. . . ,d2}, is {Ω1,. . . ,Ωfj}, where
Ωs:= U1s1· · · Umsm,withs1+· · · + sm= j,isassociated totheelement Xus11 ···usmm ofthe
(j− 1)-skeleton of Δf. Therefore, using the identification introduced inRemark 3.15,
a basis for I(0,j) is given by the monomials U1s1· · · Umsm such that s1+· · · + sm = j,
whereXus1
1 ···usmm is anelement ofthe(j− 1)-skeletonofP (m) notcontainedinΔf (for
j∈ {1,. . . ,d2});
ObservethatXiXj(f )= 0 ifi= j andXkd1+1(f )= 0= U i1
1 · · · Umim(f ) with
m j=1ij= d2+ 1,fordegreereasons.Set
β := (X0X1, . . . , Xn−1Xn, X0d1+1, . . . , Xnd1+1,U1, . . . , Um d2+1);
thisisahomogeneousidealsuchthatβ⊂ I andA∼= T β/
I β.
ByTheorem 3.16, (2),ifi= 1,. . . ,d1− 1,abasisforA(i,j) ∀j ∈ {1,. . . ,d2},is given
by Ωi,s1,...,sm s s∈{0,...,n} sk≤,r k,k=1,...,m ksk=j whereΩi,s1,...,sm
s := Xsi·U1s1· · · Umsm isassociatedtotheelementXus11 ···usmm ofthe(j
−1)-skeletonof Xgs.
AgainusingtheidentificationintroducedinRemark3.15,abasisfor I β (i,j) isgiven by • The monomials Xi
rU1s1· · · Umsm such that s1+· · · + sm = j, with r = s, where us1
1 · · · usmm dividesgs(i.e.Xus11 ···usmm isanelementofthe(j− 1)-skeletonofXgs),for
i= 1,. . . ,d1− 1,and
• The monomials XsiUr1
1 · · · Umrm such thatr1+· · · + rm = j, where ur11· · · urmm does
not divide gs (i.e. the element of the (j− 1)-skeleton of P (m), Xur11 ···urmm , is not
containedinXgs),
forj∈ {1,. . . ,d2}.
It remains to find thegenerators of I of bidegree (d1,j), with j ∈ {1,. . . ,d2}.This
is more complicated since the generators of A(d1,j) are not monomials. Let γ be the homogeneous idealgenerated by the monomials of the cases(1),(2), (3), (4) and (5), i.e.thegeneratorsthatwehavefoundsofar.Wehaveβ⊂ γ ⊂ I andtheexactsequence (1) givenbyevaluationatf becomes
0→ I γ (d1,j) → T γ (d1,j) → A(0,d2−j)→ 0,
sinceweidentifyA∼=T γ/ I γ.Then,ifρ1+· · ·+ρm= σ1+· · ·+σm= j,X d1 r U ρ1 1 · · · Umρm− Xd1 s U1σ1· · · Umσm ∈ T γ (d1,j) is in I γ (d1,j) if and only if Xd1 r U ρ1 1 · · · Umρm = Xd1 s U1σ1· · · Umσm ∈ A(0,d2−j), which means U ρ1 1 · · · Umρm(gr) = U1σ1· · · Umσm(gs). Since A(0,d2−j)isgeneratedbythemonomialsΩs:= U
s1
1 · · · Umsm,withs1+· · · + sm= d2− j,
associated totheelementsof the(d2− j − 1)-skeletonofΔf,weobtaincase(6).
As wehavedoneforTheorem 3.16,wegive, asacorollary ofTheorem 3.18thecase of the simplicial Nagata polynomial, giving an improvement of the second part of [1, Theorem 3.5]; we also correct that statement, since the authors forgot the generators XiXj,i= j. Corollary 3.19.Let f = n r=0 xd1 r gr ∈ R(d1,d2), with gr = xr1· · · xrd2, be a simplicial
Nagata polynomial whose associated simplicialcomplex is Δf,as inthe preceding theo-rem.
Then I := Ann(f ) isgeneratedby:
(1) XiXj and Xkd1+1,fori,j,k∈ {0,. . . ,n},i< j; (2) U2
1,. . . ,Um2;
(3) The monomials Us1· · · Usj, where {us1,. . . ,usj} is a (minimal) (j − 1)-cell of
2{u1,...,um} notcontainedin Δ
f (forj∈ {1,. . . ,d2});
(4) The monomialsXrUi,whereui doesnotdivide gs (i.e.{ui}∈ Δg/ r); (5) The binomials Xd1
r Uρ1· · · Uρj − X
d1
s Uσ1· · · Uσj such that gr,sGCD(gr,gs), gr =
uρ1· · · uρjgr,s, gs = uσ1· · · uσjgr,s (i.e. gr,s represents the (d2 − j − 1)-face
given by the intersection Δgr ∩ Δgs of the facets of gr and gs: Δgr,s = Δgr ∩
Δgs).
Proof. We note only that we have to add the squares of case (2) although they do not correspond to cells,sincethe polynomials gi are square-free. Therest follows from
Theorem 3.18.Weobservethatthesesquaresareincase(2) ofTheorem3.18. Example 3.20.Let
f = xd0u1u2u3+ xd1u1u2u4+ xd2u1u4u5+ xd3u1u3u5+ xd4u2u3u6+ x5du2u4u6+ xd6u4u5u6
+ xd7u3u5u6
be a bihomogeneous bidegree (d,3) polynomial with d ≥ 1; it is a simplicial Nagata polynomial,whoseassociated simplicialcomplexisinthefollowingfigure:
u1 u2 u3 u4 u5 u6 xd0 xd1 xd 2 xd 3 xd 4 xd 5 xd6 xd7 Wehave: A = A0⊕ A1⊕ . . . ⊕ Ad+3.
Wewantfirstlyto computetheHilbertvectorbyapplyingCorollary 3.17;firstofall, a1,0= 8 a0,1= 6,
andtherefore
h0= hd+3= 1
h1= hd+2= a1,0+ a0,1= 8 + 6 = 14.
Then,weanalyzethepossiblecasesdependingonthedegreed: • Ifd= 1,then
a1,1= 8· 3 = 24
a0,2= 12
h2= a1,1+ a0,2= 36
andtheHilbertvectoris(1,14,36,14,1).
• Ifd= 2,then,recallingbigradedPoincaréduality,
a2,0= a0,3= 8 a2,1= a0,2= 12
andtherefore
h2= a2,0+ a1,1+ a0,2= 8 + 8· 3 + 12 = 44,
in accordance with Poincaré duality; so the Hilbert vector is (1,14,44,44,14,1) (cfr. [1,Example3.6]).
• Ifd= 3,then, againbybigradedPoincaréduality,
a3,0= a0,3= 8, a2,1= a1,2= 8· 3 = 24, a3,1= a0,2= 12, a2,2= a1,1= 24, a1,3= a2,0= 8, therefore h2= a2,0+ a1,1+ a0,2= 44, h3= a3,0+ a2,1+ a1,2+ a0,3= 64 h4= 0 + a3,1+ a2,2+ a1,3= 44
h2= h4inaccordancewithPoincarédualityandtheHilbertvectoris(1,14,44,64,44,
14,1).
• Ingeneral,letd≥ 4;byhypothesis
hd+1 = h2= a2,0+ a1,1+ a0,2= 44,
and
hk = ak,0+ ak−1,1+ ak−2,2+ ak−3,3 ∀k ∈ {3, . . . , d},
where
ak,0= 8 ak−1,1= 8· 3 = 24 ak−2,2= 8· 3 = 24 ak,3= 8.
AgainusingthePoincarédualitywehave:
hd+3−k= hk = 64 ∀k ∈ 3, . . . , d + 3 2
andtheHilbertvectoris(1,14,44,64,. . . ,64,44,14,1).
Now,wewanttofindthegeneratorsofAnn(f ),byapplyingCorollary3.19.Behavior depends ond:
• Ifd= 1,byCorollary 3.19Ann(f ) is(minimally)generatedby: (1) X0,. . . ,X7 2= X02,X0X1,. . . ;
(2) U2
1,. . . ,U62;
(4) X0U4,X0U5,X0U6,X1U3,X1U5,X1U6,X2U2,X2U3,X2U6,X3U2,X3U4,X3U6,
X4U1,X4U4,X4U5,X5U1,X5U3,X5U5,X6U1,X6U2,X6U3,X7U1,X7U2,X7U4;
(5) X0U3−X1U4,X0U2−X3U5,X0U1−X4U6,X1U2−X2U5,X1U1−X5U6,X2U4−
X3U3,X2U1 − X6U6,X3U1 − X7U6,X4U3 − X5U4,X4U2 − X7U5, X5U2 −
X6U5,X6U4− X7U3.
• Ifd≥ 2,byCorollary3.19Ann(f ) is(minimally)generated by (1) X0,. . . ,X7 d+1 and XhXk whereh,k∈ {0,. . . ,7},h< k; (2) U2 1,. . . ,U62; (3) U1U6,U2U5,U3U4; (4) Xd 0U4,X0dU5,X0dU6,X1dU3,X1dU5,X1dU6,X2dU2,X2dU3,X2dU6,X3dU2,X3dU4,X3dU6, X4dU1,X4dU4,X4dU5,X5dU1,X5dU3,X5dU5,X6dU1,X6dU2,X6dU3,X7dU1,X7dU2,X7dU4; (5) Xd 0U3 − X1dU4,X0dU2 − X3dU5,X0dU1 − X4dU6,X1dU2 − X2dU5,X1dU1 − X5dU6, Xd 2U4 − X3dU3,X2dU1 − X6dU6,X3dU1 − X7dU6,X4dU3 − X5dU4,X4dU2 − X7dU5, Xd 5U2− X6dU5,X6dU4− X7dU3. Example3.21.Let f = xd0u1u2+ xd1u21+ xd2u2u3
beabihomogeneousbidegree(d,2) polynomial,withd≥ 1;itisaCW-Nagatapolynomial whoseCW-complexisthefollowing:
xd 0 xd 1 u1 u2 u3 xd 2 Wehave: A = A0⊕ A1⊕ . . . ⊕ Ad+2
andwewanttofinditsHilbertvector;firstofall, a1,0= 3 a0,1= 3
andtherefore
Therefore, ifd= 1,thenHilbertvectoris(1,6,6,1). Ifd= 2,wehave
a1,1= 2 + 1 + 2 = 5,
so
h2= dim A2= a2,0+ a1,1+ a0,2= 3 + 5 + 3 = 11
and theHilbertvectoris(1,6,11,6,1). Ifd= 3 then,bybigradedPoincaréduality
a3,0= a0,2= 3 a0,3= 3
so
h2= a2,0+ a1,1+ a0,2= 11
h3= a3,0+ a2,1+ a1,2+ a0,3= 3 + 5 + 3 = 11
and theHilbertvectoris(1,6,11,11,6,1). Ingeneral,letd≥ 4;byhypothesis
hd = h2= a2,0+ a1,1+ a0,2= 11,
and
hk = dim A(k,0)+ dim A(k−1,1)+ dim A(k−2,2) ∀k ∈ {3, . . . , d},
so,since
ak,0= 3 ak−1,1= 5 ak−2,2= 3
using Poincarédualitywehave:
hd+2−k= hk = ak,0+ ak−1,1+ ak−2,2= 11 ∀k ∈ 3, . . . , d + 2 2 , and theHilbertvectoris(1,6,11,. . . ,11,6,1).
Letd= 1,byTheorem3.18Ann(f ) is(minimally)generated by: • X0,X1,X2 2,U22,U32,U1U3,U13;
• X0U12,X0U3,X1U2,X1U3,X2U1;
Letd≥ 2,byTheorem3.18Ann(f ) is(minimally)generatedby: • X0,X1,X2 d+1,X0X1,X0X2,X1X2,U22,U32,U1U3,U13; • Xd 0U12,X0dU3,X1dU2,X1dU3,X2dU1; • Xd 0U2− X1dU1,X0dU1− X3dU3. References
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