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Advances
in
Applied
Mathematics
www.elsevier.com/locate/yaama
Lefschetz
properties
for
higher
order
Nagata
idealizations
Armando Cerminaraa, Rodrigo Gondimb,1, Giovanna Ilardia,∗, Fulvio Maddalonia
aDipartimentoMatematicaedApplicazioni“RenatoCaccioppoli”,UniversitàDegli
StudiDiNapoli“FedericoII”,ViaCinthia, ComplessoUniversitarioDiMonte S. Angelo80126, Napoli, Italy
b
UniversidadeFederalRuraldePernambuco,av.DonManoeldeMedeiross/n, DoisIrmãos, Recife, PE52171-900,Brazil
a r t i c l e i n f o a b s t r a c t
Articlehistory:
Received28July2018
Receivedinrevisedform5February 2019
Accepted7February2019 Availableonline15February2019
MSC:
primary13A02,05E40 secondary13D40,13E10
Keywords:
Lefschetzproperties ArtinianGorensteinalgebras Nagataidealization
We study a generalization of Nagata idealization for level algebras.ThesealgebrasarestandardgradedArtinian alge-braswhose Macaulay dualgeneratorisgivenexplicitly asa bigradedpolynomialofbidegree(1,d).Weconsiderthe alge-bra associated to polynomialsof thesame type of bidegree (d1,d2).WeprovethatthegeometryoftheNagata hypersur-faceofordere isverysimilartothegeometryoftheoriginal hypersurface.WestudytheLefschetz propertiesfor Nagata idealizationsoforderd1,provingthatWLPholdsifd1≥ d2. Wegivea completedescriptionofthe associatedalgebra in themonomialsquarefreecase.
©2019ElsevierInc.Allrightsreserved.
* Correspondingauthor.
E-mailaddresses:armando.cerminara@unina.it(A. Cerminara),rodrigo.gondim@ufrpe.br
(R. Gondim),giovanna.ilardi@unina.it(G. Ilardi),maddalonifulvio@gmail.com(F. Maddaloni).
1 PartiallysupportedbyICTP-INdAMResearchinPairsFellowship2018/2019andbyFACEPEATP
-0005-1.01/18.
https://doi.org/10.1016/j.aam.2019.02.002
0. Introduction
The Lefschetz properties are algebraic abstractions inspired by the so called Hard LefschetzTheoremaboutthecohomologyofsmoothprojectivevarietiesoverthecomplex numbers (see[18,24]).Since thecohomology ringof suchvarieties arestandardgraded ArtinianK-algebras satisfyingthePoincaréduality,from thealgebraic viewpointthese
algebras can be characterized as standard graded Artinian Gorenstein algebras, AG
algebrasforshort(see[12,19]).Inthiscontext,A=
d
k=0
Ak,theHardLefschetzTheorem
canbereformulated asapossiblepurely algebraicpropertyofthealgebraA.
It is important to highlight that nowadays the Lefschetz properties are considered in a number of distinct contexts, such as Khaler manifolds, solvmanifolds (see [16]), arithmetichyperbolicmanifolds(see [2]),Shimuravarieties(see[13]),convexpolytopes (see[17]),Coxetergroups(see[22]),matroids,simplicialcomplexes[26,25,1,9,17] among others. InthesenewcontextstheLefschetz propertiesshowedto haveinteractions with thealgebraitself,thegeometryandthecombinatorics.Thisworkliesattheintersection
of these three areas. In fact, standard graded AG algebras can be presented as A =
Q/Ann(f ) wheref ∈ K[x0,. . . ,xn]d isahomogeneousformofdegreed inapolynomial
ring, Q = K[∂0,. . . ,∂n] is the associated ring of differential operators and Ann(f ) is
the ideal of differential operators that annihilates f . Therefore, in this paper we are interested in the algebraic structure of A, together the geometry of the hypersurface
X = V (f )⊂ Pn andthecombinatoricsoftheformf in averyparticularway.
ThecornerstoneofthealgebraictheoryofLefschetzpropertiesweretheoriginal papers ofStanley[26,25,27] andtheworksofWatanabe,summarized in[12]. Averyimportant constructionthatappearsmanytimesintheseworksisthesocalledNagataidealization alsocalledtrivialextension.IngeneralNagataidealizationisausefultool,developedby Nagata,toconvertanyR-moduleM inaidealofanotherring,AM.Inourperspective the startingpointis averyinteresting isomorphismbetweentheNagataidealizationof an idealI = (g0,. . . ,gm)⊂ K[u1,. . . ,um] andalevel algebrainsuchway thatthenew
ringisanAGalgebraand wegetanexplicitformulafortheMacaulaygeneratorf (see
[12,Proposition2.77])
f = x0g0+ . . . + xngn ∈ K[x0, . . . , xn, u1, . . . , um](1,d−1). (1)
ThisbigradedpolynomialiscloselyrelatedwithGordan–NoetherandPerazzo construc-tions of forms with vanishing Hessian (see [11,23,3,8]). It is not acoincidence since in [19] theauthors present aHessiancriterion forthe SLPsaying thatthevanishing of a (higher) Hessian implies the failure of SLP. This criterion was generalizedin [10] also fortheWLPusingmixedHessians.FollowingtheoriginalideasofGordan–Noether and Perazzo,thesecondauthorin[7] constructedfamiliesofpolynomialswhosek-thHessian is zero.A naturalgeneralizationof(1) shouldbeto considerpolynomials oftheform:
f = xd1
0 g0+ . . . + xdn1gn∈ K[x0, . . . , xn, u1, . . . , um](d1,d2). (2) Thesepolynomials arecalledNagatapolynomials oforderd1 (seeDefinition3.1).
ThestudyofthehypersurfaceswithvanishingHessianbeganin1852,whenO. Hesse wrote two papers (see [14,15]), in which he claimed they must be cones. Given an
ir-reducible hypersurface X = V (f ) ⊂ PN of degree deg f = d ≥ 3, P. Gordan and
M. Noetherprovedin[11] thatHesse’sclaimistrueforN ≤ 3,butitisfalseforN ≥ 4;
infactitispossibletoconstructcounterexamplesinPNforeachN ≥ 4 andforalld≥ 3.
Moreover, they classifiedall the counterexamples to Hesse’s claim inP4 (see [3,7,11]). PerazzoclassifiedcubichypersurfaceswithvanishingHessianinPNforN ≤ 6 (see[23]),
this workwas revisitedand generalizedin[8]. Theproblem is open intheother cases. InallthecaseswheretheclassificationofhypersurfaceswithvanishingHessianaredone theysharetwoveryparticulargeometric properties(see [3,6,8,24]):
(i) there is alinear space L in thesingular locus of X, thatis the linear span of the dual varietyoftheimage ofthepolar(gradient)map,thatisL=< Z∗>;
(ii) thehypersurfaceisatangentscrollover thedual ofthepolarimage.
InthispaperwestudytheLefschetzpropertiesforthealgebras associatedtoNagata polynomials oforder d1, thegeometry oftheNagata hypersurfacesoforder d1and the
interactionbetween thecombinatoricsoff and thealgebraicstructure ofA inthecase thatthegi aresquare freemonomials, byusingasimplicialcomplextostudythis case.
WeshowthatthegeometryofNagatahypersurfacesisverysimilartothegeometry of theknownhypersurfaceswithvanishingHessian.Hencethesearehypersurfaces, satisfy-ingatleastaLaplaceequation(see[4,5]).Weprovedthattheyarescrollhypersurfaces inTheorem 2.9andCorollary2.10.These areourfirstmain results.
From the algebraic viewpoint we are interested in the Lefschetz properties and the algebraicstructure ofNagataidealizations. TheLefschetzpropertiesarestudiedintwo cases:
(1) d1 < d2, inthis casewegiveexampleswithsmall numbersofsummandswhere the
SLPholds andwerecallaresultprovedin[7] (seeProposition2.5);
(2) d1≥ d2,inthiscaseA hastheWLPasprovedinProposition2.7.Thisisoursecond
mainresult.
ThestructureofthealgebraA,includingtheHilbertvectorandacompletedescription of the ideal I was proved inTheorem 3.5 for the case inwhich the gi are square free
monomials. Thisis ourthird main result.To proveit, weuse thecombinatoricsof the simplicialcomplexassociated tothemonomialsgi inordertodescribeboth Ak andIk.
Part ofthis paper was inspiredby the discussions ofa groupworkin theworkshop
participants of thegroupworkwereM.Boij, R.Gondim, J.Migliore,U. Nagel, A.
Se-celeanu, H.SchenckandJ.Watanabe.
1. ArtinianGorensteinalgebrasandtheLefschetzproperties
InthissectionwerecallsomebasicfactsaboutArtinianGorensteinalgebras andthe Lefschetz properties.Foramoredetailedaccount,letsee[12,20,24,19,7].
1.1. StandardgradedArtinian Gorenstein algebrasandHilbertvector
InallthepaperK denotesafieldofcharacteristiczero.
Definition 1.1.Let R = K[x0,. . . ,xn] be the polynomial ring in n+ 1 variables and
I⊂ R beanhomogeneousArtinianidealsuchthatI1= 0.WesaythatagradedArtinian
K-algebraA= R/I =
d
i=0
AiisastandardgradedArtinianK-algebraifitisgeneratedin
degree1 asalgebra.Settinghi= dimKAi,theHilbertvector isHilb(A)= (1,h1,. . . ,hd).
IfI1= 0,then h1 iscalled thecodimensionofA.
Definition 1.2. A standard graded Artinian algebra A is Gorenstein if and only if
dim Ad = 1 and the restriction of the multiplication of the algebra in complementary
degree, that is, Ak × Ad−k → Ad is a perfect paring for k = 0,1,. . . ,d (see [19]). If
Aj= 0 for j > d,thend iscalledthesocledegreeofA.
Remark1.3. SinceAk× Ad−k→ Adisaperfectparingfork = 0,1,. . . ,d,itinducestwo
K-linear maps, Ad−k → A∗k, with A∗k := Hom(Ak,Ad) andAk → A∗d−k, with A∗d−k :=
Hom(Ad−k,Ad),thataretwo isomorphisms.
Let R = K[x0,. . . ,xn] be the polynomial ring in n+ 1 variables. We denote by
Rd=K[x0,. . . ,xn]d theK-vectorspaceofhomogeneouspolynomialsofdegreed.
We denote by Q = K[X0,. . . ,Xn] the ring of differential operators of R, where
Xi := ∂x∂i for i = 0,. . . ,n. We denote by Qk = Q[X0,. . . ,Xn]k the K-vector space
of homogeneousdifferentialoperators ofR ofdegreek.
For eachd≥ k ≥ 0 thereexist naturalK-bilinear mapsRd× Qk → Rd−k definedby
differentiation:
(f, α)→ fα:= α(f ).
Letf ∈ R beahomogeneouspolynomialofdegreedeg f = d≥ 1,wedefine:
Ann(f ) :={α ∈ Q|α(f) = 0} ⊂ Q.
SinceAnn(f ) isahomogeneousidealofQ,wecandefine
A = Q
Ann(f ).
A isastandardgraded ArtinianGorensteinK-algebra.
Conversely,bythetheoryofinversesystems,wegetthefollowingcharacterizationof standardgraded ArtinianGorensteinK-algebras.
Theorem 1.4(Double annihilator theorem of Macaulay).LetR = K[x0,. . . ,xn] and let
Q =K[X0,. . . ,Xn] be the ring of differential operators. Let A = d
i=0
Ai = Q/I be an
Artinian standard graded K-algebra. Then A is Gorenstein if and only if there exists f ∈ Rd suchthat A Q/Ann(f ).
Aproofofthis resultcanbefoundin[19,Theorem 2.1].
Remark1.5.Withthepreviousnotation, letA=
d
i=0
Ai = Q/I bean Artinian
Goren-steinK-algebra with I = Ann(f ),I1 = 0 andAd = 0.Thesocle degree ofA coincides
withthedegreeoftheform f .
Now we deal with standard bigraded Artinian Gorenstein algebras, i.e. Artinian
Gorensteinalgebras,A= d i=0 Ai,suchthat ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Ad= 0 Ak = k i=0
A(i,k−i) for k < d
.
Thepair(d1,d2),suchthatA(d1,d2)= 0 andd1+ d2= d,issaidthesoclebidegreeofA. Remark1.6.Since A∗k Ad−k and sincedualityis compatiblewith directsum,we get
A∗(i,j) A(d1−i,d2−j).
Byabusenotation,we denotethe polynomialring viewedas standardbigradedring intheset of variables {x0, . . . , xn} and {u1, . . . , um} byR = K[x0,. . . ,xn,u1,. . . ,um].
Ahomogeneouspolynomialf ∈ R(d1,d2)issaidtobebihomogeneouspolynomialoftotal degreedeg f = d= d1+ d2 iff canbe writteninthefollowing way:
f =
s
i=1
figi, (3)
where fi∈ K[x0,. . . ,xn]d1 andgi∈ K[u1,. . . ,um]d2,∀i≤ s.
Remark 1.7.All bihomogeneous polynomials f ∈ K[x0,. . . ,xn,u1,. . . um](d1,d2) can be writtenas(3),wherefi∈ K[x0,. . . ,xn]d1 andgi∈ K[u1,. . . ,um]d2,∀i≤ s,are monomi-als.
A homogeneousidealI⊂ R is abihomogeneousidealif
I =
∞
i,j=0
I(i,j)
where I(i,j)= I∩ R(i,j) ∀i,j.LetQ=K[X0,. . . ,Xn,U1,. . . ,Um] betheassociated ring
of differential operators and let f ∈ R(d1,d2) be abihomogeneous polynomial of total degree d= d1+ d2, thenI = Ann(f ) ⊂ Q isabihomogeneous idealand A= Q/I is a
standardbigradedArtinianGorensteinalgebraofsoclebidegree(d1,d2) andcodimension
N = n+ m+ 1.
Remark1.8.Letf ∈ R(d1,d2)be abihomogeneouspolynomialofdegree(d1,d2),andlet
A betheassociatedbigradedalgebraofsoclebidegree(d1,d2),thenfori> d1orj > d2:
I(i,j)= Q(i,j).
In fact forall α∈ Q(i,j) with i > d1 or j > d2 we get α(f ) = 0, so Q(i,j) = I(i,j). As
consequence,wehavethefollowingdecompositionforallAk:
Ak =
i≤d1,j≤d2,i+j=k
A(i,j).
Furthermore for i< d1 and j < d2, theevaluation map Q(i,j)→ A(d1−i,d2−j) given by
α→ α(f) providesthefollowingshort exactsequence:
0 −−−−→ I(i,j) −−−−→ Q(i,j) −−−−→ A(d1−i,d2−j) −−−−→ 0.
1.2. TheLefschetz propertiesandtheHessiancriterion
Definition 1.9.Let A = d i=0 Ai
The algebra A is said to have the Weak Lefschetz Property, briefly W LP , if there existsanelementL∈ A1 suchthatthemultiplicationmap
•L : Ai→ Ai+1
isofmaximalrankfor0≤ i≤ d− 1.
The algebra A is said to have the Strong Lefschetz Property, briefly SLP , if there existsanelementL∈ A1 suchthatthemultiplicationmap
•Lk: A
i→ Ai+k
isofmaximalrankfor0≤ i≤ d and0≤ k ≤ d− i.
A issaidtohavetheStrongLefschetzPropertyin thenarrowsense ifthere existsan elementL∈ A1 suchthatthemultiplicationmap
•Ld−2i: A
i→ Ad−i
isbijectivefori= 0,. . . , d2 .
Remark1.10.InthecaseofstandardgradedArtinianGorensteinalgebras thetwo con-ditionSLPandSLPinthenarrowsense areequivalent.
Definition1.11.Letf ∈ Rdbeahomogeneouspolynomial,letA= d i=0 Ai = Q Ann(f ) be theassociated ArtinianGorenstein algebraandletB = {αj|j = 1, . . . , σk} ⊂ Ak be an
orderedK-basis ofAk.Thek-th Hessianmatrix off with respecttoB is
Hesskf := (αiαj(f ))σi,j=1k .
Thek-thHessian off withrespect toB is
hesskf := det(Hesskf).
Theorem1.12.([28],[19])Letnotationbeasabove.AnelementL= a1X1+ . . . + anXn ∈
A1 is astrongLefschetzelement of A= Q/Ann(f ) ifand only ifhesskf(a1,. . . ,an)= 0
forall k = 0,. . . , d/2 .In particular,if for some k≤ d2 we have hesskf = 0, thenA doesnothave theSLP.
2. HigherorderNagataidealization
2.1. Nagataidealization
Definition2.1. LetA bearingandM beaA-module.TheidealizationofM ,A M,is theproduct setA× M inwhichadditionand multiplicationaredefinedasfollows:
(a, m) + (b, n) = (a + b, m + n) and (a, m).(b, n) = (ab, bm + an). Thefollowing isaknownresultwhoseproofcanbefoundin[12,Theorem2.77]. Theorem 2.2. Let R = K[u1,. . . ,un] and R = K[u1,. . . ,un,x0,. . . ,xn] be polynomial
rings and letQ=K[∂1,. . . ,∂n] and Q =K[∂1,. . . ,∂n,δ0,. . . ,δn] the associatedring of
differential operators. LetI = (g1,. . . ,gm)⊂ Q be anidealgenerated byforms ofdegree
d and let A= Q/Ann(g1,. . . ,gm) be the associatedlevel algebra. Letf = x0g0+ . . . +
xmgm ∈ R be a bihomogeneous polynomial and let A = Q/Ann(f ) be the associated
algebra. Considering I asanA-module, wehave A I A
2.2. Lefschetzpropertiesforhigherorder Nagataidealization
Definition 2.3.Abihomogeneous polynomial
f = s i=0 xd1 i gi∈ K[x0, . . . , xn, u1, . . . , um](d1,d2) (4) iscalledaNagatapolynomialoforderd1,ifthepolynomialsgiarelinearlyindependent
and theydepend onallvariables.
ByTheorem2.2,thealgebraA= Q/Ann(f ) canberealizedasatrivialextensionand itissaidNagataidealizationoforderd1,socledegreed1+ d2andcodimensionn+ m+ 1.
Let R = K[x0,. . . ,xn,u1,. . . ,um] be the polynomial ring and f ∈ R(d1,d2), with
d1 ≥ 1, be apolynomialof typef = n i=0 xd1 i gi,where gi is apolynomialin u1,. . . ,um
variables, for all i = 0,. . . ,m. We denote by Q = K[X0,. . . ,Xn,U1,. . . ,Um] the ring
of differential operators of R, where Xi = ∂x∂i, for i = 0,. . . ,n and Uj = ∂u∂j, for
j = 1,. . . ,m. LetA= Ann(f )Q theassociatedalgebra.
In thecased1< d2,we haveanexamplesuchthatA has theSLP, henceA has the
Example2.4. Letf = x2u3+ y2v3 be abihomogeneous polynomial.HenceA has bide-gree(2,3), Hilbertvector (1,4,6,6,4,1) and A has theSLP. Bythe Hessian criterion, Theorem1.12,therearetwo Hessianstocontrol,hess1f = 0 andhess2f= 0.
Ifthenumberofsummands inf isgreat enough,we getthefollowing Proposition: Proposition2.5([7],Proposition 2.5).Letx0,. . . ,xn andu1,. . . ,umbeindependentsets
of indeterminates with n ≥ m ≥ 2. For j = 1,. . . ,s, let fj ∈ K[x0,. . . ,xn]d1 and
gj ∈ K[u1,. . . ,um]d2 be linearlyindependent forms with 1≤ d1 < d2.If s>
m−1+d1
d1
, thentheformof degree d1+ d2 given by
f = f1g1+· · · + fsgs
satisfies
hesskf= 0
Corollary2.6. LetA beaNagataidealization oforder d1< d2,thenA fails SLP.
Ifweconsiderd1≥ d2, wehavethefollowingProposition:
Proposition 2.7.With the same notations, if d1 ≥ d2, then A has the WLP and L =
n
i=0
Xi isaweakLefschetzelement.
Proof. (Theidea ofthis result wassharedby thework groupinBanff.) We denote by
k = d1+d2
2 .Wenote thatd1≥ k.Infact,byhypothesisd1≥ d2,hence:
d1+ d1≥ d1+ d2⇒ 2d1 2 ≥ d1+ d2 2 ⇒ d1≥ d1+ d2 2 ≥ d1+ d2 2 = k. Wehave: Ak= A(k,0)⊕ A(k−1,1)⊕ · · · ⊕ A(k−d2,d2). Wewantto provethatforL= X0+ . . . + Xn∈ Q[X0,. . . ,Xn]1
•L: A(k−i,i)→ A(k−i+1,i)
has maximal rankfor all i = 0,. . . ,d2. Since A is a standardgraded AGalgebra it is
enoughtocheckitinthemiddle(see[21],Proposition2.1).
Wedenoteωj = Xjk−iαj,whereαj∈ Q[U1,. . . ,Um]i,forj = 0,. . . ,n andwesuppose
j
bjωj= 0⇒ bj = 0.
It impliesthattheαj(gj) arelinearindependentinK(x1,. . . ,xn).
LetΩj= Xjk−i+1αj=•L(ωj),wewanttoprovethat{Ω0,. . . ,Ωn} isalinear
indepen-dent systemfor A(k−i+1,i). Weconsider the following linearcombination
j cjΩj = 0. Bydefinition,weget: 0 = j cjΩj(f ) = j cjΩj i xd1 i gi = j cjxdj1−k+i−1αj(gj).
Since αj(gj) arelinearindependent inK(x1,. . . ,xn),forallj = 0,. . . ,n,wehave
cjxdj1−k+i−1= 0⇒ cj= 0.
Theresultfollows. 2
Forthis case,thereisnothingweareabletosayabouttheSLP.
2.3. Thegeometryof Nagatahypersurfaces oforder d1
Definition 2.8.Let R = K[x0,. . . ,xn,u1,. . . ,um] be the polynomial ring, with K an
algebraically closed field. Let f ∈ R be a Nagata polynomial of order d1 and degree
deg f = d= d1+ d2.
ThehypersurfaceX = V (f )⊂ PN iscalledaNagatahypersurfaceoforderd
1.
Let X = V (f ) ⊂ PN be a Nagata hypersurface of order d1. We can consider two
linear space respectively Pm−1 with coordinates u1,. . . ,um and Pn with coordinates
x0,x1,. . . ,xn. Letpα∈ Pm−1 be apoint and we consider thefollowing linear space of
dimensionn+ 1:
Lα:=pα,Pn = {pα, q : q ∈ Pn} .
IfweconsidertheintersectionLαwithX,weobtainavarietyYα.Yαisreduciblewhose
irreduciblecomponentsarethelinearspacePn andavariety,calledresidue anddenoted
byY˜α.Y˜α isaconeofvertexpαover a(n− 1)-dimensionalbasis.
Theorem 2.9. A Nagata hypersurface X = V (f )⊂ PN of order e consists of theunion of theresidueparts Y˜α,i.e.
Proof. Fixedapoint pα= (0: . . . : 0: a1: . . . : am)∈ Pm−1 andletp = (x0: . . . : xn :
0: . . . : 0) be apointinPn.Weconsider theline thatjoinsthepointspα andp:
Lα: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x0= λx0 · · · · xn = λxn u1= μa1 · · · · um= μam withλ,μ∈ K.
SinceX = V (f ) isaNagatahypersurfaceoforderd1, wehave:
f = xd1
0 g0+ . . . + xdn1gn.
Ifweconsider theintersectionbetweentheline Lαand theNagatahypersurfaceX,
weget:
fLα = λ
d1x
0d1g0(μa1, . . . , μam) + . . . + λd1xnd1gn(μa1, . . . , μam) = λd1μd2 n
i=0
xid1gi(a)
wherea isthevector(a1,. . . ,am).
Sincepα andp arepointsofX,then n i=0 xid1gi(a)= 0.Therefore ˜ Yα= V n i=0 xid1gi(a)
and,byarbitrarinessofthepointspα∈ Pm−1 andp∈ Pn,wehave∪αY˜α= X. 2
Asconsequence ofthe abovetheorem,we cansayhow manylinearspaces there are
inaNagatahypersurfaceofordere.WenotethatPm−1andPn arelinearspacesonX.
Thuswehave:
Corollary 2.10.Let X = V (f )⊂ PN be a Nagata hypersurface of order d
1.There is a
familyof linesof dimensionm+ n− 1 onX.
Proof. Let pα ∈ Pm−1 be apoint, then there is afamily of lines of dimension n that
joinspαandthelinearspacePn,forallpα∈ Pm−1.ThisfamilycoversY˜α.Thenwehave
afamilyoflinesofdimension(n)+ (m− 1)= n+ m− 1 onX.ThesingularlocusofX
Conversely, letp∈ Pn be apoint, thenthere isafamilyof linesof dimensionm− 1
thatjoinsp andallpointsq inthelinearspace Pm−1. Sotheprooffollows. 2 3. SimplicialNagataidealizationof orderk
Definition 3.1.Abihomogeneous polynomial
f =
n
i=0
xkigi∈ K[x0, . . . , xn, u1, . . . , um](k,d−k) (5)
is calledasimplicialNagatapolynomialoforderk ifallgi aresquare freemonomials.
Thefollowing combinatorialconstructionswereinspiredby[9].
Definition3.2. LetV ={u1, . . . , um} beafiniteset.Asimplicialcomplex Δ withvertex
set V is acollectionofsubsets ofV , i.e.asubsetof thepowerset 2V,suchthatfor all
A∈ Δ andforallsubsetB⊂ A,wehaveB∈ Δ.
WesaythatΔ isasimplexifΔ= 2V.
Themembers ofΔ arereferredas faces andthemaximalfaces(respectto the inclu-sion)are thefacets.Thevertex setofΔ is alsocalled 0-skeleton.IfA∈ Δ and|A|= k, it iscalled a(k− 1)-face,or aface ofdimension k− 1: the0-facesarethe verticesand the1-facesarecalled edges.
Definition3.3. Ifallthefacetshavethesamedimensiond> 0,thecomplexissaidtobe
pure.
Let Δ be a pure simplicial complex of dimension d > 0 with vertex set V =
{u1, . . . , um}, we denote by fk the number of (k− 1)-faces, hence f0 = 1, f1 = m,
fd+1 isthenumberoffacetsofΔ andfj= 0,forj > d+ 1.
Remark3.4.Thereisanaturalbijectionbetweenthesquarefreemonomials,ofdegree r, in the variables u1,. . . ,um, and the (r− 1)-faces of the simplex 2V, with vertex set
V = {u1, . . . , um}. In fact, a square free monomial g = ui1· · · uir, in the variables
u1,. . . ,um, correspondsto thefinite subsetof 2V given by{ui1, . . . , uir}.Toany finite subsetF of2V,weassociate themonomial mF =
ui∈F
ui ofsquarefreetype.
An important resultabout simplicial Nagata idealization canbe found in [9, Theo-rem 3.2].
3.1. Simplicial Nagataidealization oforder k
Letf ∈ K[x0,. . . ,xn,u1,. . . ,um](k,k+1) beasimplicialNagatapolynomialoforderk:
f =
n
r=1
xkrgr (6)
withgr monomialsinvariables u1,. . . ,umofdegreek + 1.
Wewantto characterizethe Hilbertvectorof thealgebras associated to theNagata polynomialoftype(6).
LetΔ beapuresimplicialcomplexofdimensionk,withvertexsetV ={u1, . . . , um}.
Wedenotebyfk thenumberof (k− 1)-faces,hencef0= 1,f1= m,fk+1 isthenumber
ofthefacetsofΔ andfj = 0 forj > k + 1.
Thefacetsof Δ,associated tof , correspondingto themonomials gi,will belabeled
bygi.TheassociatedalgebraisAΔ= Q/Ann(fΔ).Byabuseofnotation,wewillalways
denotefΔwithf andAΔwithA.
Ifp∈ K[u1,. . . ,um] isasquare freemonomial,we denoteby P ∈ K[U1,. . . ,Um] the
dualdifferentialoperatorP = p(U1,. . . ,Um).
Theorem3.5. Letf ∈ K[x0,. . . ,xn,u1,. . . ,um](k,k+1) beasimplicialNagata polynomial
oforder k: f = n r=1 xkrgr
with gr monomials in variables u1,. . . ,um of degree k + 1. Let Δ be a pure simplicial
complex ofdimensionk and letA= Q/Ann(f ). Then
A =
d=2k+1
i=0
Ai where Ai= A(i,0)⊕ A(i−1,1)⊕ · · · ⊕ A(0,i), Ad = A(k,k+1)
(1) forallj = 1,. . . ,k + 1: dim A(i,j)= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ fj for i = 0 (n + 1)· fj for 1≤ i < k fk+1−j for i = k
where fj isthenumberof thesubfaces,of dimensionj− 1, of thefacet, gi,ofΔ.
(2) I = AnnQ(f ) is generatedby
(a) X0, . . . , Xnk+1 andU12,. . . ,Um2;
(c) the monomialsXriPr, fori = 1,. . . ,k, such that, fixed thefacet Mr of Δ,
cor-responding to the monomial gr, Pr is the dual differential operator of pr; pr is
a monomial in the variables u1,. . . ,um, corresponding to a face M of Δ s.t.
M∩ Mr=∅;
(d) thebinomialsXrkG˜r−XskG˜swheregr= ˜grgrsandgs= ˜gsgrsandgrsrepresents
acommon subface of gr,gs.
Proof. (1) Let f beof type(6) associatedto thepure simplicialcomplex Δ of dimen-sion k.Thevariables u1,. . . ,umrepresenttheverticesofΔ.
Weconsiderthefollowing cases:
• for i = 0 and j = 1,. . . ,k + 1, A(0,j) is generated by the only monomials of
degreej,inthevariablesU1,. . . ,Uk+1,thatdonotannihilatef .Thesemonomials
represent(j− 1)-facesofΔ.Weneedtoshowthattheyarelinearlyindependent overK.
Consider{Ω1, . . . , Ων} asystemofmonomialsofQ(0,j),whereΩs,fors= 1,. . . ,ν,
isassociated toany(j− 1)-faceω. Wetakeanylinearcombination:
0 = ν r=0 crΩr(f ) = ν r=0 cr n s=0 xksΩr(gs) = n s=0 xs ν r=0 Ωr(gs). Therefore we get ν r=0
crΩr(gs) = 0, for all s = 0,. . . ,n. For each r = 0,. . . ,ν,
there is a s = 0,. . . ,n, such that if Ωr(gs) = 0, then cr = 0 for all r. Hence
dim A(0,j) = fj,where fj isthenumberof(j− 1)-facesofΔ.
• for 1≤ i< k andj = 1,. . . ,k + 1,thegeneratorsof A(i,j) arethemonomialsof
typeXi
sUr1Ur2· · · Urj fors= 0,. . . ,n,forall j.Fixs= 0,. . . ,n,and letMs be the facet of Δ, corresponding to themonomial gs, the monomial Ur1Ur2· · · Urj of Q(0,j) is the dual differential operator of the monomial ur1ur2· · · urj, that gives the (j− 1)-dimensional subfaces of Ms. The monomials XsiUr1Ur2· · · Urj for s = 0,. . . ,n, for all j are linearly independent. In fact, denotingby Ωi
s the
monomial Xi
sUr1Ur2· · · Urj,fors= 0,. . . ,n,wenotethat: Ωis(f ) = cxk−is (Ur1· · · Urj)(gs)= 0
since (Ur1· · · Urj)(gs) identifies thevertices of the (j− 1)-dimensionalface. We get:
n
s=0
csΩis(f ) = 0⇔ cs= 0 ∀s.
For s = 0,. . . ,n, in correspondence of Ωsi(f ), we can get a number of (j −
1)-dimensional faces of Δ. Denoting such number by fj, we have dim A(i,j) =
• fori= k andj = 1,. . . ,k, bydualityA∗(0,k+1−j) A(k,j),thencewehave:
dim A(k,j)= dim A∗(0,k+1−j)= fk+1−j.
(2) LetI = Ann(f ) betheannihilator.Weconsiderthefollowingexactsequence: 0 −−−−→ I(i,j) −−−−→ Q(i,j) −−−−→ A(k−i,k+1−j) −−−−→ 0. (7)
We havethefollowingcases:
• fori= 0 and1≤ j ≤ k + 1,wehaveby(7)
dim A(0,j)= fj ⇒ dim I(0,j) = dim Q(0,j)− fj.
Since A(0,j) hasa basisgiven by the(j− 1)-faces of Δ,then I(0,j) is generated
bymonomialsrepresentingallthe(j− 1)-facesofthecomplementof Δ.InI,it isenoughto considertheminimalfaces ofΔc, bydefinitionofideal.
WenotethatinI(0,2)there arealsothemonomialsU12,. . . ,Um2,sincethe
mono-mialsgi,inthevariablesu1,. . . ,umaresquare free.
• for1≤ i< k and1≤ j ≤ k + 1,fixthefacetMrofΔ,correspondingtogr,since
A(i,j) hasabasis given bythe (j− 1)-dimensional subfaces ofMr, then I(i,j) is
generatedbymonomialsXi
rPr wherePristhedualdifferentialoperatorofpr;pr
isamonomialinthevariablesu1,. . . ,um,correspondingtoa(j− 1)-dimensional
faceMr s.t.Mr∈ Δc orMr∈ Δ andMr∩ Mr=∅.
• fori= k and1≤ j ≤ k + 1,wefixtwofacetsofΔ,MrandMs,correspondingto
themonomialsgr and gs, andsuch thatMr∩ Ms= ∅. LetMrs= Mr∩ Ms; we
denotethemonomialcorrespondingtoitbygrs.WeconsiderM˜r= Mr\Mrsand
˜
Ms = Ms\Mrs. Letg˜r and g˜s be themonomials corresponding to M˜r and M˜s.
Wenote thatM˜r∩ ˜Ms = ∅.Hence thebinomials, XrkG˜r− XskG˜s, are inI(k,j),
whereG˜rand G˜s arethedualdifferentialoperators of˜grand ˜gsrespectively.
Letusconsiderthefollowingexactsequence:
0 −−−−→ I(k,j) −−−−→ Q(k,j) −−−−→ A(0,k+1−j) −−−−→ 0,
we get dim I(k,j) = dim Q(k,j)− fk+1−j. Let Q˜(k,j) be the K-space spanned by
allthemonomialsXk
rG˜r, whereG˜r isthedual differentialoperatorofgr thatis
amonomial inthe variables u1,. . . ,um, corresponding to a subface of Mr. Let
I(k,j) ⊂ I(k,j) be the K-vector space spanned by the monomials XrkPr, where
Pr isthedual differentialoperatorofthemonomial, inthevariablesu1,. . . ,um,
pr, not corresponding to a subface of Mr. They are two K-vector spaces s.t.
Q(k,j)= ˜Q(k,j)⊕ I(k,j).Weconsider theidealI˜(k,j)⊂ ˜Q(k,j).Theexactsequence
givenbyevaluationrestrictedtoQ˜(k,j) becomes:
Wenote:
dim I(k,j)= dim Q(k,j)− fk+1−j = dim ˜Q(k,j)+ dim I(k,j)− fk+1−j =
= dim ˜I(k,j)+ fk+1−j+ dim I(k,j)− fk+1−j = dim ˜I(k,j)+ dim I(k,j).
Hence I(k,j) = ˜I(k,j)⊕ I(k,j). Thegenerators of I˜(k,j) are the binomialXrkG˜r−
XskG˜s precisely.Theresultfollows.
Moreoverfori= k + 1 andj = 0,itisclearthatI(k+1,0)= (X0,. . . ,Xn)k+1.Infact
Xik+1(f )= 0,fori= 0,. . . ,n,sincethemonomialsinx0,. . . ,xn off havedegreek,
byRemark1.8. 2
Wediscuss thefollowing example:
Example 3.6.LetV ={u1,. . . ,u6} beafiniteset.Wehave:
2V ={∅, {u1}, . . . , {u6}, . . . , {u1, . . . , u6}}
LetΔ bethefollowingsimplicialcomplex: Δ ={∅, {u1}, . . . , {u6} vertices ,{u1, u2}, . . . , {u5, u6} edges ,{u1, u2, u3}, . . . , {u2, u3, u6} 2-faces }
It is given by two pyramids, with the common basis, of vertices u1,. . . u5 and u6 and
faces labeledbyg0,. . . ,g6 andg7:
u1 u2 u3 u4 u5 u6 g0 g1 g2 g3 g4 g5 g6 g7
The2-facesarethefacetsofΔ,thenΔ ispureofdimension2. Let
f = fΔ= x20u1u2u3+ x21u1u2u4+ x22u1u4u5+ x23u1u3u5+
bethebihomogeneouspolynomialofdegree5.ItisaNagatapolynomialoforder2 and the monomials g0 = u1u2u3, g1 = u1u2u4, g2 = u1u4u5, g3 = u1u3u5, g4 = u2u3u6,
g5= u2u4u6,g6= u4u5u6 andg7= u3u5u6 areofsquarefreetype.
Wehave
A = A0⊕ A1⊕ A2⊕ A3⊕ A4⊕ A5
andtheHilbertvectorisgivenby:
h0= 1 = h5 and h1= 14 = h4.
Wecalculateh2= dim A2 andh3= dim A3.
ByTheorem 3.5,wehave
h2= dim A2= dim A(2,0)+ dim A(1,1)+ dim A(0,2)=
= f3+ 8· f1+ f2= 8 + 8· 3 + 12 = 44
h3= dim A3= dim A(3,0)+ dim A(2,1)+ dim A(1,2)+ dim A(0,3)=
= 0 + f2+ 8· f2+ f3= 12 + 8· 3 + 8 = 44
HencetheHilbertvectoris(1,14,44,44,14,1). ByTheorem 3.5,I = Ann(f ) isgenerated by: • X0,. . . ,X73andU12,. . . ,Um2,bythepart(2a);
• sincethecomplementofΔ is:
Δc={{u1, u6} , . . . , {u2, u5} diagonals ,{u1, u2, u6} , . . . , {u1, u5, u6} 2-faces , {u1, u2, u3, u5} , . . . , {u2, u3, u5, u6} 3-faces , . . . ,{u1, . . . , u6}}
and sincediagonals are theminimal faces of Δc, then the monomials U
1U6, U3U4
• Fori= 1,2,fixthefacet M0={u1,u2,u3}∈ Δ,correspondingto themonomialg0,
wehavethatthemonomialp0represents:
– oneoftheremainingvertices,forexampleu4:
u1 u2 u3 u4 u5 u6 g0 g1 g2 g3 g4 g5 g6 g7
andfinallyweget:P0= p0(U1,. . . ,U8)= U4.Thencethemonomialofdegreei+1,
Xi
0U4 isinI = Ann(f ). Theother monomialsof thistype areobtainedwith the
sameprocedure.
– oneoftheremainingedges,forexampletheedgethatjoinstheverticesu5andu6:
u1 u2 u3 u4 u5 u6 g0 g1 g2 g3 g4 g5 g6 g7 Weget: P0= p0(U1, . . . , U8) = U5U6
and the monomialof degreei+ 2, Xi
0U5U6, isinI = Ann(f ),by part(2c). The
• Thefacesg0andg3havethecommonedgethatjoinstheverticesu1u3: u1 u2 u3 u5 u6 g0 g1 g2 g3 g4 g5 g6 g7
g1,3 representsthe edge that joinsthe vertices u1 and u3. g˜0 and g˜3 represent the
vertices u2 andu5respectively.Wehave:
˜
G0= ˜g0(U1, . . . , U6) = U2and ˜G3= ˜g3(U1, . . . , U6) = U5.
Thebinomial,ofdegree3,X2
0U2−X32U5,isinI = Ann(f ) bythepart(2d).Theother
binomialsofdegree3 ofthistypeareobtainedbythesameprocedure.Wenotethat thefacesg0andg2havethecommonvertexu1,hence,intheidealI = Ann(f ) there
is the binomial, of degree 4, X02U2U3− X22U4U5; the other binomials,of degree 4,
areobtainedbythesameprocedure. Acknowledgments
The second author would thank M. Boij, J. Migliore, U. Nagel, A. Seceleanu,
H. Schenck and J. Watanabe for the very stimulating discussions about the subject
inBanff.I wouldlike tothank alsoalltheorganizers andtheparticipants ofworkshop
Lefschetz Properties and Artinian Algebras at BIRS, Banff, Canada in Mach, 2016. It wasaverypleasanttimetodoMathematicsandenjoyaveryniceplace.
Thesecond authorthanksCarolinaAraujoforusefuldiscussionsaboutthegeometry ofNagatahypersurfaces,comparingitwiththeclassicalcase.
The authors want to thank P. de Poi for his import remarks and comments on a
previousversionofthemanuscript.
Thesecond author was partially supportedby ICTP-INdAM Research inPairs
Fel-lowship2018/2019andbyFACEPE ATP- 0005-1.01/18.
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