Lefschetz properties for Artinian Gorenstein algebras presented by quadrics

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by  Rodrigo Gondim

LEFSCHETZ PROPERTIES FOR ARTINIAN GORENSTEIN ALGEBRAS
PRESENTED BY QUADRICS
6
1
RODRIGOGONDIM*ANDGIUSEPPE ZAPPALA`
0
2

n Abstract. We introduce a family of standard bigraded binomial Artinian Gorenstein al-
a gebras, whose combinatoric structure characterizes the ones presented by quadrics. These
J algebras provide, for all socle degree grater than two and in sufficiently large codimension

8 with respecttothesocle degree,counter-examplestoMigliore-Nagel conjectures,see[MN1]
1 and[MN2]. OneofthempredictedthatArtinianGorensteinalgebraspresentedbyquadrat-
ics should satisfy the weak Lefschetz property. We also prove a generalization of a Hessian

] criterion fortheLefschetzpropertiesgivenbyWatanabe,see[Wa1]and[MW],whichisour
C
main tool to control the Weak Lefschetz property.
A
.
h
t
a
1. Introduction
m
[ A standard graded K algebra is said to be presented by quadrics if it is isomorphic to
the quotient of a polynomial ring over K by a homogeneous ideal generated by quadratic
1
v forms. Also called quadratic algebras, they are related with Koszul algebras and Grobner
4 basis, see for example [Co]. From a more geometric point of view quadratic ideals appear
5
as homogeneous ideals of very positive embeddings of any smooth projective varieties. As
4
pointed out in [MN2], Artinian Gorenstein algebras presented by quadrics are also related to
4
0 Eisenbud-Green-Harris conjectures motivated by the Cayley-Bacharach theorem. The Lef-
1. schetz properties have attracted a great deal of attention over the years, in a number of
0 different settings including Commutative Algebra itself, Algebraic Geometry and Combina-
6
torics, see [St, St2, MN1, HMMNWW, HMNW]. The present work lies in the border of these
1
three areas.
:
v
More recently, in [MN2] the authors studied Artinian Gorenstein algebras presented by
i
X quadrics, they presented some constructions of such algebras and described their possible
r Hilbert vectors. In [MN1] and [MN2] the authors proposed two conjectures, which, from now
a
on, we will call Migliore-Nagel conjectures. These conjectures inspired the present paper.
Conjecture 1.1. (Migliore-Nagel injective Conjecture) For any Artinian Gorenstein
algebra presented by quadrics, defined over a field K of characteristic zero, and of socle degree
atleast three, there exists L ∈ A , suchthat, the multiplication map •L :A → A isinjective.
1 1 2
Conjecture 1.2. (Migliore-Nagel WLP Conjecture) Any Artinian Gorenstein algebra
presented by quadrics, over a field K of characteristic zero, has the Weak Lefschetz Property.
That is, there exists L ∈ A such that all the maps •L :A → A have maximal rank.
1 i i+1
In [MN2] the authors proved the WLP-conjecture for complete intersection of quadratic
forms and presented computational evidence for the conjectures in low codimension. We
*Partially supported bythe CAPES postdoctoral fellowship, Proc. BEX 2036/14-2.
1
2 R.GONDIMANDG.ZAPPALA`
want to stress the fact that as soon as the codimension increases with respect to the socle
degree surprising phenomena begin to appear.
WerecallthattheLefschetzpropertiesforstandardgradedArtinianKalgebraarealgebraic
abstractions introduced by Stanley in [St], motivated by the Hard Lefschetz Theorem on
the cohomology rings of smooth irreducible complex projective varieties endowed with the
euclidean topology, see for example the survey [La] for the theorem and [Ru] for an overview.
The Poincar´e duality for these cohomology rings inspired the definition of Poincar´e duality
algebras which, in this context, is equivalent to the Gorenstein hypothesis, for more details
see, for example [MW] and [Ru]. In [Wa2] and [MW] the authors used Macaulay-Matlis
duality in characteristic zero to present the Artinian Gorenstein algebra as A= Q/Ann (f)
Q
where f ∈ R = K[x ,...,x ] a polynomial ring and Q = K[X ,...,X ] the associated ring
1 n 1 N
of differential operators. They introduced a Higher Hessian determinantal criterion using a
basis of A to identify Strong Lefschetz elements L ∈ A . In [Go] this criterion was used
k 1
by the first author to contruct explict examples of Artinian Gorenstein algebras that do not
satisfy the Strong Lefschetz property and/or the Weak Lefschetz property, producing forms
with vanishing Hessian of higher order.
Hypersurfaceswithvanishing(first)HessianisaclassicalsubjectthattracesbacktoHesse’s
wrong claim, see [He], that a form has vanihing Hessian if and only if its partial derivatives
are linearly independent. Gordan-Noether in the fundamental paper [GN] proved a criterion
to the vanishing of the Hessian, see Proposition 2.11. They showed that Hesse’s claim is true
for at most four variables and produced series of counterexamples for at least five variables,
these counterexamples exaust all the examples in five variables. A modern proof of this fact
can be found in [GR] while a very detailed account on the subject appears in [Ru]. Perazzo
in [Pe] studied cubic hypersurfaces with vanishing Hessian in at most seven variables and
found a canonial form that motivated our definition of Perazzo hypersurfaces also in degree
d ≥ 3, see Definition 2.12. More recently [CRS], [Lo], [GR], [GRu], [Wa2] and [Ru] treated
different aspects of the subject, sometimes called Gordan-Noether Theory.
In [GRu], which contain results from the first author’s thesis, Perazzo’s work has been
revisited from a modernpoint of view, re-obtaining the classification of cubics with vanishing
Hessian in PN for N ≤ 6 and describing new examples in higher dimension. An important
example of cubic with vanishing Hessian in P7, is given by a general tangent section of the
secant variety of the Segre variety P2×P2 → Seg(2,2) ⊂ P8, see [GRu, p. 803, Example 6].
From the algebraic point of view, see Example 4.8, it provides a standard graded Artinian
Gorenstein algebra presented by quadrics which does not have the WLP, being a counter
example for both Migliore-Nagel conjectures. This example was the starting point of our
work.
Our strategy to construct standard graded Artinian Gorenstein algebras presented by
quadrics is to deal with the simplest ones, which are those whose ideal contains the complete
intersection (x2,...,x2). This assumption forces all monomials that occur in f to be square
1 n
free. As a matter of fact we deal with bihomogeneous forms of bidegree (1,d−1) of special
type, here called of monomial square free type, see Definition 2.7, and note the similarity
with Perazzo polynomial. To any bihomogeneous form of monomial square free type we can
associateinabijectivewayapuresimplicialcomplexwhosecombinatoricstructuredetermine
a set of generators of the annihilator ideal, see Theorem 3.1. This combinatoric object also
characterizes when the associated algebra is presented by quadrics, see Theorem 3.3, wich
LEFSCHETZ PROPERTIES FOR ARTINIAN GORENSTEIN ALGEBRAS PRESENTED BY QUADRICS 3
summarizes the main results of the work. Inspired by the famous Turan Theorem, see [Tu],
that characterizes maximal graphs not containing a complete subgraph K , we introduce a
l
simplicial complex, here called Turan complex, whose associated algebra is always presented
by quadrics and such that, in sufficient large codimension and arbitrary degree the first
multiplication map •L : A → A is not injective for all L ∈ A . Furthermore, for a
1 2 1
particular subfamily, which occurs in very large codimension with respect to thesocle degree,
the Hilbert vector of A is not even submodal in the first step, that is dimA > dimA .
1 2
We now describe the contents of the paper in more detail. In the first section we recall the
basic definitions and constructions of standard graded Artinian Gorenstein algebras, we deal
alsowiththebigradeddcasewhichis of particularinterest, werecalltheLefschetz properties,
Macaulay-Matlis duality and the hessian criterion given in [Wa1] and [MW]. Themain result
of the firstsection is a generalization of the Hessian criterion to mixed Hessians, see Theorem
2.10. We also recall classical results about forms with vanishing Hessian that will be useful
in the sequel.
The second section is devoted to the main results and constructions. Theorem 3.1 de-
scribes the annihilator of a standard bigraded form of bidegree (1,d−1) of monomial square
free type, showing that it is a binomial ideal whose generators are closely connected with
the combinatoric of the associated simplicial complex. Theorem 3.3 is the main result. It
characterizes when such algebras are presented by quadrics. In this section we introduce the
Turan complex and as Corollary 3.9, we produce counterexamples to both Migliore-Nagel
conjectures in any socle degree and sufficient large codimension. It is surprising the fact
that if the codimension is very large with respect to the socle degree, the Hilbert vector of
the Turan algebras, that are quadratic and binomial, is non unimodal in the first step, an
unexpected and very surprising fact.
In the last section we deal with an inductive construction, see Corollary 4.4. In the second
subsection we give a complete description of bigraded algebras of monomial square free type
of socle bidegree (1,2) that are presented by quadrics (see Theorem 4.11). In Corollary 4.12
we show that for all codimension r ≥ 8 there are algebras of socle degree 3 presented by
quadrics that does not satisfy the WLP. Furthermore the bound is sharp by Corollary4.7.
For higher odd degree we prove the existence of algebras presented by quadrics that do not
satisfy the WLP in all codimensions r ≥ d+5 (see Corollary 4.14). The bound is relatively
sharp by the result for d = 3. In the last subsection we study the case of even socle degree,
showing the strong difference with the odd socle degree case. We prove the existence of
algebras of socle degree 4 and of codimension r ≥ 16 that are presented by quadrics but not
having the WLP, see Corollary 4.17. In the similar way we deal with the even socle degree
in general (see Corollary 4.19). In even socle degree we are not able to give sharp bounds
even in socle degree 4. The problem is that, since there is not a classification of quartics with
vanishing Hessian in PN for N ≥ 5, there is not an analog of Corollary 4.7.
4 R.GONDIMANDG.ZAPPALA`
2. Lefschetz properties, Gordan-Noether theory and the Hessian criteria
2.1. The Lefschetz Properties and Macaulay-Matlis duality. Let K be an infinite
field and R = K[x ,...,x ] be the polynomial ring in n indeterminates.
1 n
Definition 2.1. Let A be a standard graded K-algebra. We say that A is presented by
quadrics if A ≃ R/I, where R = K[x ,...,x ] and the homogeneous ideal I has a set of
1 n
generators formed by quadratic forms.
Let A = R/I be an Artinian standard graded R-algebra. If F ∈ R is a form, it induces a
d
K-vector spaces map ϕ : A → A , defined by ϕ (α) = Fα, for every α∈ A .
i,F i i+d i,F i
Definition 2.2. We say that A has the Weak Lefschetz property (in short WLP) if there
exists a linear form L ∈ R1 such that rkϕi,L = min{dimKAi,dimKAi+1}, for every i.
Definition 2.3. We say that A has the Strong Lefschetz property (in short SLP) if there
exists a linear form L ∈ R1 such that rkϕi,Ld = min{dimKAi,dimKAi+d}, for every i and d.
Definition2.4. LetA= R/I beanArtinianstandardgradedR-algebra. IfR = K[x ,...,x ]
1 n
andI = 0, thenn is saidto bethecodimension ofA. If A 6= 0andA = 0forall i> d, then
1 d i
discalled thesocledegreeofA. TheHilbertvector ofAish = Hilb(A) = (1,h ,h ,...,h ),
A 1 2 d
where h = dimA . We say that h is unimodal if there exists k such that 1 ≤ h ≤ ... ≤
k k A 1
h ≥ h ≥ h .
k k+1 d
d
Remark 2.5. Werecall thatanArtinian algebraA= A ,A 6= 0,isaGorenstein algebra
i d
i=0
M
if dimKAd = 1 and the bilinear pairing
A ×A → A
i d−i d
inducted by the multiplication is perfect for 0 ≤ i ≤ d. So we have an isomorphism A ≃
i
HomK(Ad−i,Ad) for i = 0,...,d. In particular dimKAi = dimKAd−i, for i = 0,...,d.
Moreover for every L ∈R , rankϕ = rankϕ , for 0≤ i ≤ d.
1 i,L d−i−1,L
Since A as a R-module is generated in degree 0, if ϕ is surjective, then ϕ is surjective
i,L j,L
for every j ≥ i. Therefore, if A is a Gorenstein Artinian algebra, if ϕ is injective, then
i,L
d
ϕ is injective for every j ≤ i. Consequently a Gorenstein Artinian algebra A = A ,
j,L i
i=0
M
Ad 6=0, has the WLP iff there exists a linear form L ∈ R1 such that ϕd−1,L is injective when
2
d is even or ϕd−1,L is injective (or surjective) when d is odd. We recall that for Gorenstein
2
Artinian algebras the SLP is equivalent to the SLP in the narrow sense, that is ϕi,Ld−2i to be
an isomorphism. Notice also that the WLP implies the unimodality of the Hilbert vector of
A. Unimodality in the Gorenstein case implies that dimA < dimA for all k ≤ d.
k−1 k 2
Now we assume that charK = 0. Let us regard the polynomial algebra R as a module over
the algebra Q = K[X ,...,X ] via the identification X = ∂/∂x . If f ∈ R we set
1 n i i
Ann (f)= {p(X ,...,X )∈ Q | p(∂/∂x ,...,∂/∂x )f = 0}.
Q 1 n 1 n
By Macaulay-Matlis duality we have a bijection:
LEFSCHETZ PROPERTIES FOR ARTINIAN GORENSTEIN ALGEBRAS PRESENTED BY QUADRICS 5
{Homogeneous ideals of R} ↔ {Graded Q modules}
Ann (M) ← M
Q
I → I−1
Let I ⊂ Q be an homogeneous ideal. It is well known that A = Q/I is a Gorenstein
standard graded Artinian algebra if and only if there exists a form f ∈ R such that I =
Ann (f) (for more details see, for example, [MW]).
Q
In the sequel we always assume that char(K) = 0, A = Q/I, I = Ann (f) and I = 0.
Q 1
Sometimes we also assume that K is algebraically closed. All arguments work over C.
d
We deal with standard bigraded Artinian Gorenstein algebras A = A , A 6= 0, with
i d
i=0
M
k
A = A , A 6= 0 for some e ,e such that e +e = d, we call (d ,d ) the socle
k (i,k−1) (d1,d2) 1 2 1 2 1 2
i=0
bidegrMee of A. Since A∗ ≃ A and since duality is compatible with direct sum, we get
k d−k
A∗ ≃ A .
(i,j) (d1−i,d2−j)
Let R = K[x ,...,x ,u ,...,u ] be the polynomial ring viewed as standard bigraded ring
1 n 1 m
in the sets of variables {x ,...,x } and {u ,...,u } and let Q = K[X ,...,X ,U ,...,U ]
1 n 1 m 1 n 1 m
be the associated ring of differential operators.
We want to stress that the bijection given by Macaulay-Matlis duality preserves bigrading,
that is, there is a bijection:
{Bihomogeneous ideals of R} ↔ {Bigraded Q modules}
Ann (M) ← M
Q
I → I−1
If f ∈ R is a bihomogeneous polynomial of total degree d = d + d , then I =
(d1,d2) 1 2
Ann (f) ⊂ Q is a bihomogeneous ideal and A = Q/I is a standard bigraded Artinian
Q
Gorenstein algebra of socle bidegree (d ,d ) and codimension r = m + n if we assume,
1 2
without lost of generality, that I = 0.
1
Remark 2.6. If f ∈ R is a bihomogeneous polynomial of bidegree (d ,d ), consider
(d1,d2) 1 2
the associated bigraded algebra A of socle bidegree (d ,d ). Notice that for all α ∈ Q
1 2 (i,j)
with i > d or j > d we get α(f) = 0, therefore, in these conditions I = Q . As
1 2 (i,j) (i,j)
consequence, we have the following decomposition for all A :
k
A = A .
k (i,j)
i+j=k,Mi≤d1,j≤d2
Furthermore,fori < d andj < d ,theevaluationmapQ → A givenbyα 7→ α(f)
1 1 i,j (d1−i,d2−j)
provides the following short exact sequence:
0 → I → Q → A → 0.
(i,j) (i,j) (d1−i,d2−j)
One of our goals is to produce bigraded algebras of socle bidegree (1,d−1) presented by
quadrics. In order to achieve this objective we study the ideal of a particular family.
6 R.GONDIMANDG.ZAPPALA`
Definition 2.7. With the previous notation, all bihomogeneous polynomials of bidegree
(1,d−1) can be written in the form
f = x g +...+x g ,
1 1 n n
whereg ∈ K[u ,...,u ] . Wesay thatf isof monomial square free type ifallg aresquare
i 1 m d−1 i
free monomials. The associated algebra, A = Q/Ann (f), is bigraded, has socle bidegree
Q
(1,d−1) and we assume that I = 0, so codimA = m+n.
1
We enhance that the combinatoric structure inward bihomogeneous polynomials of mono-
mial square free type, at least in low degree, allows us to give necessary and sufficient con-
ditions in order to the associated algebra to be presented by quadrics. On the other hand,
Gordan-NoethertheoryofformswithvanishingHessianandtheLefschetz-Hessian criteriaal-
low us to construct, in sufficiently large codimension, Artinian Gorenstein algebras presented
by quadrics faling the WLP.
2.2. The Hessian criteria for Strong and Weak Lefschetz properties. Our point of
view in order to study the Lefschetz properties for standard graded Artinian Gorenstein
algebras, A = Q/Ann (f), was inspiredby [MW], wherethe authors proved adeterminantal
Q
Hessian criterion, identifying Strong Lefschetz elements. In [Go] the first author studied and
generalized class of examples given in [MW]. Now we prove a generalization of the Strong
Lefschetz criterion using mixed Hessians. Let us introduce this natural object.
d
Definition 2.8. Let A = Q/Ann (f) = A be a standard graded Artinian Gorenstein
Q k
k=0
M
K algebra of socle degree d. Let i ≤ j ≤ d be two integers and let B = {α ,...,α }
2 k 1 s
and B = {β ,...,β } be bases of the K-vector spaces A and A respectively. The (mixed)
l 1 t k l
hessian matrix of f of order (k,l) is the matrix:
(k,l)
Hess = (α (β (f))) .
f i j s×t
We denote Hessk := Hess(k,k), hessk := det(Hessk) and hess := hess1.
f f f f f f
Remark 2.9. We observe that, under the natural assumption that Ann (f) 6= 0, the nota-
Q 1
tion hess is consistent with the classical definition of Hessian, by taking B = {X ,...,X },
f 1 1 n
the standard basis of the embedding.
If A is bigraded, and if B = {α ,...,α } and B = {β ,...,β } are bases of the K-vector
k 1 s l 1 t
((k,l),(k′,l′))
spaces A(k,l) and A(k′,l′) respectively, we can also define Hessf = (αi(βj(f)))s×t.
We now prove a generalization of [Wa1, Theorem 4] and [MW, Theorem 3.1].
Theorem 2.10. (Hessian criteria for Strong and Weak Lefschetz elements)
Let A = Q/Ann (f) be a standard graded Artinian Gorenstein algebra of codimension r
Q
and socle degree d and let L = a x +...+a x ∈ A , such that f(a ,...,a ) 6= 0. The map
1 1 r r 1 1 r
•Ll−k :A → A , for k < l ≤ d, has maximal rank if and only if the (mixed) Hessian matrix
k l 2
(k,d−l)
Hess (a ,...,a ) has maximal rank. In particular, we get the following:
f 1 r
(1) (Strong Lefschetz Hessian criterion, [Wa1], [MW]) L is a strong Lefschetz ele-
ment of A if and only if hessk(a ,...,a ) 6= 0 for all k = 1,...,[d/2].
f 1 r
LEFSCHETZ PROPERTIES FOR ARTINIAN GORENSTEIN ALGEBRAS PRESENTED BY QUADRICS 7
(2) (Weak Lefschetz Hessian criterion) L ∈ A is a weak Lefschetz element of A if
1
q
and only if either d = 2q +1 is odd and hess (a ,...,a ) 6= 0 or d = 2q is even and
f 1 r
(q−1,q)
Hess (a ,...,a ) has maximal rank.
f 1 r
Proof. For finite dimensional K-vector spaces V,W we have the isomorphism:
HomK(V,W) ≃ HomK(V⊗HomK(W,K),K).
Therefore, to any K-linear map T : V → W we can associate, in a unique natural way, a
bilinear form B : V×W∗ → K. Furthermore, T has maximal rank if, and only if, B has
T T
maximal rank. In our case of interest, the multiplication of the algebra induces the dual
pairing, A ×,A → A ≃ K via the identification A ≃ K given by ω ∈ A ↔ ω(f) ∈ K.
l d−l d d d
Hence, thelinear map•L :A → A inducesabilinear mapA ×A → Kgiven by (α,γ) 7→
k l k d−l
γ.Ll−k.α(f). If α ,...,α ∈ A and γ ,...,γ ∈ A are K-linear bases, then the matrix of
1 s k 1 t d−l
B with respect to these bases is given by [B ] = (γ .Ll−k.α (f)) = (Ll−k(γ α (f))) .
T T i j t×s i j t×s
(k,d−l)
On the other side, Hess = [γ α (f)], and we denote g := γ α (f) ∈ R . Since
f j i ij j i l−k
Ll−kg = (l−k)!g (a), the result follows. The others claim follows from Remark 2.5.
ij ij
(cid:3)
2.3. Forms with vanishing Hessian. Perazzo polynomials are the atoms of all known
constructions of polynomials with vanishing hessian (see [Go] and [CRS] for more details).
The following easy Proposition can be inferred from Gordan-Noether work. A proof of it can
be found in [Ru, Section 7.2.1].
Proposition 2.11. (Gordan-Noether criterion [GN]) Let f ∈ K[x ,...,x ] be a homo-
1 n
geneous polynomial of degree d. Then hess = 0 if and only if the partial derivatives of f are
f
algebraically dependent.
Definition 2.12. R = K[x ,...,x ,u ,...,u ], for m ≥ 3 and n ≥ 2 be the polynomial
1 n 1 m
ring and Q be the associated ring of differential operators. A Perazzo polynomial of degree
d is a reduced polynomial f ∈ R of the form
(1) f = x g +...+x g +h
1 1 n n
where g ∈ K[u ,...,u ] for i = 1,...,n are algebraically dependent but linearly inde-
i 1 m d−1
pendent and h∈ K[u ,...,u ] . The algebra A =Q/Ann (f) is called a Perazzo algebra.
1 m d Q
Remark 2.13. It is very easy to see that Perazzo polynomials have vanishing hessian, since
the first derivatives f = g for i = 1,...,n are algebraically dependent. Hence, by Gordan-
i i
Noether criterion, Proposition 2.11, hess = 0. Notice that if n > m, then all polynomial of
f
type 1 are Perazzo polynomials. By the Strong Lefschetz Hessian criterion, Theorem 2.10,
PerazzoalgebrasdonothavetheSLP.Noticealsothattheconstructionbeginsincodimention
five, this can be explained by Gordan-Noether Theorem, 2.14.
Theorem 2.14. [GN] Let X = V(f)⊂ PN be a hypersurface, not a cone, given by a reduced
polynomial f. If N ≤ 3, then hess 6= 0.
f
Corollary 2.15. [MW] Let A be an standard graded Artinian Gorenstein algebra of socle
degree d ≤ 4 and codimension r ≤ 4. Then A has the SLP.
8 R.GONDIMANDG.ZAPPALA`
3. Bigraded algebras, simplicial complexes and the main results
Let f ∈ K[x ,...,x ,u ,...,u ] be a bihomogeneous form of monomial square free
1 n 1 m (1,d−1)
n
type, f = x g . Let us consider the homogeneous (or pure) simplicial complex K = K(u)
i i
i=1
X
ofdimensiond−1 whosefacets aregiven bythemonomials g . The0-skeleton willbereferred
i
as vertex set and we write V = {u ,...,u }. We identify the 1-skeleton with a simple graph
1 m
K = (V,E), hence the 1-faces are called edges. Since X (f) = g , we identify each facet
1 i i
g with the differential operator X . We denote by e the number of k-faces, hence e = m
i i k 0
and e = n. Let A = Q/Ann (f) be the associated algebra, we suppose that I = 0. We
d−1 Q 1
identify the faces of K with the dual differential operators, u ↔ U . If p ∈ K[u ,...,u ]
i i 1 m
is a square free monomial, we denote by P ∈ K[U ,...,U ] the dual differential operator
1 m
P = p(U ,...,U ), notice that P(p)= 1.
1 m
n
Theorem 3.1. Let f = x g ∈ K[x ,...,x ,u ,...,u ] be a bihomogeneous poly-
i i 1 n 1 m (1,d−1)
i=1
X
nomial of monomial square-free type, let K be the associated simplicial complex and let
A= Q/Ann (f) be the standard bigraded Artinian Gorenstein algebra of codimension m+n
Q
d
and socle bidegree (1,d − 1). Then A = A where A = A ⊕ A . Moreover,
k k (0,k) (1,k−1)
k=0
M
A has a basis identified with the k faces of K, hence dimA = e . By duality,
(0,k) (0,k) k
A∗ ≃ A , and a basis for A can be chosen by taking, for each d − k-face
(1,k−1) (0,d−k) (1,k−1)
of K, a monomial X G˜ such that X G˜ (f) represents this d − k-face. In particular, the
i i i i
Hilbert vector of A is given by h = dimA = e + e . Furthermore, I = Ann (f) is
k k k d−k Q
a binomial ideal generated by (X ,...,X )2, all the monomials in U that do not represent
1 n i
faces of K, all monomials X F where f does not represent a subface of g and all binomials
i i i i
X G˜ −X G˜ where g = g˜g and g = g˜ g and g represents a common subface of g ,g .
i i j j i i ij j j ij ij i j
Proof. It is easy to see that A is generated by the monomials that represent k-faces,
(0,k)
since they are the only ones that do not annihilate f. Now we show that they are linearly
independent over K. For any k-face ω, let Ω be the monomial of Q , let Ω ,...,Ω be all
(0,k) 1 s
n
of them. Since Ω(f)= x Ω(g ), if we take any linear combination
i i
i=1
X
s s n n s
0 = c Ω (f) = c x Ω (g )= x c Ω (g ).
j j j i j i i j j i
j=1 j=1 i=1 i=1 j=1
X X X X X
s
We get c Ω (g ) = 0 for all i = 1,...,n. Fixed i, Ω (g ) are distinct monomials or zero,
j j i j i
j=1
X
but for each j there is a i such that Ω (g ) 6= 0, therefore c = 0 for all j = 1,...,s. The
j i j
other assertions concerning A are now clear.
Notice that I = Q for all i ≥ 2 and it is generated by I = (X ,...,X )2. Now we
(i,j) (i,j) (2.0) 1 n
LEFSCHETZ PROPERTIES FOR ARTINIAN GORENSTEIN ALGEBRAS PRESENTED BY QUADRICS 9
describe I and I . Consider the sequence giving by evaluation:
(0,k) (1,k−1)
0 → I → Q → A → 0.
(0,k) (0,k) (1,d−1−k)
Since dimA = dimA∗ = dimA = e , we get dimI = dimQ −e .
(1,d−1−k) (1,d−1−k) (0,k) k (0,k) (0,k) k
Since dimA = e and it has a basis given by the k-faces of K and since all the other
(0,k) k
dimQ −e monomials are linearly independent elements of I , they form a basis for
(0,k) k (0,k)
it. Consider the sequence giving by evaluation:
0 → I → Q → A → 0.
(1,k−1) (1,k−1) (0,d−k)
We have dimI = Q − e . Let us write Q = I˜ ⊕ Q˜ where
(1,k−1) (1,k−1) d−k (1,k−1) (1,k−1) (1,k−1)
I¯ is the K-vector space spanned by the monomials X F where F does not represent a
(1,k−1) i i i
subfaceofG . OfcourseI¯ ⊂ I andQ˜ isspannedbyallthemonomialsX G˜
i (1,k−1) (1,k−1) (1,k−1) i i
where G˜ is a subface of G . The exact sequence given by evaluation restricted to Q˜
i i (1,k−1)
becomes
0 → I˜ → Q˜ → A → 0.
(1,k−1) (1,k−1) (0,d−k)
Hence, I = I˜ ⊕I¯ , since X G˜ (f) is a face of K, I˜ is generated by the
(1,k−1) (1,k−1) (1,k−1) i i (1,k−1)
binomials X G˜ −X G˜ such that X G˜ (f) = g = X G˜ (f) where g is a common subface
i i j j i i ij j j ij
of g ,g , g = g˜g and g = g˜g . The result follows. (cid:3)
i j i i ij j j ij
Definition 3.2. Let K be a homogeneous simplicial complex of dimension d−1. We say
that K is facet connected if for any pair of facets F,F′ of K there exists a sequence of facets,
F = F,F ,...,F = F′ such that F ∩F is a (d−2)-face. We say that K is upper closed
0 1 s i i+1
if for all complete subgraphs H = K ⊂ K there is a l-face F ∈ K such that H is the first
l 1 l
skeleton of F. In particular K does not contain any K .
1 d
Theorem 3.3. Let f ∈ K[x ,...,x ,u ,...,u ] be a bihomogeneous polynomial of
1 n 1 m (1,d−1)
monomial square-free type, let K be the associated simplicial complex and letA= Q/Ann (f)
Q
be the standard bigraded Artinian Gorenstein algebra. A is presented by quadrics if and only
if K is facet connected and upper closed.
Proof. Supposethat K is upperclosed andfacet connected. By Theorem 3.1, I = Ann (f)it
Q
is enough to consider the monomials in the U that does not represent a face of K, monomials
i
X F where F is a monomial in the U that does not represent a subface of G and the
i i i j i
binomials X G˜ −X G˜ where X G˜ (f) = g = X G˜ (f) is a common subface of g ,g . Let
i i j j i i ij j j i j
M = Ue1...Uem be a monomial such that M(f) = 0, if e ≥ 0 for some i = 1,...,m, then
1 m i
M = U2M˜ and since U2 ∈ I , M ∈ I Q, hence we can consider M square free and it does
i i 2 2
not represents a face of K. In this case, M represents a K with the same vertex set of G,
l
since, by hypothesis, for 3 ≤ l ≤ d− 2 all K ⊂ G comes from a l-face of K and since G
l
does not contain a K as subgraph, there exists U U in M such that U U (f) = 0 and
d−1 i j i j
M = U U M˜ ∈ I Q.
i j 2
Let Ω = X M with M = Ue1...Uem be a monomial such that Ω(f)= M(g ) = 0, if e ≥ 2
i 1 m i i
for some i = 1,...,m, then Ω = U2Ω˜ ∈ I Q, so we can suppose that M is square free and
i 2
it does not represent a subface of g , hence there is a U in M that does not belongs to G ,
i j i
yielding Ω= X U M˜ ∈ I Q.
i j 2
Tofinishtheproof,considerthebinomials X G˜ −X G˜ whereX G˜ (f) = g = X G˜ (f)and
i i j j i i ij j j
10 R.GONDIMANDG.ZAPPALA`
g is a common subface of g ,g . If G˜ and G˜ are subfaces of the facets G ,G respectively
ij i j i j i j
and if g ⊂ G ∩G and the intersection is a d− 2-face, then there are only two vertices
ij i j
they do not share, say u ,u , G˜ = U G and G˜ = U G and finally X G˜ − X G˜ =
i j i i ij j j ij i i j j
(X U −X U )G ∈ I Q. In the general case, by the facet connection of K, there exists a
i i j j ij 2
sequence of facets G = G ,G ,...,G = G such that the intersection of two consecutive
i0 i i1 is j
facets is a d − 2-face, hence X G˜ − X G˜ ,X G˜ − X G˜ ,...,X G˜ − X G˜ ∈ I Q,
i i i1 i1 i1 i1 i2 i2 is is j j 2
summing up we get the desired result.
Conversely, if K is not facet connected, let g ,g be two facets that can not be facet
j j
connected and let g = gcd(g g ). By Theorem 3.1 it is easy to see that X G˜ −X G˜ is
ij i j i i j j
a minimal generator of I where g = g˜g and g = g˜ g . If K is not upper closed, then
i i ij j j ij
there is a complete subgraph K ⊂ G that does not came from a s-face of K. In this case, if
s
we choose s to be minimal, then by Theorem 3.1 the monomial M = v is a minimal
v∈VY(Ks)
generator of I.
(cid:3)
We introduce the following complexes inspired by the famous Turan Theorem characteriz-
ingmaximalgraphsnotcontainingaK asthe(d−1)-partitecompletegraphK(a ,...,a )
d−1 1 d−1
whith |a −a | ≤ 1. Here is the definition.
i j
Definition3.4. Let2≤ a ≤ ... ≤ a beintegers,theTurancomplexofordera ,...,a ,
1 d−1 1 d−1
K = TK(a ,...,a ), is the Homogeneous simplicial complex whose facets set is the carte-
1 d−1
d−1
sian product π = {1,2,...,a }. The Turan polynomial of order a ,...,a it the
i 1 d−1
i=1
Y
multihomogeneous polynomial f = f = x u ∈ R = K[x ,u ] ,
K α α α (i,ji) α∈π,1≤i≤d−1,1≤ji≤ai
α∈π
X
where α = (j ,...,j ) ∈ π and u = u ...u . The Turan algebra of order
1 d−1 α (1,j1) (d−1,jd−1)
(a ,...,a ) is TA(a ,...,a )= A = Q/Ann (f).
1 d−1 1 d−1 K Q
Remark 3.5. Notice that the number of k-faces of a Turan complex is e = s where
k k
s = s (a ,...,a ) is the symmetric function of order k. By Theorem 3.3, the Hilbert
k k 1 d−1
vector of the Turan algebra TA(a ,...,a ) is given by h = s +s . It is easy to verify
1 d−1 k k d−k
that Turan complexes are facet connected and upper closed.
Lemma 3.6. Let f ∈ R be a bihomogeneous polynomial of monomial square free type,
(1,d−1)
let K be the associated simplicial complex and A the associated algebra. Let e the number of
k
k-faces of K. Then the map •L : A → A , for k ≤ d, is injective for a general L ∈ A if,
k−1 k 2 1
((1,k−2),(0,d−k)) ((0,k−1),(1,d−k−1))
and only if rkHess = e and rkHess = e .
f d−k+1 f k−1
Proof. SinceA = A ⊕A ,andsincebyTheorem3.3dimA = e anddimA =
k (1,k−1) (0,k) (0,k) k (1,k−1)
dimA = e , with a choice of bases consistent with the decomposition as direct sum,
(0,d−k) d−k
we have:
((1,k−2),(0,d−k))
0 Hess
Hess(k−1,d−k) = f ,
f ((0,k−1),(1,d−k−1)) ((0,k−1),(0,d−k))
" Hess Hess #
f f (ed−k+1+ek−1)×(ek+ed−k)