Hilbert Functions of Artinian Gorenstein algebras with the Strong Lefschetz Property
aa r X i v : . [ m a t h . A C ] J u l HILBERT FUNCTIONS OF ARTINIAN GORENSTEIN ALGEBRASWITH THE STRONG LEFSCHETZ PROPERTY
NASRIN ALTAFI
Abstract.
We prove that a sequence h of non-negative integers is the Hilbert functionof some Artinian Gorenstein algebra with the strong Lefschetz property if and only if it isan SI-sequence. This generalizes the result by T. Harima which characterizes the Hilbertfunctions of Artinian Gorenstein algebras with the weak Lefschetz property. We also provideclasses of Artinian Gorenstein algebras obtained from the ideal of points in P n such thatsome of their higher Hessians have non-vanishing determinants. Consequently, we providefamilies of such algebras satisfying the SLP. Introduction
An Artinian graded algebra A over a filed k is said to satisfy the weak Lefschetz prop-erty (WLP for short) if there exists a linear form ℓ such that the multiplication map × ℓ : A i → A i +1 has maximal rank for every i ≥
0. Algebra A is said to satisfy the strongLefschetz property (SLP) if there is a linear form ℓ such that × ℓ j : A i → A i + j has maximalrank for each i, j ≥
0. Determining which graded Artinian algebras satisfy the Lefschetzproperties has been of great interest (see for example [4, 5, 11, 15, 16, 22, 24] and their refer-ences). It is known that every Artinian algebra of codimension two in characteristic zero hasthe SLP, it was proven many times using different techniques, see for example [16] and [6].This is no longer true for codimension three and higher and in general not easy to determineArtinian algebras satisfying or failing the WLP or SLP. Studying the Lefschetz propertiesof Artinian Gorenstein algebras is a very interesting problem. The h -vector of an Artinianalgebra with the WLP is unimodal. In general there are examples of Artinian Gorensteinalgebras with non-unimodal h -vector and hence failing the WLP. R. Stanley [25] gave thefirst example with h -vector h = (1 , , , , h -vector h = 5.Sequence h = ( h , h , . . . ) is a Stanley-Iarrobino sequence, or briefly SI-sequence, if it issymmetric, unimodal and its first half, ( h , h , . . . , h ⌊ d ⌋ ) is differentiable. R. Stanley [25]showed that the Hilbert functions of Gorenstein sequences are SI-sequences for h ≤
3. Bythe examples of non-unimodal Gorenstein Hilbert functions it is known that it is not truenecessarily for h ≥ h = 4. It is known that any SI-sequence is aGorenstein h -vector [7, 21]. T. Harima in [14] gave a characterization on h -vectors of Ar-tinian Gorenstein algebras satisfying the WLP. In this article we generalize this result andcharacterize h -vectors of Artinian Gorenstein algebras satisfying the SLP, see Theorem 3.2. Mathematics Subject Classification.
Key words and phrases.
Artinian Gorenstein algebra, Hilbert function, Hessians, Macaulay dual genera-tors, strong Lefschetz property, SI-sequence.
In section 4, we consider classes of Artinian Gorenstein algebras which are quotients ofcoordinate rings of a set of k -rational points in P n k . We prove that for a set X of points in P n k which lie on a rational normal curve any Artinian Gorenstein quotient of A ( X ) satisfiesthe SLP, Theorem 4.3. Higher Hessians of dual generators of Artinian Gorenstein algebraswas introduced by T. Maeno and J. Watanabe [20]. We study the higher Hessians of dualgenerators of Artinian Gorenstein quotients of A ( X ). We show Artinian Gorenstein quotientsof A ( X ) where X ⊂ P k lie on a conic satisfy the SLP, Theorem 4.5. We also prove non-vanishing of the determinants of certain higher Hessians in Theorems 4.6 and 4.7 for ArtinianGorenstein quotients of coordinate ring of points X ⊂ P k where X contains points on a conicand a line respectively. We then in Corollary 4.8 provide classes of such Artinian algebrassatisfying SLP. 2. Preliminaries
Let S = k [ x , . . . , x n ] be a polynomial ring equipped with the standard grading over afield k of characteristic zero and P n = P n k = Proj S . Let A = S/I be a graded Artinian (it hasKrull dimension zero) algebra where I is a homogeneous ideal. The Hilbert function of A indegree i is h A ( i ) = h i = dim k ( A i ). Since A is Artinian the Hilbert function of A is identifiedby its h -vector, h = ( h , h , h , . . . , h d ) such that h d = 0. The integer d is called the socledegree . The graded k -algebra is Gorenstein if it has a one dimensional socle degree. Withoutloss of generality we may assume that I does not contain a linear form (form of degree 1)so h = n + 1 and is called the codimension of A . If A is Gorenstein then the h -vector issymmetric and so h d = 1. A sequence h = ( h , . . . , h d ) is called a Gorenstein sequence if h is the Hilbert function of some Artinian Gorenstein algebra.Let h and i be positive integers. Then h can be written uniquely in the following form(2.1) h = (cid:18) m i i (cid:19) + (cid:18) m i − i − (cid:19) + · · · + (cid:18) m j j (cid:19) , where m i > m i − > · · · > m j ≥ j ≥
1. This expression for h is called the i -binomialexpansion of h . Also define(2.2) h h i i = (cid:18) m i + 1 i + 1 (cid:19) + (cid:18) m i − + 1 i (cid:19) + · · · + (cid:18) m j + 1 j + 1 (cid:19) where we set 0 h i i := 0.A sequence of non-negative integers h = ( h , h , . . . ) is called an O-sequence if h = 1and h i +1 ≤ h h i i i for all i ≥
1. Such sequences are the ones which exactly occur as Hilbertfunctions of standard graded algebras.
Theorem 2.1 (Macaulay [19]) . The sequence h = ( h , h , . . . , h d ) is an O-sequence if andonly if it is the h -vector of some standard graded Artinian algebra. We say h = ( h , h , . . . ) is differentiable if its first difference ∆ h = ( h , h − h , . . . ) is anO-sequence. Moreover, an h -vector is called unimodal if h ≤ h ≤ · · · ≤ h i ≥ h i +1 ≥ · · · ≥ h d . Sequence h = ( h , h , . . . ) is Stanley-Iarrobino sequence , or briefly
SI-sequence , if it issymmetric, unimodal and its first half, ( h , h , . . . , h ⌊ d ⌋ ) is differentiable.Now we recall the theory of Macaulay Inverse systems . Define Macualay dual ring R = k [ X , . . . , X n ] to S where the action of x i on R , which is denoted by ◦ , is partial differentiation ILBERT FUNCTIONS OF ARTINIAN GORENSTEIN ALGEBRAS WITH THE SLP 3 with respect to X i . For a homogeneous ideal I ⊆ S define its inverse system to be the graded S -module M ⊆ R such that I = Ann S ( M ). There is a one-to-one correspondence betweengraded Artinian algebras S/I and finitely generated graded S -submodules M of R , where I = Ann S ( M ) and is the annihilator of M in S , conversely, M = I − is the S -submodule of R which is annihilated by I . Moreover, the Hilbert functions of S/I and M are the same,in fact dim k ( S/I ) i = dim k M i for all i ≥
0. See [9] and [17] for more more details.By a result by F.H.S. Macaulay [18] it is known that an Artinian standard graded k -algebra A = S/I is Gorenstein if and only if there exists F ∈ R d , such that I = Ann S ( F ). Definition 2.2. [20, Definition 3.1] Let F be a polynomial in R and A = S/ Ann S ( F ) beits associated Artinian Gorenstein algebra. Let B j = { α ( j ) i + Ann S ( F ) } i be a k -basis of A j .The entries of the j -th Hessian matrix of F with respect to B j are given by(Hess j ( F )) u,v = ( α ( j ) u α ( j ) v ◦ F ) . We note that when j = 1 the form Hess ( F ) coincides with the usual Hessian. Up to a non-zero constant multiple det Hess j ( F ) is independent of the basis B j . By abuse of notation wewill write B j = { α ( j ) i } i for a basis of A j . For a linear form ℓ = a x + · · · + a n x n we denoteby Hess jℓ ( F ) the Hessian evaluated at the point P dual to ℓ that is P = ( a , . . . , a n ).The following result by T. Maeno and J. Watanabe provides a criterion for ArtinianGorenstein algebras satisfying the SLP. Theorem 2.3. [20, Theorem 3.1] Let A = S/ Ann S ( F ) be an Artinian Gorenstein quotientof S with socle degree d . Let ℓ be a linear form and consider the multiplication map × ℓ d − j : A j −→ A d − j . Pick any bases B j for A j for j = 0 , . . . , ⌊ d ⌋ . Then linear form ℓ is a strongLefschetz element for A if and only if det Hess jℓ ( F ) = 0 , for every j = 0 , . . . , ⌊ d ⌋ . Definition 2.4.
Let A = S/ Ann( F ) where F ∈ R d . Pick bases B j = { α ( j ) u } u and B d − j = { β ( d − j ) u } u be k -bases of A j and A d − j respectively. The entries of the catalecticant matrix of F with respect to B j and B d − j are given by(Cat jF ) uv = ( α ( j ) u β ( d − j ) v ◦ F ) . Up to a non-zero constant multiple det Cat jF is independent of the basis B j . The rankof the j -th catalecticant matrix of F is equal to the Hilbert function of A in degree j ,see [17, Definition 1.11].Throughout this paper we denote by X = { P , . . . , P s } the set of s distinct points in P n . Denote the coordinate ring of X by A ( X ) = S/I ( X ), where I ( X ) is a homogeneousideal of forms vanishing on X . For each point P ∈ X we consider it as a point in theaffine space A n +1 and denote it by P = ( a , . . . , a n ) and the linear form in R dual to P is L = a X + · · · + a n X n . We set(2.3) τ ( X ) := min { i | h A ( X ) ( i ) = s } . A. Iarrobino and V. Kanev [17] have proved that a general enough hyperplane in A ( X ) d for d ≥ τ ( X ) generates an Artinian Gorenstein algebra with the Hilbert function as h A ( X ) up NASRIN ALTAFI to degree ⌊ d ⌋ . M. Boij in [2] provides the special form of the dual generator for an ArtinianGorenstein quotients of A ( X ). Proposition 2.5. [2, Proposition 2.3] Let A be any Artinian Gorenstein quotient of A ( X ) with socle degree d such that d ≥ τ ( X ) and dual generator F . Then F can be written as F = s X i =1 α i L di where α , . . . , α s ∈ k , are not all zero. Proposition 2.6. [2, Proposition 2.4] Assume that d ≥ τ ( X ) − and F = P si =1 α i L di where α i = 0 for all i . Let A be the Artinian Gorenstein quotient of A ( X ) with dual generator F . Then the Hilbert function of A is given by h A ( i ) = ( h A ( X ) ( i ) 0 ≤ i ≤ ⌊ d ⌋ ,h A ( X ) ( d − i ) ⌈ d ⌉ ≤ i ≤ d. The following well known result guarantees the existence of set of points X ⊆ P n withgiven Hilbert function under an assumption on the Hilbert function. Theorem 2.7. [10, Theorem 4.1] Let h = ( h , h , . . . ) be a sequence of non-negative inte-gers. Then there is a reduced k -algebra with Hilbert function h if and only if h is a differen-tiable O-sequence. Hilbert functions of Artinian Gorenstein algebras and SI-sequences
In this section we give a characterization of the Hilbert functions of Artinian Gorensteinalgebras satisfying the SLP which generalizes Theorem 1.2 in [14]. We do so by using thehigher Hessians of the Macaulay dual form of Artinian Gorenstein algebras. We first providean explicit expression for the higher Hessians of such polynomials F = P si =1 α i L di . Lemma 3.1.
Let A be an Artinian Gorenstein quotient of A ( X ) with dual generator F = P si =1 α i L di where α i = 0 for all i and d ≥ τ ( X ) − . Then for each ≤ j ≤ τ ( X ) − wehave that (3.1) det Hess j ( F ) = X I⊆{ ,...,s } , |I| = h A ( X ) ( j ) c I Y i ∈I α i L d − ji , where c I ∈ k .Moreover, c I = 0 if and only if for X I = { P i } i ∈I we have that h A ( X ) ( j ) = h A ( X I ) ( j ) .Proof. We have that Hess j ( F ) = s X i =1 α i Hess j ( L di ) . Notice that Hess j ( L di ) is equal to L d − ji times an scalar matrix. Let T = k [ α , . . . , α s , L , . . . , L s ] to be a polynomial ring over k . For each i denote L i = a i, X + · · · + a i,n X n and define the action of S on T by x j ◦ L i = a i,j and x j ◦ α i = 0for every 1 ≤ i ≤ s and 0 ≤ j ≤ n . ILBERT FUNCTIONS OF ARTINIAN GORENSTEIN ALGEBRAS WITH THE SLP 5
We consider det Hess j ( F ) as a bihomogeneous polynomial in T having bidegree (cid:0) h A ( X ) ( j ) , ( d − j ) h A ( X ) ( j ) (cid:1) . We claim that det Hess j ( F ) is square-free in α i ’s. To showthe claim it is enough to show that α L d − j )1 Q h A ( X ) ( j ) − i =2 α i L d − ji has zero coefficient. As-sume not, then det Hess j ( F ) = 0 when we set α h A ( X ) ( j ) = · · · = α s = 0. This implies thatthe Hilbert function of the coordinate ring of { P , . . . , P h A ( X ) ( j ) − } in degree j is equal to h A ( X ) ( j ) which is a contradiction.Now assume that let I ⊆ { , . . . , s } such that |I| = h A ( X ) ( j ). If c I = 0 in Equation (3.1)setting α i = 0 for every i ∈ { , . . . , s } \ I implies that det Hess j ( F ) = 0 and therefore h A ( X ) ( j ) = h A ( X I ) ( j ). Conversely, assume that we have h A ( X ) ( j ) = h A ( X I ) ( j ) then since |I| = h A ( X ) ( j ) we pick B j = { L ji } i ∈I as a basis for A ( X ) j . Therefore, setting α i = 0 forevery i ∈ { , . . . , s } \ I implies that Hess j ( F ) with respect to B j is a diagonal matrix withdiagonal entries equal to d !( d − j )! α i L d − ji for every i ∈ I which implies that c I = 0. (cid:3) Now we are able to state and prove the main result of this section.
Theorem 3.2.
Let h = ( h , h , . . . , h d ) be a sequence of positive integers. Then h is theHilbert function of some Artinian Gorenstein algebra with the SLP if and only if h is anSI-sequence.Proof. Suppose A is an Artinian Gorenstein algebra with the Hilbert function h and thestrong Lefschetz element ℓ ∈ A that is in particular the weak Lefschetz element and using[14, Theorem 1.2] we conclude that h is an SI-sequence.Conversely, assume that h is an SI-sequence. We set h = n + 1 and h t = s where t = min { i | h i ≥ h i +1 } . So we have(3.2) h = (1 , n + 1 , . . . , s, . . . , s, . . . , n + 1 , . Define sequence of integers h = ( h , h , . . . ) such that h i = h i for i = 0 , . . . , t and h i = s for i ≥ t . Assuming that h is an SI-sequence implies that h is a differentiable O-sequenceand by Theorem 2.7 there exists X = { P , . . . , P s } ⊆ P n such that the Hilbert functionof its coordinate ring A ( X ) is equal to h , that is h A ( X ) = h . Denote by { L , . . . , L s } thelinear forms dual to { P , . . . , P s } . As in Proposition 2.5, let A be the Artinian Gorensteinquotient of A ( X ) with dual generator F = P si =1 α i L di for d ≥ τ ( X ), notice that τ ( X ) = t .By Proposition 2.6, in order to have h A = h we must have α i = 0 for all i . Let L be alinear form dual to P ∈ P n such that X ∩ P = ∅ and denote by ℓ the dual linear form to L . Therefore, β i := ℓ ◦ L i = 0 for every 1 ≤ i ≤ s . We claim that there exist α , . . . , α s such that ℓ is the strong Lefschetz element for A . First note that for every j = t, . . . , ⌊ d ⌋ the multiplication map by × ℓ d − j : A j → A d − j can be considered as the multiplication mapon A ( X ), that is × ℓ d − j : A ( X ) j → A ( X ) d − j which has trivially maximal rank.Now we prove that there is a Zariski open set for α i ’s such that for every j = 0 , . . . , t − j -th Hessian matrix of F evaluated at ℓ has maximal rank, that isrk Hess jℓ ( F ) = h A ( j ) = h j . Using Lemma 3.1 we get that(3.3) det Hess jℓ ( F ) = X I⊆{ ,...,s } , |I| = h j c I Y i ∈I α i β d − ji NASRIN ALTAFI where c I = 0 if and only if h A ( X I ) ( j ) = h j for X I = { P i } I . Notice that since there is at leastone subset I such that c I = 0 the determinant of the j -th Hessian is not identically zero.Therefore, det Hess jℓ = 0 or equivalently rk Hess jℓ ( F ) = h j for each i = 0 , . . . , t −
1, providesa Zariski open subset of P s − for α i ’s and therefore the intersection of all those open subsetsis non-empty. Equivalently, there is an Artinian Gorenstein algebra A such that h A = h andsatisfies the SLP with ℓ ∈ A . (cid:3) Higher Hessians of Artinian Gorenstein quotients of A ( X )In this section we prove the non-vanishing of some of the higher Hessians for any ArtinianGorenstein quotient of A ( X ) for X ⊂ P n under some conditions on the configuration of thepoints in X . In some cases we conclude that they satisfy the SLP. Proposition 4.1.
Let X = { P , . . . , P s } be a set of points in P n and A be any ArtinianGorenstein quotient of A ( X ) with dual generator F = P si =1 α i L di for d ≥ τ ( X ) − . Assumethat h A ( j ) = s − for some j ≥ then there is a linear form ℓ such that det Hess jℓ ( F ) = 0 . Proof.
Using Lemma 4.4 we have thatdet Hess j ( F ) = X I⊆{ ,...,s } , |I| = s − c I Y i ∈I α i L d − ji . We prove that det Hess j ( F ) = 0 as a polynomial in X i ’s. Without loss of generality assumethat for I = { , . . . , s } we have that c { ,...,s } = 0. Then if det Hess j ( F ) is identically zero weget that c { ,...,s } s Y i =2 α i L d − ji = − α L d − j X I⊆{ ,...,s } , |I| = s − c I Y i ∈I α i L d − ji . This contradicts the fact that R = k [ X , . . . , X n ] is a unique factorization domain. Weconclude that there exists ℓ such that det Hess jℓ ( F ) = 0. (cid:3) Proposition 4.2.
Let s ≥ and X = { P , . . . , P s } be a set of points in P in a generallinear position. Let A be any Artinian Gorenstein quotient of A ( X ) with dual generator F = P si =1 α i L di for d ≥ τ ( X ) − and assume that h A ( j ) = s − for some j ≥ . Thenthere is a linear form ℓ such that det Hess jℓ ( F ) = 0 . Proof. If h A ( j + 1) < s − j ( F ) = Cat jF and trivially has maximal rank.If h A ( j +1) > s − X implies that h A ( j +1) = s . Since if h A ( j +1) = s − h A ( X ) ends with at least three ones which by E. D. Davis’s theorem [8] X contains atleast three colinear points. So h A = (1 , , . . . , s − , s, . . . , s | {z } k , s − , . . . , , , for some k ≥
1. Note that for a linear form ℓ such that ℓ ◦ L i = 0 for every i we get thatthe multiplication map ℓ d − i : A i → A d − i for every j + 1 ≤ i ≤ ⌊ d ⌋ is a map on A ( X ) ILBERT FUNCTIONS OF ARTINIAN GORENSTEIN ALGEBRAS WITH THE SLP 7 in the same degrees and therefore has maximal rank. So det Hess jℓ ( F ) = 0 if and only ifdet Hess jℓ ( ℓ k ◦ F ) = 0. Denote by β i = ℓ ◦ L i = 0 for each i . So we have that G := ℓ k ◦ F = d !( d − k )! s X i =1 α i β ki L d − ki . The Artinian Gorenstein quotient of A ( X ) with dual generator G has the following Hilbertfunction (1 , , . . . , s − , s − , . . . , , . Therefore, it is enough to show that det Hess jℓ ( F ) = 0 for some ℓ in the case h A ( j ) = h A ( j + 1) = s −
2. Note that in this case d = 2 j + 1. By Lemma 4.4 we have that(4.1) det Hess j ( F ) = X I⊆{ ,...,s } , |I| = s − c I Y i ∈I α i L i , such that c I = 0 if and only if h A ( X ) ( j ) = h A ( X I ) ( j ). Without loss of generality assume that c { ,...,s } = 0. Suppose that det Hess j ( F ) = 0 then c { ,...,s } s Y i =3 α i L i = − α L X I⊆{ ,...,s } , |I| = s − c I Y i ∈I α i L i − α L X I⊆{ , ,...,s } , |I| = s − c I Y i ∈I α i L i . Common zeros of L and L correspond to the line passing through P and P . By theassumption this line does not pass through any other point in { P , . . . , P s } which means thatthe left hand side of the above equality is nonzero on the points where L = L = 0 whichis a contradiction. So this implies that det Hess j ( F ) = 0 and therefore det Hess jℓ ( F ) = 0, forsome linear form ℓ . (cid:3) Theorem 4.3 (Points on a rational normal curve) . Let X = { P , . . . , P s } be a set of pointsin P n lying on a rational normal curve. Assume that A is an Artinian Gorenstein quotientof A ( X ) with dual generator F = P si =1 α i L di for d ≥ τ ( X ) and α i = 0 for every i . Then A satisfies the SLP.Proof. Denote by Y = { Q , . . . , Q s } ⊂ P the preimage of X under the Veronese embedding ϕ : P −→ P n . For each i = 1 , . . . , s denote by K i the linear form in k [ S, T ] dual to Q i . Let B be any Artinian Gorenstein quotient of the coordinate ring of Y , k [ S, T ] /I ( Y ), with dualgenerator G = P si =1 β i K ndi .Artinian Gorenstein algebra B has the following Hilbert function h B = (1 , , , . . . , s, . . . , s | {z } k , . . . , , , , for some k ≥ d ≥ τ ( X ) and α i = 0 for every i . It is knownthat B has the SLP for some linear form ℓ ∈ B . Consider the linear form ℓ ′ := ϕ ( ℓ n ) ∈ A so rk (cid:0) × ( ℓ ′ ) d − j : A j −→ A d − j (cid:1) = rk (cid:0) × ( ℓ ) n ( d − j ) : B nj −→ B nd − nj (cid:1) = nj + 1 = dim k A j . Thus A satisfies the SLP with linear form ℓ ′ . (cid:3) NASRIN ALTAFI
The above proposition shows that every Artinian Gorenstein quotient of A ( X ) such that X ⊂ P consists of points on a smooth conic satisfies the SLP. We will show that the SLPalso holds when X ⊂ P consists of points on a singular conic.First we need to prove a lemma. Lemma 4.4.
Let A = B + C be a square matrix of size m − for m ≥ as the following (4.2) B = f f . . . f m · · · f f . . . f m +1 · · · ... ... ... ... ... f m f m +1 . . . f m · · ·
00 0 . . . · · · ... ... ... ... ... . . . · · · , C = . . . · · · . . . · · · ... ... ... ... ... . . . g m . . . g m +1 g m ... ... ... ... ... . . . g m +1 . . . g g . . . g m . . . g g . Then det A = (det B { ,...,m − } )(det C { m,..., m − } ) + (det B { ,...,m } )(det C { m +1 ,..., m − } ) , such that for a subset J ⊂ { , . . . , m − } we denote by B J and C J the square submatricesof B and C respectively with rows and columns in the index set J .Proof. We have that det A = X σ ∈ S m − sign σA σ . . . A (2 m − σ m − , where the entry A iσ i is the entry in row i and column σ i . Then we split det A in the followingwaydet A =( X σ ∈ S m − sign σA σ . . . A ( m − σ m − ) A m,m ( X τ ∈ S m − sign τ A ( m +1)( m + τ ) . . . A (2 m − m + τ m − ) )+ ( X σ ∈ S m ,σ m = m sign σA σ . . . A mσ m )( X τ ∈ S m − sign τ A ( m +1)( m + τ ) . . . A (2 m − m + τ m − ) )+ ( X σ ∈ S m − sign σA σ . . . A ( m − σ m − )( X τ ∈ S m ,τ =1 sign τ A m ( m − τ ) . . . A (2 m − m − τ m ) )=(det B { ,...,m − } )(det C { m,..., m − } ) + (det B { ,...,m } )(det C { m +1 ,..., m − } ) . (cid:3) Theorem 4.5.
Assume that X = { P , . . . , P s } is a set of points in P which lie on a conic.Let A be an Artinian Gorenstein quotient of A ( X ) with dual generator F = P si =1 α i L di , for d ≥ τ ( X ) and α i = 0 for every i . Then A satisfies the SLP.Proof. If X lies on a smooth conic applying Theorem 4.3 for n = 2 we get the desiredresult. Now suppose that X consists of points on a singular conic that is a union of twolines in P . Suppose that X := { P , . . . , P s } is a subset of X which lie on one line and X := { Q , . . . , Q s } is a subset of X with the points on the other line, so X = X ∪ X .If X ∩ X = ∅ then s + s = s otherwise s + s − s . Denote by L i the linear formdual to P i for 1 ≤ i ≤ s and by K i the linear form dual to Q i for each 1 ≤ i ≤ s . Let F = P s i =1 a i L di and F = P s i =1 b i K di for linear forms L i and K i where F = F + F . By linear ILBERT FUNCTIONS OF ARTINIAN GORENSTEIN ALGEBRAS WITH THE SLP 9 change of coordinates we may assume that L i = u ,i X + u ,i X and K i = v ,i X + v ,i X such that u ,i , u ,i , v ,i , v ,i ∈ k for every i . The Hilbert function of A is equal to h A = (1 , , , . . . , k + 1 , s, . . . , s, k + 1 , . . . , , , , where k is the largest integer such that 2 k + 1 ≤ s . If s = 2 k + 1 then τ ( X ) = k andotherwise τ ( X ) = k + 1. Let j be an integer such that 1 ≤ j ≤ τ ( X ) −
1. Consider thefollowing ordered monomial basis for A in degree j B j = { x j , x j − x , . . . , x x j − , x j , x j − x , . . . , x x j − , x j } . The j -th Hessian of F with respect to B j is the following matrixHess j ( F ) = Hess j ( F ) + Hess j ( F ) = s X i =1 a i Hess j ( L i ) + s X i =1 b i Hess j ( K i )= C j C j . . . C jj · · · C j C j . . . C jj +1 · · · C jj C jj +1 . . . C j j + D j j . . . D jj +1 D jj ... ... ... ... ...0 0 . . . D jj +1 . . . D j D j . . . D jj . . . D j D j where we set C ji = ( x j − i x i ) ◦ F and D ji = ( x j − i x i ) ◦ F for each i = 0 , . . . , j .Then using Lemma 4.4 we get that(4.3) det Hess j ( F ) = (det C j { ,...,j − } )(det D j ) + (det D j { ,...,j } )(det C j ) , where we set C j = C j C j . . . C jj C j C j . . . C jj +1 ... ... ... C jj C jj +1 . . . C j j and D j = D j j D j j − . . . D jj D j j − D j j − . . . D jj − ... ... ... D jj D jj − . . . D j and wedenote by C j { i ,...,i r } the square submatrix of C j of size r with rows and columns i , . . . , i r ,similarly for D j .Let A and A be Artinian Gorenstein quotients of A ( X ) = k [ x , x ] /I ( X ) and A ( X ) = k [ x , x ] /I ( X ) with dual generators F and F respectively. We observe that C j = Hess j ( F )and D j = Hess j ( F ). Since every Artinian algebra of codimension two has the SLP we havethat det C j = 0 and det D j = 0.We set F ′ := x ◦ F , F ′ := x ◦ F . Then C j { ,...,j − } is equal to the ( j − F ′ with respect to the ordered basis { x j − , x j − x , . . . , x j − } . Similarly, D j ,...,j = Hess j − ( F ′ ) with respect to { x j − , x j − x , . . . , x j − } . So using the result that Artinian algebras in codimension two have the SLP we get thatdet Hess j − ( F ′ ) = det C j { ,...,j − } = 0 , and det Hess j − ( F ′ ) = det D j { ,...,j } = 0 . Therefore, Equation (4.3) is equivalent to(4.4) det Hess j ( F ) = (det Hess j − ( F ′ ))(det Hess j ( F )) + (det Hess j − ( F ′ ))(det Hess j ( F )) . Note that assuming d ≥ τ ( X ) and 1 ≤ j ≤ τ ( X ) − j − ( F ′ )) < deg(det Hess j ( F )) , deg(det Hess j − ( F ′ )) < deg(det Hess j ( F )) . Therefore, det Hess j ( F ) = 0 unless when X d − j is a factor of both det Hess j ( F ) and det Hess j ( F )so we must have X ∩ X = ∅ and s = s + s −
1. On the other hand, using Lemma 3.1 we getthat det Hess j ( F ) is in fact a non-zero monomial in L i ’s and j = s . Similarly, we get j = s .So s = 2 s − s − k + 1 and therefore τ ( X ) = k and s = s = k + 1 = τ ( X ) + 1.This contradicts the assumption that j ≤ τ ( X ) − τ ( X ) ≤ j ≤ ⌊ d ⌋ the j -the Hessian of F correspond to the multiplication map on A ( X ) and then trivially has maximal rank for general enough linear forms.Note that det Hess ( F ) = F = 0. Therefore, we have proved that there is a linear form ℓ such that det Hess jℓ ( F ) = 0 for every 0 ≤ j ≤ ⌊ d ⌋ and equivalently A has the SLP. (cid:3) We now prove that if X ⊆ P contains points on a conic then higher Hessians of F aftersome point are non-zero. Theorem 4.6.
Let X = { P , . . . , P s } be a set of points in P and A be an Artinian Goren-stein quotient of A ( X ) with dual generator F = P si =1 α i L di , for d ≥ τ ( X ) and α i = 0 forevery i . Suppose that the first difference of h A is equal to ∆ h A = (1 , , h − , . . . , k , . . . , τ ( X ) ) , for some ≤ k < τ ( X ) . Then there is a linear form ℓ such that for every k − ≤ j ≤ ⌊ d ⌋ det Hess jℓ ( F ) = 0 . Proof.
E. D. Davis’s theorem [8] implies that since ∆ h A ( X ) is flat the set X is a unionof 2 τ ( X ) + 1 points on a conic and s − τ ( X ) − P , . . . , P s − τ ( X ) − lie outside the conic. Note that for each k − ≤ j ≤ ⌊ d ⌋ we have that h A ( X ) ( j ) = 2 j + 1 + s − τ ( X ) − s − τ ( X ) + 2 j . Using Lemma 3.1 we get that for each k − ≤ j ≤ ⌊ d ⌋ (4.5) det Hess j ( F ) = X I⊆{ ,...,s } , |I| = h A ( X ) ( j ) c I Y i ∈I α i L d − ji , where c I = 0 if and only if h A ( X I ) ( j ) = h A ( X ) ( j ) = s − τ ( X ) + 2 j . Notice that the Hilbertfunction of the coordinate ring of the points on a conic in degree j is at most 2 j +1. Therefore, c I = 0 if and only if I contains s − τ ( X ) + 2 j − (2 j + 1) = s − τ ( X ) − { , . . . , s − τ ( X ) − } ⊂ I .This implies that Q s − τ ( X ) − i =1 α i L d − ji is a common factor of the right hand side of Equation(4.5), so(4.6) det Hess j ( F ) = s − τ ( X ) − Y i =1 α i L d − ji X I⊆{ s − τ ( X ) ,...,s } , |I| =2 j +1 c I Y i ∈I α i L d − ji . ILBERT FUNCTIONS OF ARTINIAN GORENSTEIN ALGEBRAS WITH THE SLP 11
Let Y := { P s − τ ( X ) , . . . , P s } be the subset of X which lie on a conic. Consider ArtinianGorenstein quotient of A ( Y ) with dual generator G = P si = s − τ ( X ) α i L di . Theorem 4.5 impliesthat B has the SLP. Equivalently, for every 0 ≤ j ≤ ⌊ d ⌋ det Hess j ( G ) = X I⊆{ s − τ ( X ) ,...,s } , |I| =2 j +1 c I Y i ∈I α i L d − ji = 0 . This implies that the polynomial in Equation (4.6) is non-zero and we this completes theproof. (cid:3)
Similarly, using that all Artinian algebras in codimension two have the SLP we have thefollowing which proves non-vanishing of some of higher Hessians in the case where X ⊆ P contains points on a line. Theorem 4.7.
Let X = { P , . . . , P s } be a set of points in P and A be an Artinian Goren-stein quotient of A ( X ) with dual generator F = P si =1 α i L di , for d ≥ τ ( X ) and α i = 0 forevery i . Suppose that the first difference of h A is equal to ∆ h A = (1 , , h − , . . . , k , . . . , τ ( X ) ) , for some ≤ k < τ ( X ) . Then there is a linear form ℓ such that for every k − ≤ j ≤ ⌊ d ⌋ det Hess jℓ ( F ) = 0 . Proof.
By Davis’s theorem [8] we get that there are exactly τ ( X ) + 1 points on a line and s − τ ( X ) − P , . . . , P s − τ ( X ) − lie off the line. For each k − ≤ j ≤ ⌊ d ⌋ we have h A ( X ) ( j ) = j + 1 + s − τ ( X ) − s − τ ( X ) + j .So for each s − k ≤ j ≤ ⌊ d ⌋ by Lemma 4.4 we get(4.7) det Hess j ( F ) = X I⊆{ ,...,s } , |I| = h A ( X ) ( j ) c I Y i ∈I α i L d − ji , where c I is non-zero if and only if h A ( X I ) ( j ) = h A ( X ) ( j ) = s − τ ( X ) + j . Since the Hilbertfunction of the coordinate ring of the points on a line in degree j is at most j + 1, in order forthe coordinate ring of { P i } i ∈I to have the Hilbert function equal to s − τ ( X ) + j in degree j , I must contain all the indexes from 1 to s − τ ( X ) + j − ( j + 1) = s − τ ( X ) − j ( F ) = s − τ ( X ) − Y i =1 α i L d − ji X I⊆{ s − τ ( X ) ,...,s } , |I| = j +1 c I Y i ∈I α i L d − ji . Denote by Y := { P s − τ ( X ) , . . . , P s } the points in X which lie on a line. Consider ArtinianGorenstein quotient of A ( Y ) with dual generator G = P si = s − τ ( X ) α i L di and denote it by B .Since B is an Artinian algebra of codimension two it satisfies the SLP. Equivalently, for every0 ≤ j ≤ ⌊ d ⌋ det Hess j ( G ) = X I⊆{ s − τ ( X ) ,...,s } , |I| = j +1 c I Y i ∈I α i L d − ji = 0This implies that det Hess j ( F ) = 0 for every k − ≤ j ≤ ⌊ d ⌋ . (cid:3) As a consequence of Theorems 4.6 and 4.7 we provide a family of Artinian Gorensteinquotients of X ⊆ P satisfying the SLP. Corollary 4.8.
Let X = { P , . . . , P s } be a set of points in P and A be any ArtinianGorenstein quotient of A ( X ) with dual generator F = P si =1 α i L di , for d ≥ τ ( X ) . Then A satisfies the SLP if ∆ h A is equal to one the following vectors (1 , , , . . . , | {z } m ) , (1 , , , , . . . , | {z } m ) , (1 , , , , . . . , | {z } m ) , (4.9) (1 , , . . . , | {z } m ) , (1 , , , , . . . , | {z } m ) . (4.10) for some m ≥ .Proof. First we note that det Hess ( F ) = F and since F is assumed to be non-zero for ageneric ℓ we have det Hess ℓ ( F ) = 0. A well known result by P. Gordan and M. Noether [12]implies that the Hessian of every form in the polynomial ring with three variables is non-zero.Therefore, for a generic linear form ℓ we get that det Hess ℓ ( F ) = 0.Using Theorems Theorem 4.7 and 4.6 for the first difference vectors given in (4.9) and (4.10)respectively we conclude that det Hess jℓ ( F ) = 0 for every 2 ≤ j ≤ ⌊ d ⌋ and a generic linearform ℓ . This completes the proof. (cid:3) Remark 4.9.
J. Migliore and F. Zanello [23, Theorem 5] characterize the Hilbert function ofalgebras with the WLP. An immediate consequence of this result provides that any Artinianalgebra with the Hilbert functions listed in (4.9) and (4.10) has the WLP.
Summary.
We end the section by summarizing what we have shown.Let X = { P , . . . , P s } ⊆ P n and F = P si =1 α i L di , for d ≥ τ ( X ) and α i = 0 for every i .For n ≥ X ⊆ P n lies on a rational normal curve then any Artinian Gorenstein quotientof A ( X ) with dual generator F satisfies the SLP, Theorem 4.3. This result is more generalfor n = 2. In fact, if X ⊆ P lies on a conic (smooth or singular) then in Theorem 4.5we prove that any Artinian Gorenstein quotient of A ( X ) with dual generator F satisfiesthe SLP. When X ⊆ P , we show in Theorems 4.6 and 4.7 that if the first difference of anArtinian Gorenstein quotient of A ( X ) with dual generator F is equal to∆ h A = (1 , , h − , . . . , k , . . . , τ ( X ) ) , or ∆ h A = (1 , , h − , . . . , k , . . . , τ ( X ) )for some 1 ≤ k < τ ( X ), then there is a linear form ℓ such that det Hess jℓ ( F ) = 0 for every k − ≤ j ≤ ⌊ d ⌋ . As a consequence of these results we show that any Artinian Gorensteinquotient A of A ( X ) with with ∆ h A given in (4.9) and (4.10) satisfies the SLP, Corollary 4.8.We also show in Proposition 4.1 that for every n ≥ j -th Hilbert function ofan Artinian Gorenstein quotient of A ( X ) is equal to s −
1, that is h A ( j ) = s −
1, thendet Hess jℓ ( F ) = 0 for some ℓ . Also for X ⊆ P in a general linear position we have thatdet Hess jℓ ( F ) = 0 for some ℓ if h A ( j ) = s −
2, Proposition 4.2.5.
Acknowledgment
The author would like to thank Mats Boij for useful and insightful comments and discus-sion that greatly assisted this research. Computations using the algebra software Macaulay2
ILBERT FUNCTIONS OF ARTINIAN GORENSTEIN ALGEBRAS WITH THE SLP 13 [13] were essential to get the ideas behind some of the proofs. This work was supported bythe grant VR2013-4545.
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Department of Mathematics, KTH Royal Institute of Technology, S-100 44 Stockholm,Sweden
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