Lefschetz properties for jacobian rings of cubic fourfolds and other Artinian algebras

In this paper, we exploit some geometric-differential techniques to prove the strong Lefschetz property in degree $1$ for a complete intersection standard Artinian Gorenstein algebra of codimension $6$ presented by quadrics. We prove also some strong Lefschetz properties for the same kind of Artinian algebras in higher codimensions. Moreover, we analyze some loci that come naturally into the picture of"special"Artinian algebras: for them, we give some geometric descriptions and show a connection between the non emptiness of the so-called non-Lefschetz locus in degree $1$ and the"lifting"of a weak Lefschetz property to an algebra from one of its quotients.


Introduction
In this article we exploit and generalize some techniques that the authors have developed in previous works on these topics, in collaboration with Gian Pietro Pirola (see [FP22] and [BFP22]) in order to prove the validity of some Lefschetz properties for particular Artinian algebras. More precisely, we deal with standard Artinian Gorenstein graded algebras over a field K of characteristic 0. For brevity, we refer to such an algebra using the acronym SAGA. SAGAs are generated in degree 1 and satisfy a sort of "inner duality" property induced by the products of elements in complementary degrees (see Section 1.1 for details). We recall that the codimension of a standard Artinian graded algebra R is the dimension of R 1 , i.e. its degree-1 part, as vector space over K.
Classical examples of these algebras are jacobian rings of smooth hypersurfaces in projective spaces. These are constructed by considering the smooth hypersurface X = V (F ) ⊆ P n with F homogeneous in S = K[x 0 , · · · , x n ] and by taking the quotient R = S/I where I is the ideal generated by the partial derivatives of F (the so-called jacobian ideal of F ). The importance of these objects lies, for example, in the strong geometric relations between X and its jacobian ring. Just to mention some, in the seminal works [CG80,CGGH83,GH83,Gri83], Carlson, Griffiths, Green and Harris proved that a portion of the primitive part of the Dolbeault cohomology is codified in R and that R plays a crucial role in the study of the infinitesimal variation of Hodge structure of X.
This construction is a special case of a more general one. Indeed, one can also consider the quotient S/I, where I is generated by a regular sequence of length n + 1 of homogeneous polynomials, i.e. I = (f 0 , · · · , f n ) ⊂ K[x 0 , · · · , x n ], where V (f 0 , · · · , f n ) ⊂ P n is the empty set. This kind of algebras are SAGAs and are also complete intersection rings so it makes sense to refer to them as complete intersection SAGAs. For brevity one often says (see, for example, [MN13]) that R is a complete intersection SAGA presented by forms of degree e, if the generators of the ideal I have all the same degree, equal to e. Our main result (Theorem 3.1) will deal with the case e = 2, i.e. with complete intersection SAGAs presented by forms of degree 2.
Lefschetz properties for an Artinian algebra were defined in the 80's by taking inspiration from the Hard Lefschetz Theorem for a Kähler variety X of complex dimension n. We recall that this theorem states that for all k ≤ n the cup product map ω k · : H n−k (X, C) → H n+k (X, C) is an isomorphism for ω ∈ H 2 (X, C) generic (more precisely, for any Kähler form in H 1,1 ∂ (X)). Then, the even cohomology ring R = n i=0 R i = n i=0 H 2i (X, C) of X is a Gorenstein Artinian algebra. In particular, R is such that for x ∈ R 1 general, the multiplication map x n−2i · : R i → R n−i is an isomorphism for i ≤ n/2. This is, roughly, the definition of the strong Lefschetz property (SLP ) for an Artinian algebra (for details see Definition 1.2). Starting from this idea a weaker version has been introduced by following similar ideas: an algebra R has the weak Lefschetz property (W LP ) if the multiplication map x· : R k → R k+1 is of maximal rank for all k ≥ 0 and x ∈ R 1 general. For a comprehensive treatment of Lefschetz properties the interested reader can refer to [HMM + 13].
In the recent years, there has been a growing interest in Artinian algebras and their Lefschetz properties. We summarize here some results and conjectures relevant with respect to the topics of our article. In [HMNW03] it is shown that any Artinian standard algebra R of codimension 2 satisfies SLP . In the same article, it is also proved that if R has codimension 3 and is a complete intersection then it satisfies W LP . Again in codimension 3, if one restricts its attention to jacobian rings of curves of degree d in P 2 , the validity of SLP is known only up to degree 4 with the case for d = 4 treated in the very recent work [DGI20]. In codimension 4, again in [DGI20], the authors prove that jacobian rings of cubic surfaces have the SLP . In particular, nothing is known for Lefschetz properties for jacobian rings when the degree increases: quintic curves in P 2 (regarding SLP ) and quartic surfaces in P 3 (regarding both W LP and SLP ) are the first open cases. For what concerns the case of complete intersection SAGAs presented by quadrics, in [MN13] it is shown that W LP holds in degree 1 (i.e. x· : R 1 → R 2 is injective for x ∈ R 1 general) in any codimension. If one wonders if all SAGAs have to satisfy W LP , this is known to be false as there are counterexamples (which are not complete intersection) in higher codimension ( [GZ18]) but it is conjectured to hold for complete intersection SAGAs. For codimension bigger than or equal to 5 very little is known. For codimension 5, W LP has been proved in [AR19] for complete intersection SAGAs presented by quadrics whereas, in [BFP22] the authors proved SLP for the same SAGAs (in particular, for jacobian rings of smooth cubic threefolds). The cases of higher codimensions or higher degrees are, at the moment, completely open.
Just to mention some other papers, which treat problems related to Lefschetz properties for Artinian Algebras, we cite, for example, [MMRN11, MMRO13, Ila18, AAI + 22] and [DI22]. These articles are really interesting on their own since the approaches used to study, roughly, the "same" problems are really different and involve a lot of different techniques.
In this work, we follow the "geometric-differential" approach used in [AR19] and [BFP22] by exploiting and generalising the techniques of the latter, in order to prove, partially, the SLP for the first open case with low degree in higher codimension. More precisely, we deal with a complete intersection SAGA R presented by quadrics of codimension 6 and we prove our first main result: Theorem. (Theorem 3.1) Let R be a SAGA as above. Then R satisfies the SLP in degree 1, i.e. for x general in R 1 the multiplication map x 4 · : R 1 → R 5 is a bijection.
This result is a step towards the evidence of a well known conjecture, according to which, in characteristic 0 all complete intersection SAGAs should satisfy the Lefschetz properties (see [HMM + 13, Conjecture 3.46], for example). Beyond being the first open case, the interest for these SAGAs lies in the fact that they involve Lefschetz properties for the jacobian rings of cubic fourfolds, which are very interesting on their own for what concerns several other unrelated topics (such as, for example, the well known Kuznetsov's conjecture about the rationality of the cubic fourfolds).
We stress that Lefschetz properties, at least for a general complete intersection SAGA S/I with fixed degrees of the generators of I, are known to hold. For this reason, it is interesting to analyze "special" SAGAs. With this idea, we proceed by obtaining two interesting results concerning special SAGAs. The first one is a collection of geometric properties satisfied by the subschemes that we call Nihilpotent loci. Roughly, when R is "general", N i is empty for low values of i, then, in case of non-emptiness, we do expect that these loci reflect some properties of "special" SAGAs. The most relevant results are Proposition 2.1, which has as a consequence, a characterization of Fermat hypersurfaces (see 2.2) and Theorem 2.6 which analyzes the geometric behaviour of these loci when they contain a linear space.
For the second result, we consider what are called the non-Lefschetz loci, i.e. the subschemes parametrising linear forms for which the injectivity of the multiplication map fails (see Definition 1.4). These are really interesting for the geometry and the algebraic structure of a SAGA. Just to mention some articles which study these loci, one can consider [BMMRN18] and [AR19]. Our contribution for this topic is the following lifting criterion: Theorem. (Theorem 4.2) Let R = S/I be a complete intersection SAGA of codimension n + 1 ≥ 6 presented by quadrics and assume that z is a non-Lefschetz element for R. IfR = R/(z) satisfies the W LP in degree 2, then the same holds for R. This is a result for "special" SAGAs since, as for the Nilpotent loci, when R is general, all [x] ∈ P(R 1 ) are Lefschetz elements. This unexpected theorem gives a sort of converse for results which prove that Lefschetz properties are inherited by suitable quotients. This happens, for example, for SLP when one takes the quotient by the conductor of a Lefschetz element ([HMM + 13, Proposition 3.11]) or, for the W LP as shown in the recent article [Gue19].
The plan of the article is the following. In Section 1, we set our notation and we introduce the main characters of the paper. Moreover, we establish a general framework that will be used in several specific situations through the paper, and we prove the first general technical results. In Section 2, we analyze the nihilpotent loci N i for a SAGA R regardless of the validity of some Lefschetz properties. We study them from a geometric point of view and we recollect some examples which show how the nature of these loci can be very different from one case to another. In Section 3 we prove the first main result, i.e. Theorem 3.1. In Section 4, we analyze the weak Lefschetz property for complete intersection SAGAs presented by quadrics and prove the "lifting" of the W LP to R from suitable quotients, i.e. Theorem 4.2. Finally, one might wonder how much the different techniques developed in this paper could be exploited to study the next open cases. In Section 5 we try to answer to this question, by generalizing the validity of some strong Lefschetz properties to complete intersection SAGAs presented by quadrics of higher codimension. and to Alberto Alzati and Riccardo Re for the helpful discussions on the topics of the paper. The authors are partially supported by INdAM -GNSAGA and by PRIN "Moduli spaces and Lie theory" and by (MIUR): Dipartimenti di Eccellenza Program (2018-2022) -Dept. of Math. Univ. of Pavia.

Notations, preliminaries and technical results
Through all the article we will assume that K is an algebraically closed field of characteristic 0. First of all, let us recall two basic definitions that will play an important role in what follows (the reader can refer to [HMM + 13] for a deeper treatment).
If R is a graded Artinian algebra, having the Gorenstein duality is equivalent to ask that R is Gorenstein. In this case, R N is called socle of R and R 0 ≃ R N . A standard Artinian Gorenstein algebra will be called, for brevity, SAGA.
Examples of SAGAs are algebras that can be written as K[x 0 , . . . , x n ]/I with I an homogeneous ideal generated by a regular sequence of polynomials of length n + 1. We will be interested in the case where the generators of I have all the same degree e: for brevity, in this situation we say that R is a complete intersection SAGA presented by forms of degree e. Relevant examples of this kind of SAGAs are jacobian rings of smooth hypersurfaces of P n .
We now introduce Lefschetz properties for a SAGA R. These involve the multiplication maps by suitable elements of R. In the following we will often deal with kernels of such maps so it is convenient to set K a q := ker q· : R a → R a+s for q ∈ R s . Definition 1.2. Consider a SAGA R = N i=0 R i . It is said to satisfy the • weak Lefschetz property in degree k, (W LP k in short) if there exists x ∈ R 1 such that x· : R k → R k+1 has maximal rank; • strong Lefschetz property in degree k at range s, (SLP k (s) in short) if there exists x ∈ R 1 such that x s · : R k → R k+s has maximal rank.
The algebra R has the strong Lefschetz property in degree k (SLP k ) if SLP k (s) holds for all s. We also say that R satisfies weak (strong) Lefschetz property -W LP (respectively SLP ) in shortif it satisfies W LP k (respectively SLP k ) for all k. Elements of P(R 1 ) for which the multiplication map is not of maximal rank will be important in Section 4 so it is convenient to introduce the following subschemes of P(R 1 ).
Definition 1.4. Let R be any SAGA with socle in degree N . For 1 ≤ a ≤ N − 1 we define the Lefschetz locus in degree a to be On the contrary, elements not in L a are called non-Lefschetz elements (in degree a).
Geometric results on these loci can be found, for example, in [AR19] and [BMMRN18].
In the next sections, we will make a large use of the framework that we present here in a general setting. This kind of construction has been used in some articles (see, for example [BFP22] and [AR19]) in order to study from a geometric perspective SAGAs which do not satisfy some Lefschetz properties.
Let R be a SAGA of codimension n + 1 and socle in degree N .
We will refer to N (a) k as nihilpotent loci of P(R a ). For brevity, we will set N (1) k = N k . We will denote by p 1 and p 2 the standard projections from Γ (a,b) i,j to P(R a ) and P(R b ), respectively. Notice that when we consider Γ (a,b) s,1 , we have so all the fibers of p 1 are projective spaces (not necessarily of the same dimension).
We will often assume the failure of some Lefschetz property and this assumption will reflect on a condition on the projection p 1 . More precisely, we make the following observations.
• Assume that b ≤ N/2 and that SLP b (s) does not hold. Then, for all [x] ∈ P(R 1 ) we have that the multiplication map x s · : such that x s y = 0 in R b+s . This shows that the failure of SLP b (s) is equivalent to ask that is surjective. Then, as observed above, we have that all the fibers of p 1 are projective spaces and this implies that there exists a unique irreducible component Θ of Γ s,1 which dominates P(R a ) via p 1 . In this case, we set and Construction 1.5. To summarize, if R is a SAGA of codimension n + 1 and socle in degree N and we assume that SLP b (s) does not hold for R, we can construct the loci Γ (1,b) s,1 , Θ, Y and F y as above and we have the following diagram We stress that, in this case, as Θ is the unique irreducible component which dominates P(R 1 ), we have In [BFP22] and in [FP22] the authors have developped a technical lemma which gives strong informations on the locus Θ introduced above. We present now a generalised version of it.
for all i, j such that i+j = k +1 and j ≥ 1.
Proof. It is enough to prove the claim for a general smooth point p Then we can define β(t) such that β(0) = y and (x + tv) i (β(t)) j = 0 for all t ∈ K. We can consider the expansion of β and write this relation as In particular we have ivx i−1 y j + jwx i y j−1 = 0 for all v ∈ R a . If we multiply by y we have ivx i−1 y j+1 = 0 for all v ∈ R a which yields the first claim since i ≥ 1 by hyphotesis.
For the second claim, consider the multiplication map R a × R (i−1)a+(j+1)b → R ia+jb+b and notice that it is non degenerate by the assumption ia + bj + b ≤ N . Hence, if vx i−1 y j+1 = 0 for all v ∈ R a , then one has also x i−1 y j+1 = 0 as claimed. Proposition 1.6 gives some easy but relevant relations involving the dimension of Y, Θ and F y for [y] ∈ Y general. We summarize these results in the following Lemma 1.7. Assume that R is any SAGA of codimension n + 1 with socle in degree N and assume that SLP b (k) does not hold for some integer k such that 1 ≤ k ≤ N − 2b. Then, defining Γ (1,b) k,1 , Θ, Y and F y as in 1.5, we have the following properties: Proof. One can prove (a) and (b) by following the proof of [BFP22, Proposition 2.6] and using Proposition 1.6. The last point has been proved in [BFP22, Corollary 2.11].
Through the whole article, if p ∈ X is a smooth point we will denote by T X,p the differential tangent space. The notation T p (X) will be used when X is inside a projective space P n to mean its embedded Zariski tangent space. In particular, if [x] ∈ X ⊆ P n is a smooth point, andX is the affine cone of X, we have P(TX ,x ) = T [x] (X). Let us now introduce a simple description of the tangent space for the nihilpotent loci which generalize [BFP22, Corollary 4.2]. The proof is analogous with minimal changes.
Let us now show a technical result that holds for any SAGA when we deny the SLP 1 (k) for some 1 ≤ k ≤ N/2. Then, as in 1.5 we can construct Θ ⊆ Γ (1,1) k,1 , Y and F y . For the second claim of (a), let us suppose by contradiction that [x] ∈ T Y, [y] . LetΘ be the lifting to R 1 × R 1 of Θ ⊂ P(R 1 ) × P(R 1 ) and letπ 2 be the projection on the second factor fromΘ. By construction, we have thatΘ ⊆ {(x, y) | x k y = 0}. LetỸ be the affine cone of Y . Since π 2 : Θ → Y is surjective, we haveπ 2 (Θ) =Ỹ and that, for p = (x, y) ∈Θ general, d pπ2 : TΘ ,p → TỸ ,y is surjective.
By assumption we have also that x ∈ TỸ ,y so there exists a tangent vector toΘ at p of the form (v, x). Since points ofΘ satisfy x k y = 0 we have which yields x k+1 = 0. This is impossible by the generality of x: For (b), first of all, notice that by Proposition 1.6 we haveF y ⊆ {x ∈ R 1 | x i y j = 0} for all i+j = k+1 and i, j ≥ 1. Hence, for p general, if an element v ∈ R 1 belongs to TF y ,x then the following relation must be satisfied Hence, we have that v ∈ K 1 x α y β , with α + β = k and β ≥ 1. Let us now consider a complete intersection SAGA R: we recall another technical result, dealing with the kernels of the multiplication maps presented above, that will be used extensively in the whole paper.
Proposition 1.10 (Proposition 4.1 of [BFP22]). Let R be any complete intersection SAGA of codimension n + 1 presented in degree d − 1. Assume that 1 ≤ s ≤ N − 1. The following properties hold: Finally, let us now focus on the specific case that will be treated in the next sections, that is the case where R is a complete intersection SAGA of codimension n + 1 presented in degree d − 1 = 2.
Remark 1.11. If d = 3, by Proposition 1.10 we get that for [η] ∈ P(R s ) we have dim(K 1 η ) ≤ s. From this, together with Lemma 1.8, we also get that dim(N k ) ≤ k − 2.

Nihilpotent loci and geometric properties
In this section we will study geometric properties of the nihilpotent loci N k ⊆ P(R 1 ) where R is a complete intersection SAGA of codimension n + 1 presented by quadrics. Any R as above has socle in degree N = n + 1. We stress that we don't make any assumptions about the validity of any weak or strong Lefschetz property for R in this section.
We recall that the nihilpotent loci (in P(R 1 )) are defined as If X ⊂ P r is non-empty, we denote by Sec k (X) ⊆ P r the k-secant variety associated to X, i.e. Sec k (X) := p 1 ,...,p k ∈X p 1 , . . . , p k where p 1 , . . . , p k is the linear span of the points p 1 , . . . , p k . For brevity, we set Sec 2 (X) := Sec(X). The interested reader can refer to [Rus16, Chapter 1] for various properties of these classical loci (although the definition considered is slightly different from the one adopted by us).
We stress that, like the non-Lesfschetz loci P(R 1 ) \ L k , the nihilpotent loci N k are expected to be empty when k is small for R "general". Hence it is interesting to study these loci when R is "special". For example, these loci give a lot of information for SAGAs for which some Lefschetz properties do not hold.
Let us start by analyzing the locus N 2 ⊆ P(R 1 ) ≃ P n . We recall that by Lemma 1.8 we have that dim(N 2 ) ≤ 0 so it is either empty or it is the union of a finite number of points. These points have to satisfy the following: are in general position in P(R 1 ). In particular, #N 2 ≤ n + 1 = N .
Proof. The statement is trivially true for k = 1. If k = 2 the only statement one has to check is that t 1 t 2 = 0. This is true since K 1 t 1 = t 1 by Proposition 1.10 and [t 1 ] = [t 2 ]. We will then proceed by induction assuming that the claim is true till k − 1.
Hence, let us suppose that t k ∈ t 1 , . . . , t k−1 , so we can write t k = k−1 i=1 a i t i . Then we have If k = 3 we have 0 = t 2 3 = 2a 1 a 2 t 1 t 2 so, since t 1 t 2 = 0 by induction hypothesis, we have either a 1 = 0 or a 2 = 0. This implies either {t 1 , t 3 } or {t 2 , t 3 } linearly dependent, and we get a contradiction since this is against the induction hypothesis. If k ≥ 4, by multiplying by Π k−3 i=1 t i , we get Since Π k−1 i=1 t i = 0 by induction hypothesis, we have either a k−2 = 0 or a k−1 = 0 and we have a contradiction as in the case k = 3.
By considering the Fermat hypersurface X = V (F ) in P n , one can easily see that, for the Jacobian ring R = S/J(F ), the set N 2 consists of exactly n + 1 independent points. However, also the converse is true, as shown by the following: Corollary 2.2. Assume that #N 2 = n + 1. Then R is the Jacobian ring of a cubic hypersurface X projectively equivalent to the Fermat cubic hypersurface in P n .
Proof. By assumption we have that N 2 = {[t 0 ], . . . , [t n ]}. By Proposition 2.1, {t 0 , . . . , t n } are n + 1 linearly independent forms so R = S/I with S = K[t 0 , . . . , t n ]. On the other hand, in S we have t 2 i ∈ I and {t 2 0 , . . . , t 2 n } is a regular sequence which generates I as ideal of S. Then, if we set F = n i=0 t 3 i , we have that I is the Jacobian ideal of the Fermat cubic hypersurface X = V (F ) as claimed.
In particular, every monomial of degree k + 1 in the variables t i is identically 0. Then ( k i=1 a i t i ) k+1 ≡ 0 for all a 1 , . . . , a k ∈ K so Sec k (N 2 ) ⊆ N k+1 . More generally, On the other hand, this always happens if r > k(a − 1): if α i ≤ (a − 1) for all i, we would have which gives a contradiction. Proof. Assume that L is a line in N 3 . Since the dimension of N 3 is at most 1 by Remark 1.11, we have that L is a component of N 3 . As dim(N 2 ) ≤ 0 and dim(K 1 x ) ≤ 1 for any x ∈ R 1 by Proposition 1.10(a), we can find [v], [w] ∈ L such that [v] = [w], [v], [w] ∈ N 2 and vw = 0. By hypothesis, we have that (v + tw) 3 = 0 for all t ∈ K so v 3 = v 2 w = vw 2 = w 3 = 0. Then, K 1 v 2 , K 1 w 2 and K 1 vw contain v, w . On the other hand, these subspaces have dimension at most 2 by Proposition 1.10(a) so they coincide with v, w . By Proposition 1.10(b), there exist λ, µ ∈ K such that (2) v 2 = λw 2 and vw = µw 2 so we have (v + tw) 2 = v 2 + 2tvw + t 2 w 2 = w 2 (t 2 + 2µt + λ).
We claim that t 2 +2µt+λ has two distinct roots so L is indeed a line contained in Sec(N 2 ). Assume, on the contrary, that t 2 + 2µt + λ is a square. This implies that µ 2 = λ. Then, from the Equations We will generalize this result in Theorem 2.6 by considering suitable linear subspaces contained in N k . We need first the following technical lemma.
Lemma 2.5. Let k ≥ 2 and let T be an hypersurface in P k ⊂ P(R 1 ). Assume either that (1) 0 ≤ s ≤ k − 1 or (2) s = k and the support of T is not contained in the union of 2 different hyperplanes.
Then there exist [x 0 ], . . . , [x s ] ∈ T which are linearly independent and such that Π s i=0 x i = 0.
Proof. The statement of the lemma is clearly true for s = 0. We will proceed by induction on s ≤ k. Then assume that there are [x 0 ], . . . , [x s−1 ] ∈ T which are linearly independent and with y = x 0 · · · x s−1 = 0. Consider the linear spaces τ 1 = x 0 , . . . , x s−1 and τ 2 = P(K 1 y ). We are done if we prove that U = T \(τ 1 ∪ τ 2 ) is not empty. By construction we have dim(τ 1 ) = s − 1 and dim(τ 2 ) ≤ s − 1 by Proposition 1.10. Hence, if s < k, U is an open dense subset of T . If s = k and the support of T is not contained in the union of 2 different hyperplanes, there exists an irreducible component C of T which is different from τ 1 and τ 2 . Then C \ (τ 1 ∪ τ 2 ) is not empty so U is again not empty as claimed.
Theorem 2.6. Assume that π is a (k − 1)-plane contained in N k+1 . Then (A) T k = π ∩ N k is an hypersurface of degree k (with possible multiple components) in π; In particular, T k is non degenerate in π and π ⊆ Sec k (N k ).
Proof. Notice, first of all, that dim(N k ) ≤ k − 2 by Remark 1.11, so π \ N k = ∅. Then, we can find {x 0 , . . . , x k−1 } linearly independent which span π and such that x k−1 ∈ N k . Since π ⊆ N k+1 , we have that (α 0 x 0 + · · · + α k−1 x k−1 ) k+1 ≡ 0 ∀α 0 , . . . , α k−1 ∈ K and this is equivalent to say that all monomials of degree k + 1 in the variables x 0 , . . . , x k−1 are 0. Then, if m is a monomial of degree k in these variables, either m = 0 or K 1 m = x 0 , . . . , x k−1 , by Proposition 1.10. In particular, we have that for each monomial of degree k there exists λ m ∈ K with m = λ m x k k−1 (recall that we assumed x k−1 ∈ N k ). Then (3) where p k (α) is a homogeneous polynomial of degree k in the variables α 0 , . . . , α k−1 . It is not 0 since the coefficient of α k k−1 is 1 by construction. By Equation (3), T k = π ∩N k is described by the vanishing of p k (α). In particular, N k is not empty and we have also proved (A).
For (B), if T k has support which is not contained in 2 different hyperplanes, the thesis follows directly from Lemma 2.5 so we have to discuss only the cases where H 1 and H 2 are distinct hyperplanes.
In both cases (B 1 ) and (B 2 ), there is an hyperplane H 1 of π contained in T k . We recall that T k is contained in N k by construction. By Lemma 2.5 applied to H 1 ⊂ π we can find [x 0 ], . . . , [x k−2 ] ∈ H 1 which are linearly independent and such that y = k−2 i=0 x i = 0. Since H 1 = [x 0 ], . . . , [x k−2 ] and H 1 ⊂ N k we have that all monomials of degree k in the variables x 0 , . . . , x k−2 are 0. Then, by Proposition 1.10, K 1 y = x 0 , . . . , x k−2 so H 1 = P(K 1 y ). If we are in case (B 1 ) we can then choose x k−1 in H 2 \ H 1 and {x 0 , . . . , x k−2 , x k−1 } is a set of points with the desired properties. We claim now that case (B 2 ) can not occur. Assume, by contradiction, that Supp(T k ) is the hyperplane H 1 = P(K 1 y ). Then for any x k−1 in π \ H 1 we have that π = x 0 , . . . , x k−2 , x k−1 , x k k−1 = 0 and yx k−1 = 0. With this choice of the x i 's, the polynomial p k (α) of Equation (3) is proportional to α k k−1 since T k = π ∩ N k has support on H 1 . On the other hand the coefficient of k−1 i=0 α i can not be zero since k−1 i=0 x i = yx k−1 = 0. We conclude this section by presenting some examples in order to make the phenomenology of the nihilpotent loci clearer (some computations have been made by using the computer algebra software Magma). We set S = K[x 0 , . . . , x n ] = k≥0 S k and we define {w 0 , . . . , w n } to be the projective coordinates on P(R 1 ) induced by the basis {x 0 , . . . , x n } of R 1 = S 1 .
Example 2.7. Let X be the Fermat cubic in P n and consider the Jacobian ring R of X. For any 2 ≤ k ≤ n we have ( i w i x i ) k ∈ J k if and only if all monomials in the {w i } of degree k without multiple factors vanish. This is true whenever any set of n − k + 1 variables is zero. With these arguments one can prove that N k is the union of the coordinated planes of dimension k − 2. In particular, N k = Sec k (N 2 ) and Sing(N k ) = N k−1 for k ≥ 2.
Example 2.8. Consider the smooth cubic surface X = V (f ) with f = x 3 0 + x 3 1 + x 3 2 + x 3 3 + 6x 0 x 1 x 2 and consider the Jacobian ring R of X. One has that there are 4 points in N 2 so, by Corollary 2.2, X is the Fermat cubic up to a projective transformation. Indeed, if λ is a non trivial third root of 1, we have Example 2.9. Consider the smooth cubic surface X = V (f ) with f = x 3 0 + x 3 1 + x 3 2 + x 3 3 + 3x 0 x 1 x 2 and consider the Jacobian ring R of X. If P = [0 : 0 : 0 : 1] and C = V (w 3 , g) is the smooth plane cubic with g := w 3 0 + w 3 1 + w 3 2 − 6w 0 w 1 w 2 , we have (considering the reduced structure) In particular, N 4 is the union of the plane containing the cubic curve C and the cone with vertex P generated by C. Notice that N 3 does not have pure dimension.
Example 2.10. Consider the smooth cubic surface X = V (f ) with f = x 3 0 + x 3 1 + x 3 2 + x 3 3 + x 0 (x 2 1 + x 2 2 + x 2 3 ) and let R be its Jacobian ring. One can show that, in this case, N 2 and N 3 are both empty whereas N 4 is a smooth quartic hypersurface.
Example 2.11. Consider the regular sequence {x 2 0 , x 2 1 , x 2 2 , x 2 3 + 2x 0 x 1 } in S = K[x 0 , . . . , x 3 ], the ideal J spanned by it and set R = S/J. Notice that J is not the Jacobian ideal of a cubic surface. Let P i be the coordinated points and consider the conic C = V (w 2 , g) with g := w 2 3 − 3w 0 w 1 . Then we have In particular, N 4 is the union of two planes (the first one -V (w 3 ) -contains P 0 , P 1 and P 2 and the lines joining these points whereas the second -V (w 2 ) -is the plane containing the conic C and the line P 0 , P 1 ) and V (g) (which is a quadric cone with vertex P 2 ). Notice that, as varieties, we have Sing(N k ) = N k−1 for k = 2, 3, 4.
3. Strong Lefschetz property in degree 1 for codimension 6 In this section we prove our main result, i.e. the following: Theorem 3.1. Let R be a complete intersection SAGA presented by quadrics of codimension 6. Then R satisfies the strong Lefschetz property in degree 1 (SLP 1 ), i.e. the general element x ∈ R 1 is such that the map x 4 · : R 1 → R 5 is an isomorphism.
In order to prove the above theorem we will need some technical results (Propositions 3.4, 3.5 and 3.6) which will be treated separately in Subsection 3.1 as they are interesting on their own because they are valid in a much larger framework than the one of the theorem. Now we will assume them and we will deal with the proof of the main theorem.
Proof. (of Theorem 3.1) Let us assume by contradiction that the statement does not hold: the map x 4 · : R 1 → R 5 is never injective for x ∈ R 1 . We are then in the situation described in Construction 1.5 with Γ = Γ First of all, let us recall that, by Lemma 1.7, we have Let us now list in the following table the possible values of the pairs (dim(Y ), dim(F y )). By using the constraints (4) and the various results proved in Subsection 3.1, we can exclude all the cases: in the Since no pair as above is possible for our framework, we get a contradiction and this concludes the proof.
We have the following easy but important consequence.
Corollary 3.2. The jacobian ring of a smooth cubic fourfold satisfies the strong Lefschetz property in degree 1.
3.1. General preparatory results. In this subsection we consider a complete intersection SAGA presented in degree d − 1, i.e. R = S/I where S = K[x 0 , . . . , x n ] and I is generated by a regular sequence of forms of degree d − 1. In this case, we recall that R is a SAGA of codimension n + 1 with socle in degree N = (d − 2)(n + 1). Moreover, we will assume that R is a SAGA which does not satisfy the strong Lefschetz property in degree 1 at range k with 2 ≤ k ≤ N − 2 (SLP 1 (k)). Equivalently, the multiplication map x k · : R 1 → R k+1 is never injective. Hence, we are in the situation described more generally in Section 1. Under the above assumptions we have We recall that Θ is the unique irreducible component of Γ k that dominates P(R 1 ) via its first projection π 1 , Y = π 2 (Θ) and . For this reason in this subsection we do not consider k = 1 since in this case it is possible to construct Γ 1 , but p 1 is never dominant (so Θ, Y and F y cannot be constructed).
Now we prove some results giving restrictions on the dimensions of Y and of the general F y with [y] ∈ Y . We are ultimately interested into the case where d = 3. Nevertheless, we stress that the following proposition (3.4) holds for every d ≥ 3.
Proposition 3.4. If we assume n > k d−2 and k ≥ 2, then In particular, if d = 3, then F y cannot be an hypersurface (k ≤ N − 2).
Proof. Recall that dim(F y ) ≤ n−1 by Lemma 1.7 so we have to rule out only the case dim(F y ) = n−1. Let us assume, by contradiction, that F y is an hypersurface. Hence, by denoting withF y the affine cone over F y , we have dim(F y ) = n.
) | x k y = 0} by assumption. We will show that the multiplication map x k−1 · : R 1 → R k is never injective so we can define, as we have done for Θ k , an incidence correspondence Γ k−1 with a unique irreducible component Θ k−1 which dominates P(R 1 ) via its first projection. Moreover, we will have Θ k = Θ k−1 so F y is also the fiber of the second projection from Θ k−1 and we can iterate this process.
Then, Θ is contained in Θ k−1 since it dominates P(R 1 ). On the other hand, since Γ k−1 ⊆ Γ k , we have also the other inclusion: Θ = Θ k−1 . In particular, the varieties Y and F y defined for Θ are the same as the ones defined for Θ k−1 . Then, by reasoning as before, we obtain n = dim(TF y ,x ) ≤ dim(K 1 x k−2 y ). If we assume that x k−2 y = 0 for p general in Θ, by Proposition 1.10 we would obtain n ≤ (k − 1)/(d − 2) ≤ k/(d − 2) which is, as before, incompatible with the hypothesis on n. Then Θ ⊆ Γ k−2 and we can iterate this process.
By recursion, we reduce ourselves to the case with k = 1. We can then see Θ as a subvariety of Γ 1 which dominates P(R 1 ) via its first projection. This implies the failure of the weak Lefschetz Property in degree 1. Then, by Remark 3.3, we get a contradiction: F y has dimension at most n − 2, as claimed.
Proposition 3.5. Assume that d = 3. Then, the dimension of Y is at most n − 3.
Proof. First of all, let us notice that if k ≤ N − 3 = n − 2, then by Proposition 1.6, Y is contained in N k+1 ⊆ N n−1 , whose dimension is at most n − 3 (see Lemma 1.8). Hence, we easily get that in this case dim(Y ) ≤ n − 3 as claimed.
Let us now consider the remaining case: k = N − 2 = n − 1. Recall that dim(Y ) ≤ n − 2 by Lemma 1.7 so, to prove the proposition, we only have to rule out the case dim(Y ) = n − 2. Assume by contradiction that dim(Y ) = n − 2. Since k = n − 1, proceeding as above, we have that Y is contained in N n , which has dimension at most n − 2. Hence, we get that Y is a component of N n . Then, if [y] ∈ Y is a general point (y n−1 = 0, since Y ⊆ N n−1 , for dimension reasons), we can write T y (Y ) = T y (N n ) = P(K 1 y n−1 ) by Lemma 1.8. However, the general [x] ∈ F y belongs to P(K 1 y n−1 ) and then to T y (Y ), contradicting Lemma 1.9.
Proposition 3.6. Assume that d = 3. Then, the following conditions are not compatible: Proof. Notice that for k = 1, condition (a) cannot hold since otherwise we would have dim(Y ) = n, which is impossible. Hence we can assume that k ≥ 2.
Let us assume by contradiction that both conditions (a) and (b) hold. By (b) we have y k−1 = 0 for y ∈ Y general so for ([x], [y]) general in Θ we have xy k−1 = 0. Indeed, otherwise, F y would be contained in P(K 1 y k−1 ), whose dimension is at most k − 2, which is impossible by assumption. As a consequence, we have that (6) for ([x], [y]) ∈ Θ general, x α y β = 0 for α + β = k with α, β ≥ 1 since, otherwise, by using Proposition 1.6, we would obtain also xy k−1 = 0.
By property (6) and since dim(F y ) = k − 1 by assumption, we also have by Lemma 1.9.
Let us now claim that To show this, first of all, recall that Y ⊆ N k−1 and Y ⊆ N k+1 . Let us now consider two cases: is general in F y ). In the first case, since Y is contained in N k+1 , we have T y (Y ) ⊆ P(K 1 y k ) by Lemma 1.8. Moreover, we have that P(K 1 y k ) = T x (F y ) since y k = 0 and dim(F y ) = k − 1. Analogously, for the second case we have T y (Y ) ⊆ P(K 1 y k−1 ) ⊆ P(K 1 xy k−1 ) = T x (F y ). Here, we have used that xy k−1 = 0 since p is general (by property (6)).
Let us take any w in TỸ ,y . By Property (8), we have that its class [w] belongs to T x (F y ). Moreover, by the surjectivity of d pπ2 , we can take in TΘ ,p an element of the form (v, w).

A lifting criterion for weak Lefschetz property
It is known that, the SLP for a graded algebra is inherited to its quotients by suitable conductor ideals (see, for example, [HMM + 13, Proposition 3.11]). In this section we prove a sort of converse for W LP 2 for complete intersection SAGAs presented by quadrics. More precisely, we will give a criterion to reduce the proof of W LP 2 for a SAGA R as above to a suitable quotient of R, modulo the existence of a non-Lefschetz element. We stress, moreover, that this criterion works for any codimension.
Let us start by setting S = K[x 0 , . . . , x n ] and by proving the following result.
Lemma 4.1. Assume that R = S/J is a complete intersection SAGA of codimension n + 1 presented by quadrics so the socle is in degree N = n+1. Assume that there are w, z ∈ R 1 \{0} such that zw = 0. Then (z) = (0 : w) andR = R/(z) is a complete intersection SAGA of codimension n presented by quadrics. In particular, dim(K s Proof. By definition, we have (z) ⊆ (0 : w), so we can define an epimorphism of graded K-algebras ϕ :R := R/(z) →R := R/(0 : w).
By [FP22, Lemma 2.3], the latter is a SAGA of codimension n and socle in degreeÑ = n. Considerinḡ R, it is clearly an Artinian standard algebra of codimension n. We want to show thatR is also a complete intersection SAGA presented by quadrics. By hypothesis, we know that zw ∈ J, so we can complete {zw} to a regular sequence of the form {g 0 , · · · , g n−1 , zw} spanning J. Notice that, by construction, g 0 , · · · , g n−1 do not belong to the ideal (z) and the reductionsḡ i of g i modulo (z) are a regular sequence of quadrics in the polynomial ringS = S/(z).
soR is a complete intersection SAGA presented by quadrics. In particular, it has socle in degreē N = n =Ñ . Since ϕ is an epimorphism and preserve the degrees, the image of a generatorσ ofR n is a non-zero multiple of the generatorσ ofR n . This also implies the injectivity of ϕ. Indeed, let us take a non-zero element x ∈R i . There exists y ∈R n−i such that xy =σ. Hence, we have λσ = ϕ(σ) = ϕ(xy) = ϕ(x)ϕ(y), and so we get that ϕ(x) can not be zero and ϕ is an isomorphism. In particular, for all s, as claimed.
We assume again that R = S/I is a complete intersection SAGA presented by quadrics with codimension n + 1 so that it has socle in degree N = n + 1. We recall that [z] ∈ P(R 1 ) is a non-Lefschetz element (in degree 1), i.e. z ∈ L 1 , if and only if there exists [w] ∈ P(R 1 ) such that zw = 0 in R or, equivalently K 1 z = w .
Theorem 4.2. Let R = S/I be a complete intersection SAGA of codimension n + 1 ≥ 6 presented by quadrics and assume that z is a non-Lefschetz element for R. IfR = R/(z) satisfies the W LP 2 , then the same holds for R.
Proof. First of all, by Lemma 4.1, we have thatR = R/(z) = R/(0 : w) is a complete intersection SAGA presented by quadrics and K 1 z = w . Let pr z be the projection R →R.
Assume, by contradiction, that W LP 2 holds forR but not for R. In particular, for all x ∈ R 1 , the multiplication map x· : R 2 → R 3 has non trivial kernel, i.e. K 2 x = {0}. Consider the incidence correspondence with its projections p 1 and p 2 on the factors.
We claim that p 1 is surjective. Since Γ is a closed subset, it is enough to show that for [x] ∈ P(R 1 ) general there exists [v] ∈ P(R 1 ) such that xvz = 0. Let x be a general element of R 1 . As K 2 x = 0 we have that there exists [q] ∈ P(R 2 ) such that xq = 0 in R. Then we have also pr z (xq) = xq = 0 inR. Since [x] is general in P(R 1 ), then the same holds for [x] ∈ P(R 1 ), so we getq = 0 inR 2 , as W LP 2 holds forR by assumption. Then, by Lemma 4.1, we have q ∈ (0 : w) 2 = (z) 2 = z · R 1 so there exists [v] ∈ P(R 1 ) such that 0 = xq = xvz as claimed.
In analogy with what happens for the construction 1.5, we have that there exists a unique irreducible component Θ of Γ which dominates P(R 1 ) via π 1 , where we set π i to be the restriction of p i to Θ for i = 1, 2. We have that for [x] ∈ P(R 1 ) general so the general fiber of π 1 has dimension at most 1 by Proposition 1.10.
We will show now that having Y = P(R 1 ) gives a contradiction. First of all, by reasoning as in Proposition 1.6, one can prove that Since Y = P(R 1 ) and squares of elements of R 1 generates R 2 (as R is standard), we have that z·R 2 = 0. This is impossible by Gorenstein duality.
In the statement of Theorem 4.2 we require n + 1 ≤ 6 since for codimension 5 the W LP 2 has already been proved (in [AR19] or in [BFP22] as consequence of SLP ) and in even smaller codimension, it easily follows from W LP 1 that is known to hold. From this, we get the following consequence.
Corollary 4.3. Let R be a complete intersection SAGA of codimension 6 presented by quadrics (e.g. R is the jacobian ring of a cubic fourfold). If L 1 is not the whole P(R 1 ), R satisfies W LP .

Some results for higher codimensions
In this section, our aim is to obtain some results concerning Lefschetz properties for complete intersection SAGAs presented by quadrics with codimension equal to n + 1 ≥ 4 by using techniques and results developed in the previous sections.
In particular, our framework will be as follows. Let us consider the polynomial ring S = K[x 0 · · · , x n ], an ideal I of S, generated by a regular sequence of quadrics (g 0 , · · · , g n ) and the corresponding complete intersection SAGA R = S/I, which has codimension n + 1 and socle in degree N = n + 1. We recall that we have denoted by SLP 1 (k) the strong Lefschetz property in degree 1 at range k: a SAGA R will satisfy the property SLP 1 (k) if the multiplication map x k · : R 1 → R k+1 is injective for [x] general in P(R 1 ). We recall that we have defined the loci Γ (a,b) i,j in Section 1. For brevity, we will write Γ k to mean Γ (1,1) k,1 . We will prove the following: Theorem 5.1. Let k ∈ {2, 3, 4} and assume that R is a complete intersection SAGA of codimension n + 1 presented by quadrics. If n ≥ k + 1 we have that R satisfies SLP 1 (k).
We will show the above theorem by splitting up the proof in 3 cases, which will be treated in the Propositions 5.2, 5.3 and 5.5 respectively.
Proof. Let us assume by contradiction that the multiplication map x 2 · : R 1 → R 3 is not injective for [x] ∈ P(R 1 ). Then we can consider the locus Γ 2 = {([x], [y]) ∈ P(R 1 ) × P(R 1 ) | x 2 y = 0}, with the corresponding varieties Θ, Y and F y defined as we have usually done. By Proposition 1.6, we obtain that Y ⊆ N 3 . Then, by Lemmas 1.8 and 1.7 we have that dim(Y ) = 1: hence F y must be a hypersurface, which is absurd by Proposition 3.4.
The following result has been already proved when n = 4 in [BFP22].
Proof. Let us assume by contradiction that the multiplication map x 3 · : R 1 → R 4 is not injective for [x] ∈ P(R 1 ). As in the proof of Proposition 5.2, let us construct Γ 3 , Θ, Y and F y . In this case, we have that Y ⊆ N 4 so dim(Y ) ≤ 2. If [y] ∈ Y is general, the only possible value for (dim(Y ), dim(F y )) is (2, n − 2) since F y can not be a hypersurface by Proposition 3.4. By dimension reasons, Y ⊆ N 3 , thus we have that the general element [y] of Y is such that y 3 = 0. Then, by Proposition 1.6, we get F y must be contained in P(K 1 y 3 ), whose dimension is at most 2. Then we have proved that n − 2 = dim(F y ) ≤ 2 which is impossible for n ≥ 5.
Before showing the analogous result for the SLP 1 (4), let us prove a technical result.
Lemma 5.4. Let R be a SAGA as at the beginning of Section 5 and consider 4 ≤ k ≤ n − 1. Assume that R does not satisfy SLP 1 (k) so one can consider the varieties Γ k , Θ, Y and F y constructed as in Section 1. For [y] ∈ Y general we have the following properties.
Proof. For (a), let us assume by contradiction that for [y] ∈ Y general, dim(F y ) = h ≥ k. Then, by Lemma 1.9, we have that for [x] ∈ F y general T [x] (F y ) = P(TF y ,x ) ⊆ P(K 1 x α y β ), where α + β = k, with β ≥ 1 andF y is the affine cone over F y . But since SLP 1 (k − 1) holds for R by hypothesis, for ([x], [y]) ∈ Θ general, we have that x k−1 y = 0, and so h = dim(F y ) ≤ dim(P(K 1 x k−1 y )) ≤ k − 1, which is impossible by the assumptions over h.
For (b), let us consider [y] ∈ Y general and assume by contradiction that dim(Y ) = dim(F y ) = k −1. By Proposition 1.6 we get that Y ⊆ N k+1 , and by Lemma 1.8 we deduce that Y is an irreducible component of N k+1 and we have that y k = 0. By reasoning as in the proof of Proposition 5.3 and by Proposition 1.10, we get F y = P(K 1 y k ). Moreover, since [y] is general in Y , we have T [y] (Y ) = P(K 1 y k ) = F y . With these conditions, we can proceed as in the proof of Proposition 3.6 and consider for example the equations We conclude this section by observing how much can be easily said, by using these methods, for the SLP 1 for complete intersection SAGA of codimension 7 (i.e. n = 6) presented by quadrics (e.g. for the Jacobian ring of a smooth cubic fivefold).
As in Theorem 3.1, we put in a table the possible values of the pairs (dim(Y ), dim(F y )) and we specify which result rules out the corresponding case. This concludes the proof.