The non-Lefschetz locus
aa r X i v : . [ m a t h . A C ] S e p THE NON-LEFSCHETZ LOCUS
MATS BOIJ, JUAN MIGLIORE, ROSA M. MIR ´O-ROIG, AND UWE NAGEL
Abstract.
We study the weak Lefschetz property of artinian Gorenstein algebras andin particular of artinian complete intersections. In codimension four and higher, it is anopen problem whether all complete intersections have the weak Lefschetz property.For a given artinian Gorenstein algebra A we ask what linear forms are Lefschetzelements for this particular algebra, i.e., which linear forms ℓ give maximal rank forall the multiplication maps × ℓ : [ A ] i −→ [ A ] i +1 . This is a Zariski open set and itscomplement is the non-Lefschetz locus .For monomial complete intersections, we completely describe the non-Lefschetz lo-cus. For general complete intersections of codimension three and four we prove that thenon-Lefschetz locus has the expected codimension, which in particular means that it isempty in a large family of examples. For general Gorenstein algebras of codimensionthree with a given Hilbert function, we prove that the non-Lefschetz locus has the ex-pected codimension if the first difference of the Hilbert function is of decreasing type.For completeness we also give a full description of the non-Lefschetz locus for artinianalgebras of codimension two. Introduction If A = R/I is an artinian standard graded algebra over the polynomial ring R = k [ x , . . . , x n ], where k is a field, then A is said to have the Weak Lefschetz Property(WLP) if the homomorphism induced by multiplication by a general linear form, fromevery degree to the next, has maximal rank. In this paper we will always assume that k has characteristic zero.A famous result in commutative algebra says that an artinian monomial complete in-tersection over a field of characteristic zero has the WLP (and even a stronger conditioncalled the Strong Lefschetz Property). This was proved in [20], [21], and [19]. A conse-quence of this is that if the generator degrees are specified, a general complete intersectionwith those generator degrees has the WLP. It is an open question whether every completeintersection has the WLP. Notice that the result above fails to distinguish between amonomial complete intersection and a general one (always with fixed generator degrees).We give a finer measure of the Lefschetz property that does distinguish between these(conjecturally in all cases, and we give a proof in ≤ A is given. For any pair of consecutivecomponents A i and A i +1 , we can consider the locus L i of linear forms that fail to inducea homomorphism of maximal rank on these components. We will observe that for each i the variety L i is a determinantal variety, so depending on the absolute value of the difference dim[ A ] i +1 − dim[ A ] i , there is an expected codimension. If the variety achievesthis codimension, its degree (as a possibly non-reduced scheme) is also known. One canthen ask further questions about L i , such as what are its irreducible components. If L i fails to have the expected codimension, it is still determinantal but its degree is less clear.We define the non-Lefschetz locus L I to be the union of these loci L i , viewed as subva-rieties of the corresponding projective space ( P n − ) ∗ , over all possible sets of consecutivecomponents. The algebra A fails to have the WLP if and only if L I = ( P n − ) ∗ . The ariety L I is thus a union of determinantal varieties in general. If A is Gorenstein (e.g. acomplete intersection), there is a natural sequence of inclusions of the L i , so L I is in factitself a determinantal variety. (See Proposition 2.5.)In this paper we will study the non-Lefschetz locus for specific algebras (monomial al-gebras) and we will consider it in the case of the general element of an irreducible family(complete intersections of prescribed generator degrees). Much more difficult is the ques-tion of whether every element of an irreducible family (specifically complete intersections)has the WLP, i.e. whether the non-Lefschetz locus is always of positive codimension forsuch algebras.In Section 3 we completely characterize the non-Lefschetz locus of monomial completeintersections (Proposition 3.1) and we also find all the possible Jordan types of linearforms in such algebras (Proposition 3.7).In Section 4 we conjecture that the non-Lefschetz locus of a general complete intersec-tion has the expected codimenion in the sense that will be made precise in Section 2. Weprove this conjecture for complete intersections of codimension three (Theorem 4.10) andcodimension four (Theorem 4.13).In Section 5 we study the non-Lefschetz locus of a general artinian Gorenstein algebraof codimension three with a given Hilbert function. In Theorem 5.1 we prove that thenon-Lefschetz locus has the expected codimension if the g -vector associated to the Hilbertfunction is of decreasing type, while it is of codimension one otherwise.In Section 6 we give a complete description of the situation for algebras in codimensiontwo. Acknowledgements.
Part of this work was carried out during a visit to IHP in Parisand part of the work was carried out during a visit to CIRM in Trento. We are verygrateful for these opportunities. We are also grateful to Anthony Iarrobino and JunzoWatanabe for useful discussions related to this work.The first author was partially supported by the grant VR2013-4545, the second au-thor by the National Security Agency under Grant H98230-12-1-0204 and by the SimonsFoundation under Grant
Preliminaries
Let R = k [ x , x , . . . , x n ] where k is an algebraically closed field of characteristic zero.Let M be a graded R -module of finite length. We first briefly recall an idea, originallydue to Joe Harris, dealing with an isomorphism invariant of M . For further details see[15].The module structure of M is determined by a collection of homomorphisms φ i : [ R ] → Hom k ( M i , M i +1 ) as i ranges from the initial degree of M to the penultimate degree where M is not zero. Since φ i is trivial if either [ M ] i or [ M ] i +1 is zero, we assume that this is notthe case (we do not assume that M is generated in the first degree, so a zero componentcould lie between non-zero ones). Let ℓ = a x + · · · + a n x n , and let us refer to the a i as the dual variables . If we choose bases for [ M ] i and for [ M ] i +1 , we can view φ i as a(dim[ M ] i +1 ) × (dim[ M ] i ) matrix B i whose entries are linear forms in the dual variables.For any fixed t we can thus consider the ideal of ( t + 1) × ( t + 1) minors of B i , and this is anisomorphism invariant of M . However, for our purposes it is enough to consider the ideal ofmaximal minors of B i . Denoting by Y i the scheme defined by the ideal of maximal minorsof B i , we can view Y i as lying in the dual projective space ( P n − ) ∗ = Proj( k [ a , . . . , a n ]). e have an expected codimension for Y i , and if that codimension is achieved then we alsohave a formula for deg Y i : Lemma 2.1.
Without loss of generality assume that dim[ M ] i ≤ dim[ M ] i +1 (otherwiseconsider the transpose of B i ). For sufficiently general entries of B i , the codimension of Y i is dim[ M ] i +1 − dim[ M ] i + 1 . If this codimension is achieved, then deg Y i = (cid:0) dim M i +1 dim M i − (cid:1) .Example . Harris’s motivation was to apply this machinery to liaison theory. Forinstance, let C ⊂ P be the union of four general lines. Let M ( C ) = M t ∈ Z H ( P , I C ( t )) , the Hartshorne-Rao module of C . We havedim M ( C ) t = t = 0;4 if t = 1;2 if t = 2;0 otherwise . Taking M = M ( C ), the expected codimension of Y is 4 − (cid:0) (cid:1) = 6. One can show that in fact Y is the curve in ( P ) ∗ obtained as the dualsof the four components of C together with the duals of the two 4-secant lines of C . Itthen follows from the fact that Y is an isomorphism invariant, and some now-classicalresults of liaison theory (with a small argument), that C is the only union of skew linesin its even liaison class.Our idea now is to apply this machinery to the study of the Weak Lefschetz property.Traditionally, we say that an artinian algebra A = R/I has the
Weak Lefschetz property(WLP) if there is a linear form ℓ ∈ [ A ] such that, for all integers i , the multiplicationmap × ℓ : [ A ] i → [ A ] i +1 has maximal rank, i.e. it is injective or surjective. In this case, the linear form ℓ is called a Lefschetz element of A . (We will often abuse terminology and say that the correspondingideal has the WLP.) The Lefschetz elements of A form a Zariski open, possibly empty,subset of [ A ] , which as above we will projectivize and view in ( P n − ) ∗ . This open set isnothing but ( P n − ) ∗ \ L I . This is our primary focus in this paper, but we note that A issaid to have the Strong Lefschetz property (SLP) if the analogous statements are true forthe multiplication maps × ℓ d : [ A ] i → [ A ] i + d for all i and d .If we consider A as an R -module, to say that A satisfies the WLP is equivalent tosaying that none of the varieties Y i is all of ( P n − ) ∗ . We first relabel the Y i with a moredescriptive notation for our application. Definition 2.3.
Given an artinian graded algebra A = R/I , we define L I := { [ ℓ ] ∈ P ([ A ] ) | ℓ is not a Lefschetz element } ⊂ ( P n − ) ∗ and we call it the non-Lefschetz locus of I (or of A ). For any integer i ≥
0, we define L I,i := { ℓ ∈ [ A ] | × ℓ : [ A ] i −→ [ A ] i +1 does not have maximal rank } ⊂ ( P n − ) ∗ . n order to study the non-Lefschetz locus from a scheme-theoretic perspective, we view L I,i not as a set but rather as the subscheme of ( P n − ) ∗ defined by the maximal minors of asuitable matrix, as explained above, taking M = A . The size of this matrix is determinedby the Hilbert function of A . More precisely, we introduce S = k [ a , a , . . . , a n ] as thehomogeneous coordinate ring of the dual projective space ( P n − ) ∗ , where we think of thecoordinates a , a , . . . , a n as the coefficients in ℓ = a x + a x + · · · + a n x n . For eachdegree i , the multiplication by ℓ on S ⊗ k A gives the map × ℓ : S ⊗ k [ A ] i −→ S ⊗ k [ A ] i +1 of free S -modules which is represented by a matrix of linear forms in S given a choice ofbases for [ A ] i and [ A ] i +1 . The locus L I,i ⊆ ( P n − ) ∗ is scheme-theoretically defined by theideal of maximal minors of this matrix and we denote this ideal by I ( L I,i ). Observe thatthis ideal is independent of the choice of bases. In this way, we have L I = S i ≥ L I,i , and L I ⊆ ( P n − ) ∗ is defined by the homogeneous ideal I ( L I ) = T i ≥ I ( L I,i ). Definition 2.4.
If codim L I,i takes the value prescribed by Lemma 2.1, where nowdim[ M ] i is the value of the Hilbert function of A in degree i , (and hence the degreeof L I,i is also determined by the Hilbert function), then we say that L i has the expectedcodimension and the expected degree .Since in this article we are studying Gorenstein algebras, especially complete intersec-tions, it will be useful to know that the non-Lefschetz locus is determined by the failureof injectivity of the multiplication by linear forms in a single degree. It is clear on aset-theoretical level that this is true (cf. [16, Proposition 2.1]). We will now look at thequestion when there is an inclusion of the ideals I ( L I,i +1 ) ⊆ I ( L I,i ) which will ensure thatwe only have to consider the middle degree even when we look at the non-Lefschetz locusdefined scheme-theoretically and not only set-theoretically.
Proposition 2.5. If h A ( i ) ≤ h A ( i + 1) ≤ h A ( i + 2) and [soc A ] i = 0 , then I ( L I,i +1 ) ⊆ I ( L I,i ) .Proof. The ideal I ( L I,i +1 ) is generated by the maximal minors of the matrix representingthe map × ℓ : S ⊗ k [ A ] i +1 −→ S ⊗ k [ A ] i +2 , where ℓ = a x + a x + · · · + a n x n . Each suchminor equals the determinant of the matrix representing the map × ℓ : S ⊗ k [ B ] i +1 −→ S ⊗ k [ B ] i +2 where B = A/J and J is an ideal generated by h A ( i + 2) − h A ( i + 1) forms of degree i + 2.Since [ A ] i = [ B ] i and [ A ] i +1 = [ B ] i +1 , we can prove the inclusion I ( L I,i +1 ) ⊆ I ( L I,i ) for A by proving the inclusion for all such quotients B = A/J . Therefore, we will now assumethat h A ( i + 1) = h A ( i + 2).Suppose that L I,i +1 = ( P n − ) ∗ . Then I ( L I,i +1 ) = h i and the inclusion of ideals istrivial. If L I,i = ( P n − ) ∗ we will also have that L I,i +1 = ( P n − ) ∗ since A by assumptiondoes not have socle in degree i and the inclusion of ideals is again trivial. Thus we onlyhave to consider the case when L I,i = ( P n − ) ∗ and L I,i +1 = ( P n − ) ∗ . In this case, we canchange coordinates so that × x n : [ A ] i −→ [ A ] i +1 and × x n : [ A ] i +1 −→ [ A ] i +2 both havemaximal rank. Consider the diagram ⊗ k [ A ] i S ⊗ k [ A ] i +1 S ⊗ k [ A ] i +1 S ⊗ k [ A ] i +2 × x n × ℓ × ℓ × x n The injectivity of the two vertical maps shows that we can choose monomial cobasesfor [ A ] i , [ A ] i +1 and [ A ] i +2 in such a way that the matrix representing the map × ℓ : S ⊗ k [ A ] i −→ S ⊗ k [ A ] i +1 is a submatrix of the matrix representing the map × ℓ : S ⊗ k [ A ] i +1 −→ S ⊗ k [ A ] i +2 . The ideal I ( L I,i +1 ) is principal, generated by the determinant of the matrixrepresenting the map × ℓ : S ⊗ k [ A ] i +1 −→ S ⊗ k [ A ] i +2 . Since the two matrices have thesame number of rows, the Laplace expansion of the determinant of the larger matrix showsthat this determinant is in the ideal generated by the maximal minors of the submatrix,which proves the inclusion I ( L I,i +1 ) ⊆ I ( L I,i ). (cid:3) Corollary 2.6. If A = R/I is Gorenstein of socle degree e then L I = L I,i scheme-theoretically, where i = ⌊ e − ⌋ .Proof. If A does not have the WLP, we have L I = L I,i = ( P n − ) ∗ . If A has the WLP theHilbert function is unimodal and by Proposition 2.5 and the duality of the Gorensteinalgebra we get the equality. (cid:3) Remark . If A = R/I has socle in degree i , we need not have the inclusion I ( L I,i +1 ) ⊆ I ( L I,i ) since we then have that I ( L I,i ) = h i while I ( L I,i +1 ) might be non-trivial.If A is not Gorenstein, but level , we can get a similar result as Corollary 2.6 but insome cases we will have to use two degrees instead of one since we cannot apply duality.(cf. [16, Proposition 2.1] for the set-theoretic statement.)3. The non-Lefschetz locus of a monomial complete intersection
In this section and the next we will restrict ourselves to the case of complete intersec-tions. In this section we study monomial complete intersections.Notice that to say that an artinian ideal I ⊂ R has the WLP is equivalent to saying thatcodim L I ≥
1. The aim of this section is to study codim L I when I = h F , · · · , F n i ⊂ R isa monomial complete intersection. We know that for a monomial complete intersection,hence for a general choice of F , · · · , F n , R/I has the WLP, thanks to the main result of[20], [21] and [19]; and the same holds for any choice of F i if n ≤ Proposition 3.1.
Let I = h x d , · · · , x d n n i ⊂ R := k [ x , · · · , x n ] be an artinian monomialcomplete intersection, with socle degree e = d + · · · + d n − n . Assume without loss ofgenerality that d n ≥ · · · ≥ d ≥ . Then the following characterization of the Lefschetzelements holds. (1) If d n > ⌊ e +12 ⌋ then ℓ = a x + a x + · · · + a n x n is a Lefschetz element if and onlyif a n = 0 . (2) If e is even and d n ≤ ⌊ e +12 ⌋ then ℓ = a x + a x + · · · + a n x n is a Lefschetzelement if and only if a i = 0 for at most one index i and a j = 0 for all indices j with d j > . (3) If e is odd and d n ≤ ⌊ e +12 ⌋ then ℓ = a x + a x + · · · + a n x n is a Lefschetz elementif and only if a a · · · a n = 0 . roof. We start by fixing the linear form ℓ = a x + a x + · · · + a n x n . Let A = R/I , let S = { i : a i = 0 } ⊆ { , , . . . , n } and define the subrings A ′ and A ′′ of A as the subringsgenerated by { x i } i ∈ S and by { x i } i/ ∈ S , respectively. Both A ′ and A ′′ are monomial completeintersections and ℓ acts trivially on A ′′ while it is a Lefschetz element on A ′ .In order to determine whether or not ℓ is a Lefschetz element on A , it is sufficient toconsider the injectivity of the multiplication map in the middle degree, i.e.,(3.1) × ℓ : [ A ] ⌊ e − ⌋ −→ [ A ] ⌊ e +12 ⌋ . For any integer j , we have that[ A ] j = ([ A ′ ] j ⊗ [ A ′′ ] ) ⊕ ([ A ′ ] j − ⊗ [ A ′′ ] ) ⊕ · · · ( ⊕ [ A ′ ] ⊗ [ A ′′ ] j )and since ℓ acts trivially on A ′′ , the injectivity of (3.1) is equivalent to injectivity in eachcomponent × ℓ : [ A ′ ] ⌊ e − ⌋− i ⊗ [ A ′′ ] i −→ [ A ′ ] ⌊ e +12 ⌋− i ⊗ [ A ′′ ] i , for all i ≥ × ℓ : [ A ′ ] ⌊ e − ⌋ −→ [ A ′ ] ⌊ e +12 ⌋ implies injectivity in thelower degrees of A ′ . Since ℓ is a Lefschetz element on A ′ , we have injectivity of the lattermap if and only if dim k [ A ′ ] ⌊ e − ⌋ ≤ dim k [ A ′ ] ⌊ e +12 ⌋ . Since ⌊ e +12 ⌋ is above the middle degree if S = { , , . . . , n } , we must have a flat top inthe Hilbert function of A ′ between degree e ′ − ⌊ e +12 ⌋ and degree ⌊ e +12 ⌋ in this situation,where e ′ is the socle degree of A ′ . If this forced flat top has length two, we must have ⌊ e +12 ⌋ − e ′ − ⌊ e +12 ⌋ which is only possible if e is even and e ′ = e −
1. In this case A ′′ is generated by one variable x i with d i = 2 and ℓ is a Lefschetz element on A .If there is a flat of length at least three, it follows from [19, Theorem 1] that one of thegenerators of the defining ideal of A ′ must have a degree which is above the end of theflat. There can be at most one d i which is greater than ⌊ e +12 ⌋ , so in this case we musthave d n > ⌊ e +12 ⌋ . In this case, ℓ is a Lefschetz element of A .We now relate what we have shown with the statements of our proposition.In the case (1), we get that ℓ is a Lefschetz element if and only if d n is the degree ofone of the generators of the defining ideal of A ′ , which is equivalent to a n = 0.If d n ≤ ⌊ e +12 ⌋ , the only case when ℓ is a Lefschetz element and A ′ = A is when e is evenand A ′′ = k [ x i ] / h x i i . This shows (2) and (3). (cid:3) Remark . Case 2 of Proposition 3.1 shows that the non-Lefschetz locus does not needto be unmixed. The smallest example is for d = d = 2 and d = d = 3 where we get I ( L I ) = h a a a , a a a , a a a a , a a a , a a a , a a a , a a a a , a a a , a a a a ,a a a a , a a a , a a a , a a a , a a a , a a a a , a a a a , a a a , a a a ,a a a , a a a a , a a a , a a a a , a a a a , a a a a , a a a a , a a a a i with radical p I ( L I )) = h a a a , a a a i = h a , a i ∩ h a i ∩ h a i . Example . Proposition 3.1 only gives us that L I is defined set-theoretically by theequation a · · · a n = 0 in the cases given by (3). Scheme-theoretically, L I is defined by anideal generated by maximal minors of certain matrices as seen in Section 2. For instance,if n = 3 and d = d = d = 4, the Hilbert function is (1 , , , , , , , , ,
1) andthe defining polynomial of L I is a a a . More generally, if n = 3 and d = d = d = d where d is even, then the Hilbert function of R/I is (1 , h , . . . , h e ) with e = 3 d − h d − = h d − = 3 (cid:0) d (cid:1) and the defining polynomial of L I is ( a a a )( d ) . his example leads to the following two immediate corollaries. Corollary 3.4. If h F , . . . , F n i is any complete intersection in k [ x , . . . , x n ] with deg F = · · · = deg F n = d , and if n ( d − is odd (i.e. if n is odd and d is even), then the value ofthe Hilbert function in degrees n ( d − − and n ( d − is divisible by n . Corollary 3.5.
Let I = h x d , . . . , x d n n i . If d = · · · = d n = 2 and n is even thenthe non-Lefschetz locus L I has codimension 2. In all other cases, it has codimension 1.Furthermore, if d = · · · = d n = d where n is odd and d is even, then I ( L I ) = ( a α · · · a αn ) where α = n h n ( d − − and h n ( d − − = h n ( d − . When d = 2 , this is equal to (cid:0) n n − (cid:1) .Proof. The ideas are contained in the proof of Proposition 3.1. In particular, under thehypothesis d = · · · = d n = d where n is odd and d is even, the expected codimensionof the non-Lefschetz locus is achieved, namely codimension 1. In this case the degree ofthe non-Lefschetz locus is equal to h n ( d − − , and the generating polynomial has to besymmetric with respect to all n variables. The fact that α is an integer is guaranteed byCorollary 3.4. (cid:3) Remark . One can also study the non-Lefschetz locus with respect to the Strong Lef-schetz Property. Junzo Watanabe has communicated to us that he has extended Corol-lary 3.5 for the question of the Strong Lefschetz Property, showing that a x + · · · + a n x n is a Strong Lefschetz element for R/ ( x , . . . , x n ) if and only if a a . . . a n = 0. Thus thenon-Lefschetz locus for R/ ( x , . . . , x n ) for the Strong Lefschetz Property has codimension1, not 2 as it was for the non-Lefschetz locus for the Weak Lefschetz Property.Notice that if a linear form ℓ is a non-Weak-Lefschetz element for R/I then of courseit is a non-Strong-Lefschetz element, so Watanabe’s case is the only one left open byCorollary 3.5.3.1.
Jordan types.
Multiplication by a linear form ℓ corresponds to a nilpotent linearoperator on the artinian algebra A . The Jordan type of this nilpotent operator is aninteger partition P L of dim k A .The study of Jordan types refines the study of Lefschetz properties as we have thefollowing: • ℓ is a weak Lefschetz element if and only if the number of parts of P L equals themaximal value of the Hilbert function of A . • ℓ is a strong Lefschetz element if and only if P L equals the dual partition thepartition given by the Hilbert function of A .Here we investigate the possible Jordan types of linear forms for the case when A is amonomial complete intersection.For a degree sequence d , d , . . . , d n let P d ,d ,...,d n denote the dual partition to the parti-tion given by the Hilbert function of an artinian complete intersection of type ( d , d , . . . , d n ).For a partition P we denote by P k the partition given by repeating all parts of P k times.
Proposition 3.7.
Let A = k [ x , x , . . . , x n ] / h x d , x d , . . . , x d n n i be a monomial completeintersection in characteristic zero. The possible Jordan types for linear forms ℓ are P md i ,d i ,...,d ik , where m = Q nj =1 d j / Q kj =1 d i j , for all non-empty subsequences d i , d i , . . . , d i k of d , d , . . . , d n .Proof. From the action of the torus ( k ∗ ) n we see that the Jordan type of a linear form ℓ = a x + a x + · · · + a n x n depends only on which coefficients are non-zero. Let { i , i , . . . , i k } e the indices for which the coefficients are non-zero and let { j , j , . . . , j n − k } be theremaining indices.Let A ′ be the artinian monomial complete intersection of type ( d i , d i , . . . , d i k ) and let A ′′ be the artinian mononomial complete intersection of type ( d j , d j , . . . , d i n − k ). We nowhave A ∼ = A ′ ⊗ A ′′ and ℓ = P kj =1 a i j x i j is a strong Lefschetz element acting on the firstfactor while it acts trivially on the second factor. Thus the Jordan type of ℓ is P md i ,d i ,...,d ik ,where m = dim k A ′′ == Q nj =1 d j / Q kj =1 d i j . (cid:3) Example . The situation is easiest to summarize when all degrees are equal. Considerfor example the case n = 4 and d = d = d = d = 2. There are combinatorially justfour possible subseqences and the four possible Jordan types are[5 3 ] , [4 2 ] = [4 ] , [3 1] = [3 ] and [2] , corresponding to the linear forms x + x + x + x , x + x + x , x + x and x , respectively.4. The non-Lefschetz locus of a general complete intersection
In the previous section we considered the non-Lefschetz locus of a monomial completeintersection, and saw that it has codimension 1. We also know that a general completeintersection has a non-Lefschetz locus of positive codimension (since the complete inter-section has the WLP, thanks to the main result of [20], [21] and [19]). The purpose ofthis section is to describe the precise codimension of this locus for a general completeintersection.
Notation . We begin in the setting of R = k [ x , . . . , x n ], and then turn to the case n = 3 ,
4. Throughout this section we will fix integers 2 ≤ d ≤ · · · ≤ d n , and I will be acomplete intersection ideal, I = h F , . . . , F n i , where deg F i = d i and F i is a general formof degree d i . We will denote by e the socle degree of R/I , namely e = ( P ni =1 d i ) − n . Wewill denote by (1 , h , . . . , h e − , h e ) the h -vector (i.e. Hilbert function) of R/I .We will describe the expected codimension of the non-Lefschetz locus in Conjecture 4.3.One of our goals is to prove that for n = 3 or 4, and for a general choice of F i , 1 ≤ i ≤ n ,the non-Lefschetz locus L I of I = h F , . . . , F n i has the expected codimension. Remark . When the F i are general, we know that R/I has the WLP, so L I = ( P n − ) ∗ .In the case where the socle degree e is odd, the Hilbert function of R/I has at least twovalues in the middle that are equal. Thanks to Corollary 2.6, this means that L I is definedby the vanishing of the determinant of a square matrix of size h e − × h e +12 , hence (since L I = ( P n − ) ∗ ) L I is a hypersurface of degree δ I = h e − . So the case of odd socle degree iscompletely understood, and from now on we will assume without loss of generality that e is even.Based on computer experiments [8] and our results in four or fewer variables, we makethe following conjecture. Conjecture 4.3.
Let I = h F , · · · , F n i ⊂ R be a complete intersection ideal of generalforms as in Notation 4.1, and assume that e is even (see Remark 4.2). Then codim L I = min { h e − h e − + 1 , n } where we consider the empty set to have codimension n in P n − . In particular, L I ⊂ ( P n − ) ∗ is non-empty if and only if h e − h e − ≤ n − and in that case δ I := deg( L I ) = (cid:0) h e h e − h e − +1 (cid:1) . emark . Notice that in Conjecture 4.3 the hypothesis that the complete intersectionartinian ideal I ⊂ R is generated by general forms cannot be dropped. In fact, a completeintersection I ⊂ k [ x , x , x ] of type (3 , ,
3) has h -vector (1 , , , , , , L I is 2; and we will see later that indeed it istrue for a general choice of 3 cubics F , F , F ∈ k [ x , x , x ] (cf. Theorem 4.10). Butunfortunately it is not true for every choice. For instance, we saw in the last sectionthat if we take I = h F , F , F i = h x , x , x i we get that codim L I = 1 since a line a x + a x + a x ∈ k [ x , x , x ] fails to be a Lefschetz element of k [ x , x , x ] / h x , x , x i if and only if a a a = 0. Therefore, if we fix coordinates a , a and a in ( P ) ∗ , thesupport of L I is the union of the lines ℓ : a = 0, ℓ : a = 0 and ℓ : a = 0. Remark . We will see shortly that to measure the non-Lefschetz locus in ( P ( n − ) ∗ , itwill be enough to measure how many such algebras fail the WLP in a suitable irreducibleparameter space. As noted in Section 2, if R/I is a complete intersection and the WLPfails, it must fail ”in the middle”, and possibly also in other degrees. By semicontinuityand under the hypothesis that e is even, to measure the dimension of the set of algebrasfailing the WLP (in an irreducible parameter space) we can assume that WLP fails fromdegree h e − to h e (and, by duality, from h e to h e +1 ), and that the failure is just by one. Remark . We have d ≤ · · · ≤ d n . For large values of d n the question of the non-Lefschetz locus for a general complete intersection with generator degrees d , . . . d n isclear.(1) If d n ≥ d + · · · + d n − − ( n −
1) + 2 = d + · · · + d n − − n + 3 then h e − = h e (remembering that we are assuming e even), and the conjecture is clear (with thenon-Lefschetz locus consisting of the linear forms through individual points).(2) If d n = d + · · · + d n − − ( n −
1) + 1 = d + · · · + d n − − n + 2 then R/ ( F . . . F n − ) isthe coordinate ring of the reduced complete intersection set of points, Z , in P n − defined by ( F , . . . , F n − ), which reaches the multiplicity in degree d + · · · + d n − − ( n − { h i } is the Hilbert function of R/ ( F , . . . , F n ), then clearly • d + · · · + d n − − ( n −
1) = e ; • h e − h e − = 1; • h e = d d . . . d n − . • The Hilbert function of
R/I agrees with that of
R/I Z in degrees ≤ e .Notice that Z has the Uniform Position Property, since the F i are general. Weclaim that a linear form ℓ fails to have maximal rank from degree e − to degree e if and only if ℓ vanishes on (any) two points of Z . Indeed, if P , P ∈ Z , notice first that the Hilbert function of Z \{ P } agrees withthat of Z up to and including degree e −
1, and is one less than that of Z from thenon. The Hilbert function of Z \{ P , P } agrees with that of Z up to and includingdegree e −
2, is one less than that of Z in degree e −
1, and is two less than thatof Z from degree e on. In particular, there is a form of degree e − Z except P ∪ P , but the same is not true for all of Z except only P .Since R/I Z has depth 1, a linear form ℓ not vanishing on any point of Z is anon-zerodivisor, so the resulting multiplication from degree e − e isinjective. If ℓ vanishes at just one point, P , of Z , then for a form F of degree − ℓ · F = 0 in R/I means that F vanishes at all points of Z except P . Butwe know that any form of degree e − Z , so F = 0 in R/I . On the other hand, any linear form vanishingon the line spanned by P and P lies in the non-Lefschetz locus, which then hascodimension 2 and degree (cid:0) h e (cid:1) as claimed in Conjecture 4.3.(3) If d n = d + · · · + d n − − n + 1 then R/I has odd socle degree, so the non-Lefschetzlocus has codimension 1 and degree d . . . d n − − d n = d + · · · + d n − − n . In this case h F , . . . , F n − i defines acomplete intersection set of d · · · d n − points, Z , and its Hilbert function reachesits multiplicity in degree d + · · · + d n − − n + 1 = deg F n + 1. More precisely,letting s = d + · · · + d n − and d = d · · · d n − , its Hilbert function isdegree 0 1 2 . . . ( s − n −
1) ( s − n ) ( s − n + 1) ( s − n + 2) . . . n h . . . d − n d − d d . . . and the Hilbert function of R/I isdegree 0 1 2 . . . ( s − n −
1) ( s − n ) ( s − n + 1) . . . e − e n h . . . d − n d − d − n . . . n ℓ ∈ R , the failure of × ℓ : [ R/I ] s − n − → [ R/I ] s − n to be injective isequivalent to the condition that the restriction ¯ F n of F n to R/ h ℓ i is in the restrictedideal h ¯ F , . . . , ¯ F n − i . Since I is artinian, it follows then that h ¯ F , . . . , ¯ F n − i is acomplete intersection. In particular, ℓ is a non-zerodivisor on R/ h F , . . . , F n − i .We also note that in this situation, the conjectured codimension of L I is ( d − − ( d − n ) + 1 = n − P n − ) ∗ , i.e. there should only be a finite number of linearforms failing to induce an injective homomorphism from degree s − n − s − n .Thus from now on we may assume that d n ≤ d + · · · + d n − − n , and if equality holdswe have an equivalent condition for failure to have maximal rank.Our goal in this section is to prove Conjecture 4.3 in the cases n = 3 and n = 4. Webegin with a description of the approach that we will take except for Theorem 4.10. Fixdegrees d , . . . , d n for the complete intersections in R = k [ x , . . . , x n ], with d ≤ d ≤· · · ≤ d n . Let CI ( d , . . . , d n ) be the irreducible space parametrizing all such completeintersections. Let ( P n − ) ∗ be the projective space parametrizing the linear forms of R (upto scalar multiple). For each complete intersection I and linear form ℓ , we consider thepair ( ℓ, I ) ∈ ( P n − ) ∗ × CI ( d , . . . , d n ). Let X be the set of such pairs such that ℓ is not aLefschetz element for A = R/I .Since the d i are given, there is a precise degree where this latter condition must bechecked: ( ℓ, I ) ∈ X if and only if × ℓ : [ R/I ] e − → [ R/I ] e fails to be injective (recallthat the socle degree e is assumed to be even, thanks to Remark 4.2). Since the generalelement of CI ( d , . . . , d n ) has the WLP, there are expected values for the Hilbert functionof R/ ( I, ℓ ) in degrees e and e + 1 (the latter being 0), and ( ℓ, I ) ∈ X if and only if thesevalues are not achieved.Consider the projections φ and φ :(4.1) ( ℓ, I ) ∈ ( P n − ) ∗ × CI ( d , . . . , d n ) ⊃ X ( P n − ) ∗ CI ( d , . . . , d n ) φ φ e need to show that there is a non-empty open set U ⊂ CI ( d , . . . , d n ) such that if I ∈ U then the closure of φ ( φ − ( I ) ∩ X ) has the expected codimension as described inConjecture 4.3. Thus we want to show that the intersection of X with the generic fibreof φ has the expected dimension (computed from Conjecture 4.3). More precisely, let m = ( n − − min { h e − h e − + 1 , n } , the expected dimension of L I , and let I be a generalelement of CI ( d , . . . , d n ). Then Conjecture 4.3 says that(4.2) dim( φ − ( I )) ∩ X = m. We will reformulate this. Let p = dim CI ( d , . . . , d n ). We want to show that there is anopen subset U ⊂ CI ( d , . . . , d n ) such thatdim( φ − ( U ) ∩ X ) = m + p. Now, φ is surjective, and the fibres all have the same dimension (since we can always doa change of variables). Thus we want to show that for any linear form ℓ (viewed as anelement of ( P n − ) ∗ ), dim( φ − ( U ) ∩ X ∩ φ − ( ℓ )) = m + p − ( n − . So from now on we fix a linear form ℓ . We denote by ACI ℓ ( d , . . . , d n ) the irreduciblespace of ideals in S = R/ ( ℓ ) with generators in degrees d , . . . d n . We note that an ideal in ACI ℓ ( d , . . . , d n ) may have only n − d n > d + · · · + d n − − ( n −
1) and ℓ is a non-zerodivisor on R/ h F , . . . , F n − i , but wehave assumed this not to be the case in Remark 4.6. But even avoiding this situation, itmay happen that an ideal in CI ( d , . . . , d n ) restricts to an ideal in ACI ℓ ( d , . . . , d n ) withonly n − V ⊂ ACI ℓ ( d , . . . , d n ) be the open subset consistingof restricted ideals ( ¯ F , . . . , ¯ F n ) such that all the ¯ F i are minimal generators.Consider the morphism(4.3) CI ( d , . . . , d n ) ACI ℓ ( d , . . . , d n ). φ We want to study a certain subvariety, Y ⊂ ACI ℓ ( d , . . . , d n ). The precise definition of Y will depend on the value of d n , breaking into two cases, but the treatment of Y will bethe same in both cases.Case 1: d n = d + · · · + d n − − n . We have seen in Remark 4.6 (4) that in this case m = 0,and that failure of maximal rank is equivalent to ¯ F n ∈ h ¯ F , . . . , ¯ F n − i , which then is acomplete intersection. By Remark 4.5, or by direct observation in this case, we can assumethat the Hilbert function of the restricted ideal differs by one, in degrees d + · · · + d n − − n and d + · · · + d n − − ( n − Y ⊂ ACI ℓ ( d , . . . , d n ) be thesubset in the complement of V consisting of those ideals such that the first n − d n < d + · · · + d n − − n . In this case we let Y ⊂ V be the set of ideals ¯ I such that h S/ ¯ I ( e + 1) >
0. (The distinction between the cases is that the ideals of Y are completeintersections in Case 1, and are not complete intersections in Case 2.)Notice that in both cases,dim φ − ( Y ) = dim( φ − ( U ) ∩ X ∩ φ − ( ℓ )) . otice also that the fibres of φ over V ∪ Y all have the same dimension, namely p − dim ACI ℓ ( d , . . . , d n ) . So we want to show thatdim Y + p − dim ACI ℓ ( d , . . . , d n ) = m + p − ( n − , i.e. that dim Y = m − ( n −
1) + dim
ACI ℓ ( d , . . . , d n ) . Equivalently, we want to show that(4.4)
The codimension of Y in ACI ℓ ( d , . . . , d n ) is min { h e − h e − + 1 , n } . This is what we will prove in the results below.We have noted above that without loss of generality we can assume that d n ≤ d + · · · + d n − − n , and that the case of equality is handled slightly differently from the caseof strict inequality. We now consider equality. Proposition 4.7.
Let I = h F , . . . , F n i ⊂ R = k [ x , . . . , x n ] be a complete intersectiongenerated by general forms of degrees ≤ d ≤ d ≤ · · · ≤ d n . Assume that d n = d + · · · + d n − − n . Then Conjecture 4.3 is true.Proof. We have defined the quasi-projective variety Y in Case 1 above. From what wesaid in Remark 4.6 (4) and in Case 1 of the discussion above, we want to show that thecodimension of Y in ACI ℓ ( d , . . . , d n ) is n −
1. We recall that a complete intersection oftype ( d , . . . , d n − ) in R/ h ℓ i with d ≥ n − d n = d + · · · + d n − − n .Now, let M ℓ ( d , . . . , d n − ) be the variety parametrizing the ideals with generator de-grees d , . . . , d n − , and let U ′ ⊂ M ℓ be the dense open subset consisting of completeintersections of type ( d , . . . , d n − ). Consider(4.5) Y ⊆ ACI ℓ ( d , . . . , d n ) U ′ ⊆ M ℓ ( d , . . . , d n − ) . φ We have that Y is contained in φ − ( U ′ ), and φ − ( U ′ ) is a dense open subset of ACI ℓ ( d , . . . , d n ). For any J ∈ U ′ , the codimension of φ − ( J ) ∩ Y in φ − ( J ) is n − (cid:3) Thus from now on we can assume that d n < d + · · · + d n − − n , and that we are inCase 2 above. To fix the ideas for most of the rest of the paper in a simple first case, wefirst state the case n = 3 and deg( F i ) = d for 1 ≤ i ≤ n = 4 and deg( F i ) = d for 1 ≤ i ≤ Proposition 4.8.
Let I = h F , F , F i ⊂ R = k [ x , x , x ] be a complete intersectiongenerated by general forms of degree ( d, d, d ) , d ≥ . Then we have (1) If e is odd then codim L I = 1 and L I ⊂ ( P ) ∗ is a curve of degree d ) . (2) If e is even then codim L I = h e − h e − + 1 = 2 and deg L I = (cid:0) h e (cid:1) = (cid:0) d (cid:1) . Proposition 4.9.
Let I = h F , F , F , F i ⊂ R = k [ x , x , x , x ] be a complete intersec-tion generated by general forms of degree ( d, d, d, d ) , d ≥ . If d = 2 then codim L I = 3 , and in particular L I ⊂ ( P ) ∗ is a set of differentpoints. (2) If d ≥ then L I = ∅ .Proof. (1) The h -vector of R/I is (1 , , , ,
1) and L I is a scheme defined by the maximalminors of a 4 × L I ⊂ ( P ) ∗ has codimension3 and consists of 20 different points which shows that L I is a standard determinantalscheme. If ℓ fails to give an injection from degree 1 to degree 2, then there is a linearform M such that ℓM ∈ I . So we first want to know how many reducible quadrics lie inthe projectivization of the 4-dimensional vector space generated by F , F , F , F inside P [ R ] = P . The dimension of the space of such reducible quadrics is 6, and its degree is10 ([11], top of page 300). Thus its intersection with a general 3-dimensional linear spacein P is a set of 10 points in P . Such a linear space avoids the locus of double planes,and each of the 10 points is of the form ℓ ℓ where either ℓ or ℓ could play the role of ℓ for us. Thus there are 20 such linear forms, or 20 points in ( P ) ∗ .(2) Let (1 , h , h , · · · , h e − , h e ) be the h -vector of R/I . Therefore, e = 4 d − h e − h e − = d. We will prove a more general result in Lemma 4.11, but here we give a completelydifferent proof to illustrate a different approach.Proof of the Claim: We consider the rank 3 vector bundle E on P E := ker( O P ( − d ) F ,F ,F ,F ) −→ O P ) . Using the exact sequences0 −→ E −→ O P ( − d ) −→ O P −→ , and0 −→ O P ( − d ) −→ O P ( − d ) −→ O P ( − d ) −→ E −→ , we get H ( P , E ( t )) = 0 for all t < dH ( P , E ( t )) = 0 for all t ∈ Z H ( P , E ( t )) = 0 for all t ≥ d − . Therefore, we have h e − h e − = h ( P , E (2 d − − h ( P , E (2 d − − χ ( E (2 d − χ ( E (2 d − d where the last equality follows applying the Riemann-Roch Theorem, and the Claim isproved.Since h e − h e − = d , L I is expected to be empty and this is what we will prove. To thisend, we set S = k [ x , x , x , x ] / ( ℓ ) ∼ = k [ x , x , x ] where ℓ = a x + a x + a x + a x ∈ [ R ] is a linear form. Call A d,d,d,d the set of almost complete intersection ideals J ⊂ S oftype ( d, d, d, d ). It holds thatdim A d,d,d,d = dim Gr (cid:18) , (cid:18) d + 22 (cid:19)(cid:19) ) = 4 (cid:18) d + 22 (cid:19) −
16 = 2 d + 6 d − . Denote by B d,d,d,d the set of almost complete intersection ideals J ⊂ S of type ( d, d, d, d )and h -vector (1 , h − , h − h , · · · , h e − − h e − , h e − h e − + 1 = d + 1 , deal J in B d,d,d,d can be linked by means of a complete intersection J ′ of type ( d, d, d ) toa Gorenstein ideal J with socle degree 2 d − h -vector(1 , , , · · · , (cid:18) d − (cid:19) , (cid:18) d (cid:19) − , (cid:18) d (cid:19) − , (cid:18) d − (cid:19) , · · · , , , . Observe that(i) the dimension of the Gorenstein ideals J with h -vector (cid:18) , , , · · · , (cid:18) d − (cid:19) , (cid:18) d (cid:19) − , (cid:18) d (cid:19) − , (cid:18) d − (cid:19) , · · · , , , (cid:19) is (cid:0) d − (cid:1) − d − d − d − J ′ of type ( d, d, d ) contained in J isdim Gr (3 , d ) = 3(3 d −
3) (note that dim[ J ] d = (cid:0) d +22 (cid:1) − (cid:0) d − (cid:1) = 3 d ), and(iii) the dimension of complete intersections of type ( d, d, d ) contained in J is dim Gr (3 ,
4) =3.To compute dim B d,d,d,d we use liaison. The computation isdim B d,d,d,d = (cid:18)(cid:18) d − (cid:19) − d − (cid:19) + (9 d − − d + 5 d − . We have only to justify subtracting the value from (iii) in this computation. Indeed, thisis to remove over-counting, since the same ideal J can be reached from many differentideals J using different complete intersections in J . Now subtracting, we see that thedifference of the dimensions is(2 d + 6 d − − (2 d + 5 d −
13) = d + 1 = h e − h e − + 1 . Since this is > n − d ≥
3, the locus is empty according to (4.4). (cid:3)
Theorem 4.10.
Let I = h F , F , F i ⊂ R = k [ x , x , x ] be a complete intersectionartinian ideal generated by general forms of degree ( d , d , d ) . Assume that d ≤ d ≤ d .Let e be the socle degree of R/I and let (1 , h , · · · , h e − , h e ) be the h -vector of R/I . Then (1) If e is odd then codim L I = 1 and L I ⊂ ( P ) ∗ is a plane curve of degree d d if d ≥ d + d d d − ( d + d − d ) d d + 2 d d + 2 d d − d − d − d if d < d + d . (2) If e is even then codim L I = h e − h e − + 1 = ( if d ≥ d + d + 1 , if d ≤ d + d − . Moreover, if d ≥ d + d + 1 then L I ⊂ ( P ) ∗ is a plane curve of degree d d ; andif d ≤ d + d − then L I ⊂ ( P ) ∗ is a finite set of (cid:0) n I (cid:1) points, where n I = 2 d d + 2 d d + 2 d d + 1 − d − d − d roof. It is well known that I has WLP and hence codim L I ≥ d ≥ d + d arguing as in Remark 4.6 we see that J = h F , F i is the ideal of aset of d d different points in P , h e − = h e +12 = d d and × ℓ : ( R/I ) e − −→ ( R/I ) e +12 with ℓ = ax + by + cz fails to be injective if and only if ℓ passes through one of the d d points defined by J . Therefore, codim( L I ) = 1 and deg( L I ) = d d .Assume d < d + d . In this case we consider the syzygy bundle associated to I , i.e.the rank 2 vector bundle E on P defined by E := ker( ⊕ i =1 O P ( − d i ) ( F ,F ,F ) −→ O P ) . By [1, Corollary 2.7], E is µ -stable. By [3, Theorem 2.2], the linear form ℓ = ax + by + cz fails to be a Lefschetz element of I if and only if ℓ = 0 is a jumping line of E if and only if E | ℓ ∼ = O ℓ ( a ℓ ) ⊕ O ℓ ( a ℓ ) with | a ℓ − a ℓ | ≥
2. Since the first Chern class c ( E ( d + d + d )) = 0,we can apply [17, Theorem 2.2.3], and we get that the set J E of jumping lines of E is acurve of degree c ( E ( d + d + d )) in ( P ) ∗ . Therefore, the non-Lefschetz locus L I of I is aplane curve of degree c (cid:18) E ( d + d + d (cid:19) = ( d + d − d )( d − d + d )4 + ( d + d − d )( − d + d + d )4+ ( d − d + d )( − d + d + d )4 = 2 d d + 2 d d + 2 d d − d − d − d . (2) If d ≥ d + d + 1 the result follows from Remark 4.6. So, let us assume that d ≤ d + d −
1. Let (1 , h , h , · · · , h e − , h e ) be the h -vector of R/I .Claim: h e − h e − = 1 . To prove the claim, we consider the rank 2 vector bundle E on P E := ker (cid:16) ⊕ i =1 O P ( − d i ) ( F ,F ,F ) −→ O P (cid:17) . By [1, Corollary 2.7], E is µ -stable. Using the fact that E is a µ -stable rank 2 vectorbundle on P , c ( E ) = − d − d − d and E norm = E ( d + d + d − ), we get H ( P , E ( h e ) = H ( P , E ( h e )) = H ( P , E ( h e − )) = H ( P , E ( h e − )) = 0 . Therefore, we have h e − h e − = h ( P , E ( h e )) − h ( P , E ( h e − ))= − χ ( E ( h e )) + χ ( E ( h e − )) = 1where the last equality follows applying the Riemann-Roch Theorem, and the claim isproved.Thanks to the claim, the expected codimension of L I ⊂ ( P ) ∗ is two and, in fact, we aregoing to prove that L I ⊂ ( P ) ∗ is a set of (cid:0) n I (cid:1) , n I := d d +2 d d +2 d d +1 − d − d − d , differentpoints. To this end, we consider the rank 2 vector bundle E on P E := ker (cid:16) ⊕ i =1 O P ( − d i ) ( F ,F ,F ) −→ O P (cid:17) . By [1, Corollary 2.7], E is µ -stable. By [3, Theorem 2.2], the linear form ℓ = ax + by + cz fails to be a Lefschetz element of I if and only if ℓ = 0 is a jumping line of E if and only if E | ℓ ∼ = O ℓ ( a ℓ ) ⊕ O ℓ ( a ℓ ) with | a ℓ − a ℓ | ≥
2. Since the first Chern class c ( E ( d + d + d − )) =
1, we can apply [12, Corollary 10.7.1], and we get that E has exactly (cid:0) c ( E ( d d d − ))2 (cid:1) jumping lines. Let us compute c ( E ( d + d + d − )). From the exact sequence0 −→ E −→ ⊕ i =1 O P ( − d i ) −→ O P −→ c ( E ) = − d − d − d and c ( E ) = d d + d d + d d . Since c (cid:18) E (cid:18) d + d + d − (cid:19)(cid:19) = c ( E ) + c ( E ) (cid:18) d + d + d − (cid:19) + (cid:18) d + d + d − (cid:19) , we have c (cid:18) E (cid:18) d + d + d − (cid:19)(cid:19) = 2 d d + 2 d d + 2 d d + 1 − d − d − d J E of jumping lines of E is a set of (cid:0) n I (cid:1) points in ( P ) ∗ , where n I := 2 d d + 2 d d + 2 d d + 1 − d − d − d , which proves what we want. (cid:3) Our next goal is to prove Conjecture 4.3 for n = 4. To this end the following lemmaswill be very useful. Lemma 4.11.
Let I = h F , F , F , F i ⊂ R = k [ x , x , x , x ] be a complete intersectionartinian ideal generated by general forms of degree ( d , d , d , d ) . Assume that d ≤ d ≤ d ≤ d . Let e = d + d + d + d − be the socle degree of R/I and let (1 , h , · · · , h e − , h e ) be the h -vector of R/I . Then (1) If e is odd then h e − = h e +12 = d d d − (cid:18) d + d + d − d + 13 (cid:19) + 14 (cid:18) − d + d + d − d + 13 (cid:19) (2) If e is even then h e − h e − = if d ≥ d + d + d d + d + d − d if − d + d + d ≤ d ≤ d + d + d d if d ≤ − d + d + d . Proof.
Let h ′ i be the h -vector of R/ h f , f , f i . Then for e odd we get h e − = d − X i =0 h ′ e − − i = e − X j = e +12 − d h ′ j = d d d − e − − d X j =0 h ′ j − d + d + d − X j = e +12 h ′ j = d d d − d d d − d − X j =0 h ′ j − d + d + d − − e +12 X j =0 h ′ j = d d d − d d d − d − X j =0 h ′ j − d d d − d − X j =0 h ′ j n the range 0 ≤ j ≤ d + d + d − d − we have that h ′ j = (cid:0) j +22 (cid:1) − (cid:0) j − d +22 (cid:1) since d > d + d + d − d − . For any m we have that m X j =0 (cid:18) j + 22 (cid:19) + m − X j =0 (cid:18) j + 22 (cid:19) = (cid:18) m + 33 (cid:19) + (cid:18) m + 23 (cid:19) = 14 (cid:18) m + 43 (cid:19) . Thus we conclude that the maximum value of the h -vector is h e − = h e +12 = d d d − (cid:18) d + d + d − d − (cid:19) + 14 (cid:18) − d + d + d − d − (cid:19) . When e is even we want to compute the difference h e − h e − = d − X i =0 h ′ e − i − h ′ e − − i = h ′ e − h ′ e − − d +1 = h ′ e − h ′ e − d = h ′ d d d − + d − − h ′ d d d − d − = h ′ d d d − − d − − h ′ d d d − d − = h ′ d d d − d − − h ′ d d d − d − . Again, we are in a range where we can use the expression h ′ j = (cid:0) j +22 (cid:1) − (cid:0) j − d +22 (cid:1) to concludethat h e − h e − = (cid:20) d + d + d − d (cid:21) + − (cid:20) − d + d + d − d (cid:21) + where [ x ] + = x for x ≥ x ] + = 0 for x <
0. For d ≤ − d + d + d this expressionequals d and for d ≥ − d + d + d it equals the first term, which concludes the proof ofthe lemma. (cid:3) We denote by
Gor ( H ) the scheme parametrizing artinian Gorenstein codimension 3algebras R/I with h -vector H = (1 , h , · · · , h e − , h e ) [6]. We have Lemma 4.12.
Let H = (1 , h , · · · , h e − , h e ) and H ′ = (1 , h ′ , · · · , h ′ e − , h ′ e ) be the h -vectors of two artinian Gorenstein codimension 3 algebras with odd socle degree e = 2 r +1 .Assume that ( h ′ i = h i , for i = r, r + 1 , h ′ i = h i − , for i = r, r + 1 .Then, it holds: dim Gor ( H ) − dim Gor ( H ′ ) = h r +1 − h r +3 + h r +4 + 1 . Proof.
By [4, Example 5.2], we havedim
Gor ( H ) = 12 (3 h r + h r − − e X i =0 h i p i )where p i = h i − h i − + 3 h i − − h i − . Therefore after a long but routine calculation weobtaindim
Gor ( H ) − dim Gor ( H ′ ) = (3 h r + h r − − P ei =0 h i p i ) − (3 h ′ r + h ′ r − − P ei =0 h ′ i p ′ i )= h r +1 − h r +3 + h r +4 + 1which proves what we want. (cid:3) heorem 4.13. Let I = h F , F , F , F i ⊂ R = k [ x , x , x , x ] be a complete intersectionartinian ideal generated by general forms of degree ( d , d , d , d ) . Assume that d ≤ d ≤ d ≤ d . Let e be the socle degree of R/I and let (1 , h , · · · , h e − , h e ) be the h -vector of R/I . Then (1) If e is odd then the non-Lefschetz locus L I ⊂ ( P ) ∗ is a surface of degree h e − = d d d − (cid:18) d + d + d − d − (cid:19) + 14 (cid:18) − d + d + d − d − (cid:19) . (2) If e is even then codim L I = min { h e − h e − + 1 , } . In particular, L I ⊂ ( P n − ) ∗ is non-empty if and only if h e − h e − ≤ if and onlyif d ≥ d + d + d or − d + d + d ≤ d ≤ d + d + d and d + d + d − d ≤ or d ≤ − d + d + d and d ≤ . In these cases δ I := deg( L I ) = (cid:0) h e h e − h e − +1 (cid:1) .Proof. Part (1) follows from Lemma 4.11 taking into account that if
R/I has the WLPand the socle degree is odd then the non-Lefschetz locus is a surface of degree h e − .We now consider (2). By Remark 4.6 (1) , if d ≥ d + d + d (remembering that e iseven, so d = d + d + d − h e = h e − = d d d and L I ⊂ ( P n − ) ∗ is a surface ofdegree d d d . By Remark 4.6 (2) , if d = d + d + d −
2, then h e − h e − = 1, h e = d d d and L I ⊂ ( P ) ∗ is an arithmetically Cohen-Macaulay curve of degree (cid:0) d d ··· d n − (cid:1) . ByProposition 4.7 if d = d + d + d − L I = min { h e − h e − + 1 , } = 3 (wherethe last equality follows from Lemma 4.11 (2)) and has degree (cid:0) h e (cid:1) .From now on we assume d ≤ d + d + d −
6. We fix a linear form ℓ and we set S = R/ ( ℓ ). We denote by A d ,d ,d ,d the set of almost complete intersection ideals J ⊂ S of type ( d , d , d , d ) and by B d ,d ,d ,d the set of almost complete intersection ideals J ⊂ S of type ( d , d , d , d ) and h -vector(1 , h − , h − h , · · · , h e − − h e − , h e − h e − + 1 , . A general ideal J in A d ,d ,d ,d can be linked by means of a complete intersection K oftype ( d , d , d ) to a Gorenstein ideal G with socle degree s := d + d + d − d − h -vector H G = (1 , f , · · · , f s ). A general ideal J ′ in B d ,d ,d ,d can be linked by means ofa complete intersection K of type ( d , d , d ) to a Gorenstein ideal G ′ with socle degree s and h -vector H G ′ = (1 , f ′ , · · · , f ′ s ). Moreover, we have: f ′ i = f i for i = s − , s +12 f ′ i = f i − i = s − , s +12 . According to (4.4), to finish the proof it is enough to demonstrate thatdim
Gor ( H G ) − dim Gor ( H G ′ ) = h e − h e − + 1(see also the end of the proof of Proposition 4.9 for the equivalence). Let us prove it. Tothis end, we denote by (1 , ˜ h , · · · , ˜ h w ) the h -vector of the complete intersection ideal K in S of type ( d , d , d ). So, w = d + d + d −
3. Applying Lemma 4.12, we obtaindim
Gor ( H G ) − dim Gor ( H G ′ ) = f s − +1 − f s − +3 + f s − +4 + 1= ( f s − +1 − f s − +2 ) + ( f s − +2 − f s − +3 + f s − +4 ) + 1= − ∆ f s − +2 − ∆ f s − +4 + 1 . ince G (resp. G ′ ) is linked to J (resp. J ′ ) by a complete intersection K of type d , d , d we have f i = ˜ h i + d for i = s − + 1 , s − + 2. So, we get: − ∆ f s − +2 − ∆ f s − +4 + 1 = − ∆˜ h d d d d + ∆ ˜ h d d d d +2 + 1= d ≥ d + d + d d + d + d − d + 1 if − d + d + d ≤ d ≤ d + d + d d + 1 if d ≤ − d + d + d . Therefore, applying Lemma 4.11, we conclude thatdim
Gor ( H G ) − dim Gor ( H G ′ ) = h e − h e − + 1 . (cid:3) The non-Lefschetz locus of a general height three Gorensteinalgebra
When the Hilbert function is fixed, the height three Gorenstein algebras with thatHilbert function lie in a flat family [6], so it makes sense to talk about the general
Goren-stein algebra in this family. From now on, we will abuse terminology and refer to a generalGorenstein algebra, and assume that it is understood that we have fixed the Hilbert func-tion; we will also assume that it is understood that in this section we refer only to theheight three situation, except for a small remark at the end of the section. In this sec-tion we will describe the codimension of the non-Lefschetz locus of a general Gorensteinalgebra, and in particular describe exactly when it is of the expected codimension (giventhe Hilbert function) in the sense of the earlier sections. One might expect that just aswith complete intersections, the general Gorenstein algebra has non-Lefschetz locus of theexpected codimension, but this is not always the case. We give a classification of thoseHilbert functions for which the general Gorenstein algebras fail to have non-Lefschetzlocus of the expected codimension.The Hilbert functions of height three Gorenstein algebras are well-understood. Theyare the so-called
Stanley-Iarrobino (SI) sequences of height three. They are characterizedas follows. A sequence h = (1 , , h , . . . , h e − , h e ) is an SI-sequence if and only if(i) h is symmetric.(ii) Setting g i = h i − h i − for 1 ≤ i ≤ ⌊ e ⌋ , the sequence g = (1 , , g , . . . , g ⌊ e ⌋ ) satisfiesMacaulay’s growth condition.Condition (ii) says that the sequence (1 , , h , . . . , h ⌊ e ⌋ ) is the beginning of the Hilbertfunction of some zero-dimensional scheme in P of degree h ⌊ e ⌋ . It is important to note thatit does not mean that for every Gorenstein algebra R/I with this Hilbert function, thecomponents of
R/I up to degree ⌊ e ⌋ actually coincide with the corresponding componentsof a zero-dimensional scheme. If such a condition does hold, and if the zero-dimensionalscheme is reduced, we will say that R/I “comes from points.” For any SI-sequence, bytaking a suitable Gorenstein quotient of the coordinate ring of a suitable reduced setof points, there is always a subfamily (of the Gorenstein family corresponding to theSI-sequence) that does come from points.We say that a sequence (1 , , g , g , . . . , g k ) is of decreasing type if begins with (1 , , , . . . )(growing with the polynomial ring k [ x, y ]), then is possibly flat, then is strictly decreasing. Theorem 5.1.
Fix an SI-sequence h = (1 , , h , . . . , h e − , , of socle degree e . i) If there are two or more consecutive values of h i that are equal then the generalGorenstein algebra with Hilbert function h has non-Lefschetz locus of the expectedcodimension, namely one. This holds, in particular, when e is odd. (ii) Assume e is even and (5.1) h = (1 , , h , . . . , h e − , h e , h e +1 , . . . , h e − , , where h e − < h e > h e +1 . Let g be the sequence of positive first differences, asabove. Then the general Gorenstein algebra with this Hilbert function has non-Lefschetz locus of the expected codimension if and only if g is of decreasing type.If g is not of decreasing type then the non-Lefschetz locus has codimension one.Proof. Suppose h ⌊ e ⌋ = h ⌊ e ⌋ +1 , for instance if the socle degree e is odd. Then the ex-pected codimension of the non-Lefschetz locus is one. Since it is known that the generalheight three artinian Gorenstein algebra with any given Hilbert function has the WLP [9],the non-Lefschetz locus of the general Gorenstein algebra with odd socle degree has theexpected codimension. So from now on assume that e is even, and that h e − < h e > h e +1 .We first assume that g is of decreasing type. Our strategy will be to construct anexplicit Gorenstein algebra having such a Hilbert function and non-Lefschetz locus ofexpected codimension; then by semicontinuity the general Gorenstein algebra with thisHilbert function has non-Lefschetz locus of the expected codimension.So consider the SI-sequence (5.1), and assume that its first difference is of decreasingtype. Let Z be a reduced set of h e points in P with Hilbert function given by(1 , , h , . . . , h e − , h e , h e , . . . ) . The h -vector of Z is given by the first difference sequence g . Let I be an artinian Goren-stein ideal obtained as a suitable quotient of R/I Z , so that the Hilbert function of R/I isprecisely h . (See [2].) This means that [ I ] i = [ I Z ] i for i ≤ e . We want to show:(i) L I = ∅ if g e ≥ L I = 2 if g e = 1;We have already seen that codim L i = 1 if g e = 0, so we have assumed h e − < h e .Because g is of decreasing type, we can assume that Z has the Uniform Position Property(UPP) by a result by Maggioni and Ragusa [14]. In particular it has the 2-Cayley-Bacharach Property: the Hilbert functions of Z minus a point are all the same, and theHilbert functions of Z minus two points are all the same. We consider the multiplicationon R/I Z from degree e − e by a linear form ℓ . Notice that by UPP, ℓ vanisheson at most two points since h = 3 (so not all points lie on a line).Case 1: ℓ does not vanish on any point of Z . Then ℓ is a non-zerodivisor, so the multiplication is injective and ℓ is a Lefschetzelement.Case 2: ℓ vanishes at exactly one point, P , of Z . Let Y = Z \ P , defined by I Y = I Z : ℓ . Notice that since h e − < h e , and Z has theUPP, we have [ I Y ] e − = [ I Z ] e − . From the diagram[ R/I ] e − × ℓ −→ [ R/I ] e || || → [ I Y /I Z ] e − → [ R/I Z ] e − × ℓ −→ [ R/I Z ] e || e see that multiplication by ℓ is again injective, i.e. ℓ is a Lefschetz element.Case 3: ℓ vanishes at exactly two points, P and Q , of Z . We obtain the same diagram as in Case 2. In this case, though, we have [ I Y /I Z ] e − = 0if and only g e ≥
2. If g e = 1, then [ I Y /I Z ] e − = 0. Thus L I = ∅ if g e ≥
2, andcodim L I = 2 if g e = 1, both of which correspond to the expected codimension. In thelatter case, the degree formula gives deg L I = (cid:0) h e (cid:1) , which can be seen directly as allchoices of two points of Z .Thus we have constructed an explicit Gorenstein algebra with Hilbert function h andnon-Lefschetz locus of the expected codimension, so as noted above, by semicontinuitythe general Gorenstein algebra has non-Lefschetz locus of the expected codimension.It remains to consider the case where g is not of decreasing type. In this case the sameapproach will not work, since it is a priori possible that the Gorenstein algebras comingfrom points in this case fail to have non-Lefschetz locus of the expected codimension, butnevertheless the general one does. We will show by a different method that this is notthe case.So assume that g i − = g i for some i ≤ e , and that i − g i − > g i − = g i . Assume also that R/I is general in the flat family of Gorenstein algebraswith this Hilbert function. By a result of Ragusa and Zappal`a [18] the generators of I ofdegree ≤ i − F , of degree g i . Furthermore, the generators ofthe ideal I : F of degree ≤ i − Z , in P .In order to prove our statement on the non-Lefschetz locus, let ℓ be a linear form andlet Y be defined by I Z : ℓ . This time we consider the multiplication from degree i − i . We have a diagram [ R/I ] i − × ℓ −→ [ R/I ] i || || → [ I Y · F/I Z · F ] i − → [ R/I Z · F ] i − × ℓ −→ [ R/I Z · F ] i But dim k [ R/I Z ] j reaches its multiplicity in degree i − − g i [5], so whenever ℓ vanishesat a point of Z , [ I Y · F/I Z · F ] i − is not zero and the multiplication fails to have maximalrank. Thus the non-Lefschetz locus in degree i − (cid:3) Remark . Using the ideas from the proof of Theorem 5.1, it is easy to construct anartinian Gorenstein algebra whose non-Lefschetz locus is non-reduced, for almost any SI-sequence h . We simply relax the generality condition on Z and allow three points to lieon a line. The only obstacle is when the h -vector of Z does not allow this, i.e. when it is(1), (1 , Remark . If g is not of decreasing type and A has odd socle degree, then even though L I is of the expected codimension, it is still true that its behavior is not the expected onebecause in earlier degrees it is a hypersurface when we expect it not to be. Remark . Since complete intersections of codimension three have g -vectors of decreas-ing type, Theorem 5.1 is a generalization of Theorem 4.10. However, since the method ofproof is completely different, we prefer to give the proof of Theorem 4.10 in Section 4.6. The non-Lefschetz locus in codimension two
In this short section we describe the situation in codimension two. Let R = k [ x, y ]and let I be an artinian ideal in R . Now the Hilbert function of R/I has the form , , h , . . . , h e ), where h i = i +1 until the initial degree of h R/I , and then is non-decreasingfrom then on. Furthermore, if h i = h i +1 for some i , this represents maximal growth of theHilbert function, so Macaulay’s theorem [13] together with Gotzmann’s theorem [7] givesthat the greatest common divisor of all the elements in I of degree i and degree i + 1 hasdegree h i .For fixed Hilbert function, the algebras having that Hilbert function form an irreduciblefamily. For a general such algebra, if h i +1 < h i then the elements of I in degree i + 1 donot have a common divisor. Lemma 6.1.
Let I be any artinian graded ideal in R = k [ x, y ] . Let { h i } be the h -vectorof R/I . Fix any degree i . There exists a linear form ℓ such that × ℓ : [ R/I ] i − → [ R/I ] i fails to have maximal rank if and only if I has a common factor, say F , between all formsof degree i . We have deg F ≤ h i .Proof. The fact that the degree of a GCD in degree i must be ≤ h i is well known. Assumethat the forms of degree i in I have a GCD, say F , of positive degree. If F ∈ I then [ I ] i isthe degree i part of a principal ideal ( F ), we have h i − = h i = deg F . But in R , F factorsinto linear factors. Thus clearly the non-Lefschetz locus consists precisely of the factorsof F (counted with multiplicity). That is, the locus in ( P ) ∗ of linear forms ℓ for which × ℓ : [ R/I ] i − → [ R/I ] i fails to have maximal rank is zero-dimensional of degree equal todeg F = h i . (This is not quite the same as the non-Lefschetz locus since we are lookingonly in degrees i − i .)Suppose instead that the GCD, F , is not in I and has degree d ≤ h i . We havedim[ I ] i = i + 1 − h i = m , say. Choose a basis for [ I ] i of the form { F A , . . . , F A m } . Say F factors as F = ℓ · · · ℓ d . For each factor of F , for instance ℓ , we have m independentelements of [ R ] i − such that multiplication by ℓ is zero in R/I . Now, h i − − h i = ( i − dim[ I ] i − ) − ( i + 1 − dim[ I ] i ) = m − − dim[ I ] i − < m so multiplication by ℓ has a larger kernel than expected (surjectivity implies a kernel ofdimension h i − − h i ) and so fails to have maximal rank.Conversely, assume that I does not have a GCD in degree i . We want to show thatmultiplication by any linear form ℓ gives a homomorphism of maximal rank from degree i − i . In degrees smaller than the initial degree of I , R/I agrees with thepolynomial ring, so the result is clear. If h i − = h i then by the result of Davis [5] I hasa GCD in degree i . Thus we may assume that h i − > h i . Suppose that there exists alinear form ℓ for which the corresponding multiplication from degree i − i isnot surjective. Consider the exact sequence0 → [ R/ ( I : ℓ )( − i × ℓ −→ [ R/I ] i → [ R/ ( I, ℓ )] i → . By assumption, [ R/ ( I, ℓ )] i = 0. But R/ ( ℓ ) ∼ = k [ x ]. This means that the restriction of[ I ] i modulo ℓ is zero. This can only happen if ℓ is a GCD for [ I ] i , contradicting ourassumption. The result follows. (cid:3) Proposition 6.2.
Let R = k [ x, y ] . Fix a Hilbert function { h i } that exists for artiniangraded quotients of R . Let R/I be a general algebra with this Hilbert function. For any i , there exists a linear form ℓ such that × ℓ : [ R/I ] i − → [ R/I ] i fails to have maximalrank if and only if h i − = h i . In particular, if R/I is a general complete intersection oftype ( d , d ) , with d ≤ d , then the non-Lefschetz locus is empty if and only if d = d .Otherwise, the degree of the non-Lefschetz locus is d . eferences [1] G. Bohnhorst and H. Spindler. The stability of certain vector bundles on P n . In Complex algebraicvarieties (Bayreuth, 1990) , volume 1507 of
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Adv. Stud. Pure Math. , pages 303–312.North-Holland, Amsterdam, 1987. epartment of Mathematics, KTH Royal Institute of Technology, S-100 44 Stock-holm, Sweden E-mail address : [email protected] Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA
E-mail address : [email protected] Facultat de Matem`atiques, Departament de Matem`atiques i Inform`atica, Gran Via desles Corts Catalanes 585, 08007 Barcelona, Spain
E-mail address : [email protected] Department of Mathematics, University of Kentucky, 715 Patterson Office Tower,Lexington, KY 40506-0027, USA
E-mail address : [email protected]@uky.edu