On the Weak Lefschetz Property for Vector Bundles on P 2
aa r X i v : . [ m a t h . AG ] M a r ON THE WEAK LEFSCHETZ PROPERTY FOR VECTOR BUNDLES ON P GIOIA FAILLA, ZACHARY FLORES, CHRIS PETERSON
Abstract.
Let R = K [ x, y, z ] be a standard graded polynomial ring where K is an algebraicallyclosed field of characteristic zero. Let M = ⊕ j M j be a finite length graded R -module. We saythat M has the Weak Lefschetz Property if there is a homogeneous element L of degree one in R such that the multiplication map × L : M j → M j +1 has maximal rank for every j . The main resultof this paper is to show that if E is a locally free sheaf of rank 2 on P then the first cohomologymodule of E , H ∗ ( P , E ), has the Weak Lefschetz Property. Introduction
Let K be an algebraically closed field of characteristic zero and let R = K [ x, y, z ] be the polynomialring with standard grading R = ⊕ j ≥ R j . Let M = ⊕ j ∈ Z M j be a graded R -module. We say that M is of finite length if all but a finite number of the M j are equal to 0 (with each M j a finite dimensional K -vector space). We say that M has the Weak Lefschetz Property if there is a linear form L ∈ R suchthat the K -linear map × L : M j → M j +1 has maximal rank for all j . Stanley and others showedhow the Weak Lefschetz Property, a property that is geometric/algebraic in nature, ties in withseveral interesting problems of a combinatorial nature [6, 16, 22, 23]. In particular, Stanley utilizedthe property to complete the proof of McMullen’s conjecture on the characterization of f -vectorsof simplicial polytopes. In honor of the influential works of Stanley, the Weak Lefschetz Propertyis also referred to as the Weak Stanley Property in the literature. There has been a rich body ofresearch establishing the existence or non-existence of the Weak Lefschetz Property for various typesof Artinian algebras, in particular for Artinian Gorenstein algebras [2,9,11,14,24] and other Artinianalgebras with special structure [17, 18, 25]. Within this rapidly growing body of research involvingthe Weak Lefschetz Property, we found the following survey type works to be very helpful [10, 19].There were two papers that played a major role in inspiring us to utilize the vector bundle basedtechniques that ultimately led to a proof of our main result. The first was the paper by Harimaet al [11] which made use of the Grauert-M¨ulich theorem to gain further insight into the WeakLefschetz Property of a height three Artinian complete intersection. The Grauert-M¨ulich theoremenabled them to pinpoint the generic splitting type of a stable, normalized, rank two vector bundleon P which enabled precise homological conclusions to be made. The second influential work forus was the paper by Brenner and Said [3] which made further use of the Grauert-M¨ulich theoremfor higher rank bundles on P and solidified the connection between the generic splitting type of abundle and the Weak Lefschetz Property.It is very natural to study height three Artinian complete intersections via the Koszul complex.First of all, the Koszul complex is exact for complete intersections. Second, by sheafifying theKoszul complex, one can identify the first cohomology module of an associated rank two locally freesheaf as the Artinian module R/ ( F , F , F ) where F , F , F is a regular sequence of homogeneouspolynomials in R defining the complete intersection. A natural generalization can be obtained viathe Buchsbaum-Rim complex associated to an R -graded map φ : F → G where F is a free R -moduleof rank n + 2 and G is a free R -module of rank n . In particular, if the cokernel of φ is of codimensionthree, which over R = K [ x, y, z ] corresponds to the cokernel being a module of finite length, thenthe Buchsbaum-Rim complex is exact. By sheafifying this complex we can again identify the firstcohomology module of an associated rank two locally free sheaf, E , as the cokernel of φ . As in thepapers [3, 11], it is crucial to understand the generic splitting type of E and its relationship to the Mathematics Subject Classification.
Primary 13E10, 13F20, 13D02; Secondary 14F05, 14M12. multiplication between consecutive graded components of the cokernel of φ induced by a generallinear form.The paper is broken into four short sections. In section two of this paper we provide backgroundmeant to clarify the connection between the Buchsbaum-Rim complex for a certain class of finitelength K [ x, y, z ]-modules and rank 2 vector bundles on P . The third section contains the statementand proofs of the main results of the paper. In particular, we show that the first cohomology moduleof any rank 2 vector bundle on P satisfies the Weak Lefschetz Property. The final section consistsof examples, some potential paths for future research, and concluding remarks.2. The Buchsbaum-Rim complex
Let R = K [ x, y, z ], let F = ⊕ n +2 i =1 R ( − a i ), let G = ⊕ ni =1 R ( − b i ), let a = a + · · · + a n +2 , and let b = b + · · · + b n . Given a graded degree zero map φ : F → G we have a kernel E , cokernel M andan exact sequence(1) 0 → E → F → G → M → . In addition, we have a Buchsbaum-Rim complex associated to φ : F → G [4, 5]. If the cokernelof φ has the “expected codimension”, which in this case corresponds to requiring that M has finitelength, then the Buchsbaum-Rim complex is exact and has the form:(2) 0 → G ∨ ( b − a ) → F ∨ ( b − a ) → F → G → M → R -modules. These complexes are exact if a certain genericity condition is met and theycan be derived by considering “strands” of a particular Koszul complex (see [7] Appendix A2.6 fordetails).If we sheafify (2) then we get an exact sequence of locally free sheaves(3) 0 → G ∨ ( b − a ) → F ∨ ( b − a ) → F → G → → G ∨ ( b − a ) → F ∨ ( b − a ) → E → → E → F → G → . Note that (5) is also the sheafification of (1). The locally free sheaf E has rank two and is an exampleof a (first) Buchsbaum-Rim sheaf. The apparent symmetry of the Buchsbaum-Rim complex is closelyrelated to the fact that a rank 2 locally free sheaf is self dual (up to a twist by a line bundle). Ingeneral, the structure found in the Buchsbaum-Rim complex is reflected in properties of E , inproperties of its sections, and in properties of its cohomology modules [15, 20]. In particular, therigidity of the Buchsbaum-Rim complex, when it is exact, suggests that properties of the objectsinvolved reduce to combinatorial considerations of the a i and b i involved in the definitions of F and G . In the next section, we will see that this is indeed the case. Let H ∗ ( P , E ) denote the module ⊕ i ∈ Z H ( P , E ( i )). If we apply the global section functor to the short exact sequence0 → E → F → G → → H ∗ ( P , E ) → H ∗ ( P , F ) → H ∗ ( P , G ) → H ∗ ( P , E ) → H ∗ ( P , F ) → . . . . Note that H ∗ ( P , F ) = 0 since F = O P ( − a i ) and that (6) is actually a recovery of (1). In particular,we have H ∗ ( P , E ) = M . In general, finite length R = K [ x, y, z ]-modules that can be expressed as cokernels of maps of theform φ : ⊕ n +2 i =1 R ( − a i ) → ⊕ ni =1 R ( − b i ) correspond to finite length modules of the form H ∗ ( P , E )where E is a rank 2 locally free sheaf on P . N THE WEAK LEFSCHETZ PROPERTY FOR VECTOR BUNDLES ON P Main Results
In this section, we collect the key definitions and theorems that form the heart of the paper.Many of the needed tools can be found in the book by Hartshorne on
Algebraic Geometry [13] and inthe book by Okonek, Schneider, and Spindler on
Vector Bundles on Complex Projective Spaces [21].
Definition 3.1.
Let E be a torsion free sheaf on P n . Let c ( E ) denote the first Chern class of E andlet rank ( E ) denote its rank.1) The slope of E is the rational number µ ( E ) = c ( E ) /rank ( E )2) E is said to be stable if for any non-zero subsheaf F ⊂ E the slopes satisfy µ ( F ) < µ ( E )3) E is said to be semistable if for any non-zero subsheaf F ⊂ E the slopes satisfy µ ( F ) ≤ µ ( E )4) E is unstable if it is not semistable.In various contexts, the definition of stability given above is sometimes referred to by other namesincluding slope stability, µ -stability, Mumford stability, or Mumford-Takemoto stability.Let E be a vector bundle on P n and let r denote the rank of E . We say that E is a normalized bundle if c ( E ) ∈ {− r + 1 , . . . , } . In general, there exists a unique a ∈ Z such that E ( a ) is anormalized bundle. In particular, if E is a normalized rank 2 vector bundle, then c ( E ) ∈ {− , } .The following lemma is a quick application of the definition of stability, see [21] (Chapter II for amore detailed discussion of stability and Lemma 1.2.5 on Pg 166-167 for the statement and proof ofthe lemma). Lemma 3.2.
Let E be an normalized rank 2 vector bundle on P n . E is stable if and only if H ( P n , E ) = 0 . If c ( E ) = − then E is semistable if and only if E is stable If c ( E ) = 0 then E is semistable if and only if H ( P n , E ( − . The following is the Grauert-M¨ulich Theorem for rank 2 bundles on P n . For a more detaileddiscussion of the Grauert-M¨ulich theorem and its role in the classification of vector bundles, see [8]for the original result or see [21] (Chapter II, section 2 for a general discussion of the splitting behaviorof vector bundle and Corollary 2 on Pg 206 for the specifics of the Grauert-M¨ulich theorem). Proposition 3.3.
Let E be a semistable, normalized, rank 2 vector bundle on P n . Let L be a generalline. If c ( E ) = 0 then the restriction to L splits as E| L ∼ = O P ⊕ O P . If c ( E ) = − then the restriction to L splits as E| L ∼ = O P ( − ⊕ O P . Definition 3.4. If E is an unstable, normalized, rank 2 vector bundle on P n then the largest a suchthat H ( P n , E ( − a )) = 0 is called the index of instability of E .From the above lemma, if E is an unstable, normalized, rank 2 vector bundle on P n and c ( E ) = 0then its index of instability is greater than zero. Similarly, if c ( E ) = − E is a vector bundle on P , we can make a stronger statement: Proposition 3.5.
Let E be an unstable, normalized, rank 2 vector bundle on P . Let k be the indexof instability of E . Let L be a general line in P . Then Every nonzero section s ∈ H ( P , E ( − k )) is regular. If c ( E ) = 0 then k > and E| L = O P ( − k ) ⊕ O P ( k ) . If c ( E ) = − then k ≥ and E| L = O P ( − k − ⊕ O P ( k ) .Proof. Let s be a nonzero section in H ( P , E ( − k )). Using s we can build a short exact sequence ofsheaves 0 → O P → E ( − k ) → Q ( − k ) → O P ( k ) to get the short exact sequence of sheaves0 → O P ( k ) → E → Q → . If s is not regular (i.e. its vanishing locus is not of codimension 2 or greater), then the vanishinglocus of s contains a curve component. This curve is of codimension 1 in P thus can be identifiedwith a form F ∈ R . If we factor out F from s we obtain a nonzero section s ′ ∈ H ( P , E ( − k − d )) GIOIA FAILLA, ZACHARY FLORES, CHRIS PETERSON where d is the degree of F (see [1], Lemma 2 on page 128). Since k is the largest integer such that H ( P , E ( − k )) = 0, we get a contradiction. Therefore s is regular.Suppose first that c ( E ) = 0. If L = P is a general line in P then L does not meet the zerolocus of s . As a consequence, the restriction of the short exact sequence to L is still a short exactsequence and by Chern class considerations, the restriction of Q to L is O P ( − k ). Thus, restrictingthe exact sequence to L leads to0 → O P ( k ) → E| P → O P ( − k ) → . Since E has rank 2, is unstable, and has c = 0, we know that H ( P , E ( − = 0 thus we canconclude that k >
0. Using this fact, we can conclude that
Ext ( O P ( − k ) , O P ( k )) = 0. As aconsequence, E| P = O P ( − k ) ⊕ O P ( k ).Now suppose that c ( E ) = −
1. Like before, the restriction of the short exact sequence to L isstill a short exact sequence except now, by Chern class considerations, the restriction of Q to L is O P ( − k − → O P ( k ) → E| P → O P ( − k − → . Since E has rank 2, is unstable, and has c = −
1, we know that H ( P , E ) = 0 thus we can concludethat k ≥
0. Using this fact, we can conclude that
Ext ( O P ( − k − , O P ( k )) = 0. As a consequence, E| P = O P ( − k − ⊕ O P ( k ). (cid:3) Proposition 3.6. If E is an unstable, normalized, rank 2 vector bundle on P with index of instability k then • If c ( E ) = 0 then h ( P , E ( t )) = (cid:0) k + t +22 (cid:1) for t < k . • If c ( E ) = − then h ( P , E ( t )) = (cid:0) k + t +22 (cid:1) for t ≤ k .Proof. Consider the exact sequence0 → E ( t − → E ( t ) → E ( t ) | L → . If we apply the global section functor we get the exact sequence0 → H ( P , E ( t − → H ( P , E ( t )) → H ( P , E ( t ) | L ) → . . . From this exact sequence, we have h ( P , E ( t )) ≤ h ( P , E ( t − h ( P , E ( t ) | L ) . If L is a general line then from the previous proposition we have thatif c ( E ) = 0 then h ( P , E ( t ) | L ) = h ( P , O P ( − k + t ) ⊕ O P ( k + t ))if c ( E ) = − h ( P , E ( t ) | L ) = h ( P , O P ( − k − t ) ⊕ O P ( k + t ))As a consequenceif c ( E ) = 0 and if t < k then h ( P , E ( − k + t ) | L ) = max { , t + 1 } if c ( E ) = − t ≤ k then h ( P , E ( − k + t ) | L ) = max { , t + 1 } Since there exists a nonzero section s ∈ H ( P , E ( − k )), we can tensor this section by forms ofdegree t and produce sections in H ( P , E ( − k + t )). As a consequence, we have h ( P , E ( t )) ≥ (cid:18) k + t + 22 (cid:19) or equivalently h ( P , E ( − k + t )) ≥ (cid:18) t + 22 (cid:19) We can now establish the claim of the proposition by an inductive approach. In the interest of space,we let h ( E ) denote h ( P , E ). Recalling that h ( E ( − k − h ( P , E ( − k + t ) | L ) = t + 1(provided t is in the proper range) we have the following inequalities: N THE WEAK LEFSCHETZ PROPERTY FOR VECTOR BUNDLES ON P ≤ h ( E ( − k + 0)) ≤ h ( E ( − k − h ( E ( − k + 0) | L ) = 0 + 1 = 13 ≤ h ( E ( − k + 1)) ≤ h ( E ( − k + 0)) + h ( E ( − k + 1) | L ) = 1 + 2 = 36 ≤ h ( E ( − k + 2)) ≤ h ( E ( − k + 1)) + h ( E ( − k + 2) | L ) = 3 + 3 = 6 . . . (cid:0) t +22 (cid:1) ≤ h ( E ( − k + t )) ≤ h ( E ( − k + t − h ( E ( − k + t ) | L ) = (cid:0) t +12 (cid:1) + t + 1 = (cid:0) t +22 (cid:1) . . . For c ( E ) = 0, following the inequalities through one at a time leads to the constraint (cid:18) t + 22 (cid:19) ≤ h ( E ( − k + t )) ≤ (cid:18) t + 22 (cid:19) for t < k or equivalently (cid:18) k + t + 22 (cid:19) ≤ h ( E ( t )) ≤ (cid:18) k + t + 22 (cid:19) for t < k. Thus we conclude thatif c ( E ) = 0 then h ( P , E ( t )) = (cid:18) k + t + 22 (cid:19) for t < k. In a similar manner, we can also conclude thatif c ( E ) = − h ( P , E ( t )) = (cid:18) k + t + 22 (cid:19) for t ≤ k. (cid:3) Theorem 3.7.
Let E be a normalized, rank 2, locally free sheaf on P . Let L ∈ K [ x, y, z ] be a generallinear form. Let H ∗ ( P , E ) = ⊕ t ∈ Z H ( P , E ( t )) . Let φ L ( t ) : H ( P , E ( t − → H ( P , E ( t )) be thelinear map induced by L .1) H ∗ ( P , E ) has the Weak Lefschetz Property2) Let E be stable. • If c ( E ) = 0 then φ L ( t ) is injective for t ≤ − and surjective for t ≥ − • If c ( E ) = − then φ L ( t ) is injective for t ≤ − and surjective for t ≥ .3) Let E be unstable with index of instability k . • If c ( E ) = 0 then φ L ( t ) is injective for t ≤ k − and surjective for t ≥ − k − • If c ( E ) = − then φ L ( t ) is injective for t ≤ k and surjective for t ≥ − k − Proof.
In order to prove the theorem, we will first prove 2) and 3) which immediately imply 1).Consider the short exact sequence of sheaves(7) 0 → E ( t − → E ( t ) → E ( t ) | L → . If we apply the global section functor we get the long exact sequence
GIOIA FAILLA, ZACHARY FLORES, CHRIS PETERSON H ( P , E ( t − H ( P , E ( t )) H ( P , E ( t ) | L ) H ( P , E ( t − H ( P , E ( t )) H ( P , E ( t ) | L ) H ( P , E ( t − H ( P , E ( t )) H ( P , E ( t ) | L ) = 0To show that H ∗ ( P , E ) has the Weak Lefschetz Property, we need to show that for each t ∈ Z , themap H ( P , E ( t − → H ( P , E ( t )) is either injective or surjective. From the long exact sequenceabove, we have the following observations: • The map is injective if and only if h ( P , E ( t − − h ( P , E ( t )) + h ( P , E ( t ) | L ) = 0 . • The map is injective if h ( P , E ( t ) | L ) = 0 . • The map is surjective if and only if h ( P , E ( t ) | L ) − h ( P , E ( t − h ( P , E ( t )) = 0 . • The map is surjective if h ( P , E ( t ) | L ) = 0 . If the generic splitting type of E is O P ( a ) ⊕ O P ( b ) then, by Serre Duality, h ( P , E| L ) = h ( P , O P ( − a − ⊕ O P ( − b − h ( P , E ( t ) | L ) = h ( P , O P ( − a − t − ⊕ O P ( − b − t − h ( P , E ( t ) | L ). In particular, if − a − t − ≤ − − b − t − ≤ − h ( P , E ( t ) | L ) = 0. We collect the following facts:A) Since E is locally free on P , by duality we have h ( P , E ( t )) = h ( P , E ∨ ( − t − E to a general line L we have h ( P , E ( t ) | L ) = h ( P , E ∨ ( − t − | L ).C) Since E has rank two, if c ( E ) = 0 then E ∨ ∼ = E and if c ( E ) = − E ∨ ∼ = E (1).We now assume that E is stable and use the above considerations to establish a range of valuesof t where the map, φ L ( t ) : H ( P , E ( t − → H ( P , E ( t )), is injective and a range of values wherethe map is surjective. It is important to note that the following shows that for every value of t , themap is either injective or surjective.Suppose E is stable and that c ( E ) = 0. By Proposition 3.3, E splits on L as O P ⊕ O P . In thiscase, h ( P , E ( t ) | L ) = 0 for t ≤ − h ( P , E ( t ) | L ) = 0 for t ≥ −
1. Thus φ L ( t ) is injective for t ≤ − t ≥ − E is stable and that c ( E ) = −
1. By Proposition 3.3, E splits on L as O P ( − ⊕ O P .In this case, h ( P , E ( t ) | L ) = 0 for t ≤ − h ( P , E ( t ) | L ) = 0 for t ≥ E is unstable and that c ( E ) = 0. If the index of instability is k then by Proposition 3.5, k > E| L = O P ( − k ) ⊕ O P ( k ). In this case, Proposition 3.6 allows us to conclude that h ( P , E ( t − − h ( P , E ( t )) + h ( P , E ( t ) | L ) = 0 for t ≤ k − . This implies that φ L ( t ) is injective for t ≤ k −
1. Using A) and B) above, we note that h ( P , E ( t ) | L ) − h ( P , E ( t − h ( P , E ( t ))can be expressed as h ( P , E ∨ ( − t − | L ) − h ( P , E ∨ ( − t − h ( P , E ∨ ( − t − . Using C) and rearranging, we can then express this as h ( P , E ( − t − − h ( P , E ( − t − h ( P , E ( − t − | L ) . By Proposition 3.6 this quantity is equal to 0 for − t − ≤ k −
1. In other words, φ L ( t ) is surjectivefor − k − ≤ t .Suppose E is unstable and that c ( E ) = −
1. If the index of instability is k then by Proposition 3.5, k ≥ E| L = O P ( − k − ⊕ O P ( k ). In this case, Proposition 3.6 allows us to conclude that h ( P , E ( t − − h ( P , E ( t )) + h ( P , E ( t ) | L ) = 0 for t ≤ k. N THE WEAK LEFSCHETZ PROPERTY FOR VECTOR BUNDLES ON P This implies that φ L ( t ) is injective for t ≤ k . Using A) and B) above, we note that h ( P , E ( t ) | L ) − h ( P , E ( t − h ( P , E ( t ))can be expressed as h ( P , E ∨ ( − t − | L ) − h ( P , E ∨ ( − t − h ( P , E ∨ ( − t − . Using C) and rearranging, we can then express this as h ( P , E ( − t − − h ( P , E ( − t − h ( P , E ( − t − | L ) . By Proposition 3.6 this quantity is equal to 0 for − t − ≤ k . In other words, φ L ( t ) is surjective for − k − ≤ t .In each of these cases, we see that for each t ∈ Z , the map H ( P , E ( t − → H ( P , E ( t )) iseither injective or surjective. Thus H ∗ ( P , E ) has the Weak Lefschetz Property for any rank 2 vectorbundle E on P . (cid:3) Corollary 3.8. If F , F , F is a regular sequence in R = K [ x, y, z ] then R/ ( F , F , F ) has theWeak Lefschetz Property. Corollary 3.9. If E is a rank 2 vector bundle on P then H ∗ ( P , E ) is unimodal.Proof. In the proof of Theorem 3.7, we saw that for any rank 2 vector bundle E on P , there existsan r such that for t < r the map × L : H ( P , E ( t − → H ( P , E ( t )) is injective and for t ≥ r themap × L : H ( P , E ( t − → H ( P , E ( t )) was surjective. This fact establishes unimodality. (cid:3) An Example and Further Remarks
In this section, we first give an example to illustrate the theorems of the paper and the structureof the Buchsbaum-Rim complexes. In each of the following two examples, the associated locally freesheaf is unstable. After giving the two examples, we conclude the paper with a few remarks andconsiderations for possible further research.
Example 4.1.
Consider a map φ : R ( − ⊕ R ( − → R ( − ⊕ R whose cokernel is a finite lengthmodule M . An elementary computation show that M = M ⊕ · · · ⊕ M has Hilbert function(1 , , , , , , , , , φ is:(8) 0 → R ( − ⊕ R ( − → R ( − ⊕ R ( − → R ( − ⊕ R ( − → R ( − ⊕ R → M → O P (6) we get the exact sequence(9) 0 → O P ( − ⊕ O P ( − → O P ( − ⊕ O P (1) → O P ( − ⊕ O P (4) → O P (5) ⊕ O P (6) → → O P ( − ⊕ O P ( − → O P ( − ⊕ O P (1) → E → → E → O P ( − ⊕ O P (4) → O P (5) ⊕ O P (6) → E is a normalized rank 2 locally free sheaf with c ( E ) = 0. From the exact sequence (10), wesee that H ( P , E ( − > H ( P , E ( − E is unstable with index of instability k = 1. By Theorem 3.7, φ L ( t ) is injective for t ≤ t ≥ − × L : M d − → M d is injective for d ≤ d ≥
4. Note that thisimplies bijectivity for 4 ≤ d ≤ M , M , M and M all have the same dimension. Furthernote that for every value of d , the map × L : M d − → M d is either injective or surjective thus M has the Weak Lefschetz Property. GIOIA FAILLA, ZACHARY FLORES, CHRIS PETERSON
Example 4.2.
Consider a map φ : R ( − ⊕ R ( − → R ( − ⊕ R whose cokernel is a finite lengthmodule M . An elementary computation show that M = M ⊕ · · · ⊕ M has Hilbert function(2 , , , , , , , , , , , , φ is:(12) 0 → R ( − ⊕ R ( − → R ( − ⊕ R ( − → R ( − ⊕ R ( − → R ( − ⊕ R → M → O P (7) we get the exact sequence(13) 0 → O P ( − ⊕ O P ( − → O P ( − ⊕ O P → O P ( − ⊕ O P (5) → O P (6) ⊕ O P (7) → → O P ( − ⊕ O P ( − → O P ( − ⊕ O P → E → → E → O P ( − ⊕ O P (5) → O P (6) ⊕ O P (7) → E is a normalized rank 2 locally free sheaf with c ( E ) = −
1. From exact sequence (14), wesee that H ( P , E ) > H ( P , E ( − E is unstable with index of instability k = 0. By Theorem 3.7, φ L ( t ) is injective for t ≤ t ≥ − × L : M d − → M d is injective for d ≤ d ≥
6. Note thatthis implies bijectivity for 6 ≤ d ≤ M , M , M all have the same dimension. Further notethat for every value of d , the map × L : M d − → M d is either injective or surjective thus M has theWeak Lefschetz Property.In this paper, we have shown that the first cohomology module of a rank two locally free sheafon P must have the Weak Lefschetz Property. This is equivalent to showing that if M is a finitelength module arising as the cokernel of a map of the form φ : F → G with F = ⊕ n +2 i =1 R ( − a i ) and G = ⊕ ni =1 R ( − b i ), then M has the Weak Lefschetz Property. As a special case, every height threeArtinian complete intersection has the Weak Lefschetz Property (proved first in [11] and provedagain in [3]).The key piece needed in the proofs of the main theorems is that E is a rank two locally freesheaf on a surface. Many of the key conclusions ultimately follow from this fact. This suggests thatthere may be generalizations of Theorem 3.7 to the case of rank two locally free sheaves on weightedprojective planes and on P × P . We note the interesting paper by Harima and Watanabe wherethey considered the strong Lefschetz property for Artinian algebras with non-standard grading [12].It is hoped that additional progress may be made in the understanding of Lefschetz Properties byconsidering the more general problem for modules over non-standard graded rings. Acknowledgments : The third author wishes to express thanks to a brief but helpful conversationwith Aaron Bertram which helped crystallize some ideas in the proof of Proposition 3.5. This paperis based on research partially supported by a grant of the group GNSAGA of INdAM and by theNational Science Foundation under grant
References [1] Wolf Barth. Some properties of stable rank-2 vector bundles on P n . Mathematische Annalen , 226(2):125–150,1977.[2] Mats Boij. Components of the space parametrizing graded Gorenstein Artin algebras with a given Hilbert function.
Pacific Journal of Mathematics , 187(1):1–11, 1999.[3] Holger Brenner and Almar Kaid. Syzygy bundles on P and the weak Lefschetz property. Illinois Journal ofMathematics , 51(4):1299–1308, 2007.[4] David A Buchsbaum. A generalized Koszul complex I.
Transactions of the American Mathematical Society ,111(2):183–196, 1964.[5] David A Buchsbaum and Dock S Rim. A generalized Koszul complex II. Depth and multiplicity.
Transactions ofthe American Mathematical Society , 111(2):197–224, 1964.[6] David Cook II and Uwe Nagel. The weak Lefschetz property, monomial ideals, and lozenges.
Illinois Journal ofMathematics , 55(1):377–395, 2011.[7] David Eisenbud.
Commutative Algebra: with a view toward algebraic geometry , volume 150. Springer Science &Business Media, 2013.
N THE WEAK LEFSCHETZ PROPERTY FOR VECTOR BUNDLES ON P [8] Hans Grauert and Gerhard M¨ulich. Vektorb¨undel vom Rang 2 ¨uber dem n-dimensionalen komplex-projektivenRaum. manuscripta mathematica , 16(1):75–100, 1975.[9] Tadahito Harima. Characterization of Hilbert functions of Gorenstein Artin algebras with the weak Stanleyproperty. Proceedings of the American Mathematical Society , 123(12):3631–3638, 1995.[10] Tadahito Harima, Toshiaki Maeno, Hideaki Morita, Yasuhide Numata, Akihito Wachi, and Junzo Watanabe.Lefschetz properties. In
The Lefschetz Properties , pages 97–140. Springer, 2013.[11] Tadahito Harima, Juan C Migliore, Uwe Nagel, and Junzo Watanabe. The weak and strong Lefschetz propertiesfor Artinian K-algebras.
Journal of Algebra , 262(1):99–126, 2003.[12] Tadahito Harima and Junzo Watanabe. The strong Lefschetz property for Artinian algebras with non-standardgrading.
Journal of Algebra , 311(2):511–537, 2007.[13] Robin Hartshorne.
Algebraic geometry , volume 52. Springer Verlag, New York, 1977.[14] Anthony Ayers Iarrobino. Associated graded algebra of a Gorenstein Artin algebra.
Memoirs of the AMS , 514,1994.[15] Martin Kreuzer, Juan C Migliore, Chris Peterson, and Uwe Nagel. Determinantal schemes and Buchsbaum–Rimsheaves.
Journal of Pure and Applied Algebra , 150(2):155–174, 2000.[16] Jizhou Li and Fabrizio Zanello. Monomial complete intersections, the weak Lefschetz property and plane parti-tions.
Discrete Mathematics , 310(24):3558–3570, 2010.[17] Emilia Mezzetti, Rosa M Mir´o-Roig, and Giorgio Ottaviani. Laplace equations and the weak Lefschetz property.
Canadian Journal of Mathematics , 65(3):634–654, 2013.[18] Juan Migliore, Rosa Mir´o-Roig, and Uwe Nagel. Monomial ideals, almost complete intersections and the weakLefschetz property.
Transactions of the American Mathematical Society , 363(1):229–257, 2011.[19] Juan Migliore and Uwe Nagel. Survey article: a tour of the weak and strong Lefschetz properties.
Journal ofCommutative Algebra , 5(3):329–358, 2013.[20] Juan C Migliore, Uwe Nagel, and Chris Peterson. Buchsbaum–Rim sheaves and their multiple sections.
Journalof Algebra , 219(1):378–420, 1999.[21] Christian Okonek, Michael Schneider, and Heinz Spindler.
Vector bundles on complex projective spaces , volume 3.Springer, 1980.[22] Richard P Stanley. The number of faces of a simplicial convex polytope.
Advances in Mathematics , 35(3):236–238,1980.[23] Richard P Stanley. Weyl groups, the hard Lefschetz theorem, and the Sperner property.
SIAM Journal onAlgebraic Discrete Methods , 1(2):168–184, 1980.[24] Junzo Watanabe. A note on complete intersections of height three.
Proceedings of the American MathematicalSociety , 126(11):3161–3168, 1998.[25] Fabrizio Zanello. A non-unimodal codimension 3 level h-vector.
Journal of Algebra , 305(2):949–956, 2006.
University of Reggio Calabria, Department DIIES Via Graziella, Feo di Vito, Reggio Calabria
E-mail address : [email protected] Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA
E-mail address : [email protected] Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA
E-mail address ::