It has been proved by various authors that a normalized, 1-Buchsbaum rank 2
vector bundle on P3 is a nullcorrelation bundle, while a normalized,
2-Buchsbaum rank 2 vector bundle on P3 is an instanton bundle of charge 2. We
find that the same is not true for 3-Buchsbaum rank 2 vector bundles on P3, and
propose a conjecture regarding the classification of such objects.Comment: Corrected previous result

Jardim, Marcos

Marchesi, Simone

English

arXiv

It has been proved by various authors that a normalized,
l-Buchsbaum rank 2 vector bundle on P3 is a nullcorrelation bundle,
while a normalized, 2-Buchsbaum rank 2 vector bundle on P3 is an
instanton bundle of charge 2. We find that the same is not true for
3-Buchsbaum rank 2 vector bundles on P3, and propose a conjecture
regarding the classification of such objects

OpenstarTs

Rend. Istit. Mat. Univ. TriesteVolume 47 (2015), 17–25DOI: 10.13137/0049-4704/11216p-Buchsbaum rank 2 bundles on theprojective spaceMarcos Jardim and Simone MarchesiAbstract. It has been proved by various authors that a normalized,1-Buchsbaum rank 2 vector bundle on P3 is a nullcorrelation bundle,while a normalized, 2-Buchsbaum rank 2 vector bundle on P3 is aninstanton bundle of charge 2. We find that the same is not true for3-Buchsbaum rank 2 vector bundles on P3, and propose a conjectureregarding the classification of such objects.Keywords: Buchsbaum bundles, Instanton bundles.MS Classification 2010: 14F05, 14J60, 32L10.IntroductionA coherent sheaf E on P3 is said to be p-Buchsbaum if p is the minimal power ofthe irrelevant ideal which annihilatesH1∗ (E). The complete list of p-Buchsbaumrank 2 bundles on P3 for p ≤ 2 has been established by several authors, see forexample [7, 9, 14, 15, 16]. More precisely, we have the following.Theorem 1. Let E be a normalized p-Buchsbaum rank 2 vector bundle on P3.Then• p = 0 if and only if E is direct sum of line bundles;• p = 1 if and only if E is a null correlation bundle, i.e. an instantonbundle of charge 1;• p = 2 if and only if E is an instanton bundle of charge 2.After examining this list, two questions natually arise. First, is every rank 2instanton bundle of charge k on P3 k-Buchsbaum? Second, since every bundleis p-Buchsbaum for some sufficiently high p, for which values of p can we finda p-Buchsbaum rank 2 bundle which is not instanton?The goal of this paper is to provide partial answers to these questions.In particular, we show that every rank 2 instanton bundle of charge 3 is 3-Buchsbaum. However, this is false for instantons of higher charge. On the other18 M. JARDIM AND S. MARCHESIhand, we show that the generic instanton of charge 4 or 5 is also 3-Buchsbaum.In addition, we provide an explicit example of a 3-Buchsbaum bundle of rank 2which is not an instanton, and conjecture that every 3-Buchsbaum rank 2bundle on P3 is one of these.1. PreliminariesIn this section we will fix the notation and recall the basic definitions usedthroughout this paper.1.1. Buchsbaum sheavesLet K be an algebraically closed field of characteristic zero. Let us denote byS = K[x0, x1, x2, x3] the ring of polynomials in four variables, so that P3 :=Proj(S), and let m = (x0, x1, x2, x3) denote the irrelevant ideal.Let V be a K-vector space of dimension m+ 1, with V ∗ denoting its dual.The projective space P(V ) = Pm is understood as the set of equivalence classesof m-dimensional subspaces of V , or, equivalently, the equivalence classes ofthe lines of V ∗.Given a coherent sheaf E on P3, consider the following graded S-module:H1∗ (E) =⊕n∈ZH1(E(n)).Definition 1.1. A coherent sheaf E on P3 is said to be p-Buchsbaum if andonly if p is the minimal power of the irrelevant ideal which annihilates theS-module H1∗ (E), i.e.p = min{t |mtH1∗ (E) = 0}.In this work, we will only consider locally free sheaves on P3.1.2. Monads and regularityRecall that a monad on a projective variety X of dimension n is a complex oflocally free sheaves on X of the formM• : Aα−→ B β−→ Csuch that the map α is injective and the map β is surjective. It follows thatE := kerβ/ Imα is the only nontrivial cohomology of the complex M•. Thecoherent sheaf E is called the cohomology of M•; it is locally free if and only ifthe map α is injective in every fiber.The monad M• is called a Horrocks monad if, in addition:P -BUCHSBAUM RANK 2 BUNDLES 19i) A and C are direct sum of invertible sheaves,ii) H1∗ (B) = Hn−1∗ (B) = 0.Furthermore, the monad is also called minimal if it satisfiesiii) no direct sum of A is isomorphic to a direct sum of B,iv) no direct sum of C is the image of a line subbundle of B.Let us recall the following result on minimal Horrocks monads, cf. [12,Theorem 2.3].Theorem 1.2. Let X be an arithmetically Cohen–Macaulay variety of dimen-sion n ≥ 3, and let E be a locally free sheaf on X. Then there is a 1-1correspondence between collections{n1, . . . , nr,m1, . . . ,ms} with ni ∈ H1(E∨ ⊗ ωX(ki)) and mj ∈ H1(E(−lj))for integers ki’s and lj’s, and equivalence classes of Horrocks monads of theformM• :r⊕i=1ωX(ki)α−→ F β−→s⊕j=1OX(lj),whose cohomology is isomorphic to E.Moreover, the correspondence is such that M• is minimal if and only if theelements mj generate H1∗ (E) and the elements ni generate H1∗ (E∨ ⊗ ωX) asmodules.Recall that a coherent sheaf E on Pn is said to be m-regular in the senseof Castelnuovo–Mumford if Hi(Pn, E(m − i)) = 0 for i > 0. Costa and Miró-Roig studied in [3] the Castelnuovo–Mumford regularity of the cohomology ofa certain class of monads which include monads of the following form:OP3(−l)⊕kα−→2+2k⊕j=1OP3(bj)β−→ OP3(d)⊕k, (1)where l, k, c ≥ 1 and −l < b1 ≤ · · · ≤ b2+2k < d. Specializing [3, Theorem 3.2]to monads of the form (1), one obtains the following result.Proposition 1.3. If E is the cohomology of a monad of the form (1), then Eis m-regular for any integer m such thatm ≥ max{(k + 2)d− (b1 + · · ·+ bk+3)− 2, l}.20 M. JARDIM AND S. MARCHESI1.3. Cohomology of generic instanton bundlesRecall that a bundle E of rank 2 on P3 is called an instanton bundle if it isisomorphic to the cohomology of a monad of the following form:OP3(−1)⊕kα−→ O⊕2+2kP3β−→ OP3(1)⊕k (2)The integer k is called the charge of E; notice that c1(E) = 0 and c2(E) =k. Note also that nullcorrelation bundles are precisely instanton bundles ofcharge 1.Alternatively, an instanton bundle can also be defined as a bundle E on P3with c1(E) = 0 and satisfying the following cohomological conditions:h0(E(−1)) = h1(E(−2)) = h2(E(−2)) = h3(E(−3)) = 0.The Hilbert polynomial of an instanton bundle is given byPE(t) = 2(k + 1)χ(OP3(t))− kχ(OP3(t− 1))− kχ(OP3(t+ 1)) (3)=13(t+ 2)((t+ 3)(t+ 1)− 3k)=13(t+ 2)(t+ 2 +√3k + 1)(t+ 2−√3k + 1).Note also that PE(t) = h0(E(t))− h1(E(t)) for t ≥ −2.On another direction, recall that a coherent sheaf F on P3 is said to havenatural cohomology if for each t ∈ Z, at most one of the cohomology groupsHp(F (t)), where p = 0, . . . , 3, is nonzero; every torsion free coherent sheaf withnatural cohomology is in fact locally free [10, Lemma 1.1]. In addition, everyrank 2 locally free sheaf with c1 = 0, c2 > 0 and natural cohomology is aninstanton bundle [10, p. 365].Hartshorne and Hirschowitz have shown in [10] that the generic instantonbundle has natural cohomology. More precisely, let I(k) denote the modulispace of rank 2 locally free instanton sheaves of charge k; this is known tobe an affine [4], irreducible [18, 19], nonsingular variety of dimension 8k − 3[13]. Let N (k) denote the subset of I(k) consisting of instanton bundles withnatural cohomology; it is easy to see that N (k) is open within I(k), and [10,Theorem 0.1 (a)] tells us that it is nonempty.More recently, Eisenbud and Schreyer have introduced the notion of super-natural bundles, see [6, p. 862]: a locally free sheaf on P3 is called supernaturalif it has natural cohomology and its Hilbert polynomial has distinct integralroots. Therefore we see that there exists a rank 2 supernatural bundle withc1 = 0 and c2 = k > 0 if and only if 3k + 1 is a perfect square; the first threepossible values for k are k = 1, 5, 8.P -BUCHSBAUM RANK 2 BUNDLES 212. Instanton vs BuchsbaumWe start by introducing the following function on the positive integersm(k) =⌊√3k + 1− 2⌋,where b·c denotes the largest positive integer which is smaller than or equal tothe argument.Proposition 2.1. A rank 2 instanton bundle E is p-Buchsbaum if and only ifh1(E(p − 2)) 6= 0 and h1(E(p − 1)) = 0. In addition, every rank 2 instantonbundle of charge k is p-Buchsbaum for some m(k) + 2 ≤ p ≤ k.Proof. By Theorem 1.2, we get that H1∗ (E) is generated in H1(E(−1)). Thusif h1(E(p − 2)) 6= 0 and h1(E(p − 1)) = 0 (and hence h1(E(t)) = 0 for everyt ≥ p−1), thenH1∗ (E)must be p-Buchsbaum. Conversely, if E is p-Buchsbaum,then h1(E(p−2)) 6= 0 (otherwise, H1∗ (E) would be annihilated by the (p−1)-thpower of the irrelevant ideal) and h1(E(p− 1)) = 0.By Proposition 1.3, we have that E is k-regular (cf. also [3, Corollary 3.3]).Hence H1(E(k − 1)) = 0, and it follows that every rank 2 instanton bundle isat most k-Buchsbaum.Finally, note from (3) that for −1 ≤ t ≤ m(k) we have χ(E(t)) < 0. Sinceh3(E(t)) = 0 in this range, it follows that h1(E(t)) 6= 0 for t = m(k). Thusevery rank 2 instanton bundle is at least (m(k) + 2)-Buchsbaum.Since m(3)+2 = 3, the first immediate consequence of the previous Propo-sition is given by the following Corollary.Corollary 2.2. Every rank 2 instanton bundle of charge 3 is3-Buchsbaum.However, it is not true that every rank 2 instanton bundle of charge 3 hasnatural cohomology, as observed in [10, Example 1.6.1]. Indeed, recall that aninstanton bundle E is called a ’t Hooft instanton if h0(E(1)) 6= 0, cf. [1]; moreformally, consider the setH(k) := {E ∈ I(k) | h0(E(1)) 6= 0} ,which is known to be a locally closed subvariety of I(k) of dimension 5k + 4,irreducible and rational [1, Theorem 2.5]. On the other hand, let U(k) :=I(k) \ N (k), the subvariety of “unnatural" instanton bundles.Lemma 2.3. For every k ≥ 3, we have H(k) ⊂ U(k), while H(3) = U(3).22 M. JARDIM AND S. MARCHESIProof. If E is a rank 2 instanton bundle of charge k ≥ 3, then h1(E(1)) 6= 0(because χ(E(−1)) < 0). Hence if E is a ’t Hooft instanton, then it does nothave natural cohomology, showing that H(k) ⊂ U(k).Conversely, let now E be a rank 2 instanton bundle of charge 3 which doesnot have natural cohomology. We then know that(i) h0(E(t)) = 0 for t ≤ 0;(ii) h1(E(t)) = 0 for t 6= −1, 0, 1;(iii) h2(E(t)) = 0 for t 6= −5,−4,−3;(iv) h3(E(t)) = 0 for t ≥ −4.The last two claims are obtained by Serre duality and the fact E ' E∗.Therefore the only way in which E may fail to have natural cohomology isif h0(E(1)) = h3(E(−5)) 6= 0. It follows that U(3) ⊂ H(3).It would be interesting to determine properties of the U(k) for k ≥ 4,particularly its dimension and number of irreducible components. The previouslemma tells us that dimU(k) ≥ 5k + 4.Another immediate consequence of Proposition 2.1 is the following.Corollary 2.4. The generic rank 2 instanton bundle of charge k is (m(k)+2)-Buchsbaum.In particular, since m(4) + 2 = m(5) + 2 = 3, the generic rank 2 instantonbundle of charges 4 and 5 are 3-Buchsbaum, while instanton bundles of chargek ≥ 6 are at least 4-Buchsbaum.3. A 3-Buchsbaum rank 2 bundle with c1 = −1Theorem 1 tells us, in particular, that the first Chern class of every 1- and2-Buchsbaum rank 2 bundle on P3 is zero. In this section, we show that thesame is not true for p-Buchsbaum bundles with p ≥ 3, providing an exampleof a 3-Buchsbaum rank 2 bundle with c1 = −1.Indeed, consider the monadOP3(−2)α−→ O⊕2P3 ⊕OP3(−1)⊕2 β−→ OP3(1) , (4)which is the simplest example of a class of monads originally introduced byEin in [5, eq. 3.1.A]. The existence of such monads can be easily established;consider for instance the following explicit mapsα =−z2−w2xy and β =xyz2w2P -BUCHSBAUM RANK 2 BUNDLES 23where [x : y : z : w] are homogeneous coordinates on P3.Let F denote the locally free cohomology of a monad of the form (4); it isa rank 2 bundle with c1(F ) = −1 and c2(F ) = 2. Ein claims in [5, p. 21],without proof, that F is µ-stable. For the sake of completeness, we include aproof below.Lemma 3.1. Every locally free sheaf F obtained as the cohomology of a monadof the form (4) is µ-stable.Proof. First consider the kernel bundle K := kerβ defined by the sequence0→ K → O⊕2P3 ⊕OP3(−1)⊕2 β−→ OP3(1).It follows from [2, Theorem 2.7] that K is µ-semistable (but not µ-stable).Therefore, since µ(K) = −1, we must have h0(K) = 0. Now, from the sequence0→ OP3(−2)α−→ K → F → 0we have that h0(F ) = 0, which implies that F is µ-stable.We now show that the bundles considered in this Section are 3-Buchsbaum.Proposition 3.2. Every locally free sheaf F obtained as cohomology of a monadof the form (4) is 3-Buchsbaum.Proof. By Theorem 1.2, we get that H1∗ (F ) is generated in H1(F (−1)). Onthe other hand, Proposition 1.3 tells us that F is 3-regular, thus h1(F (2)) = 0.If we also had h1(F (1)) = 0, F would be 2-Buchsbaum, which, by Theorem 1cannot happen. Therefore F must be 3-Buchsbaum.Note also that, since h0(F (1)) = h1(F (1)) = 1 [11, 2.2], such bundles donot have natural cohomology.Based on the evidence here presented and also motivated by results due toRoggero and Valabrega in [17], specially Propositions 5 and 6 and Theorem 2there, we propose the following classification of 3-Buchsbaum rank 2 bundleson P3.Conjecture 3.3. Every normalized, 3-Buchsbaum rank 2 bundle on P3 is ei-ther an instanton bundle of charge 3, 4 or 5, if c1 = 0, or the cohomology of amonad of the form (4), if c1 = −1.Finally, let us comment on p-Buchsbaum rank 2 bundles on P3 for p ≥ 4.An interesting, possible source of examples of such bundles is provided by Ein’sgeneralized nullcorrelation bundles, described in [5]. These are bundles obtainedas cohomologies of monads of the following two types:24 M. JARDIM AND S. MARCHESIOP3(−d) −→ OP3(−b)⊕OP3(−a)⊕OP3(a)⊕OP3(b) −→ OP3(d) , (5)andOP3(−d−1) −→ OP3(−b−1)⊕OP3(−a−1)⊕OP3(a)⊕OP3(b) −→ OP3(d) , (6)where d > b ≥ a ≥ 0. Let us denote the cohomology of such monads by Ea,b,dand Fa,b,d, respectively.Note that, by Theorem 1.2 and Proposition 1.3, H1∗ (Ea,b,d) is generated indegree −d, and that Ea,b,d is (3d − 2)-regular when d ≥ 1. Therefore, suchbundles are at most (4d− 3)-Buchsbaum, being precisely (4d− 3)-Buchsbaumprovided h1(Ea,b,d(3d− 4)) 6= 0.Similarly, note that H1∗ (Fa,b,d) is generated in degree −d, and that Fa,b,d is3d-regular. Therefore, such bundles are at most (4d− 1) – Buchsbaum, beingprecisely (4d− 1) –Buchsbaum provided h1(Fa,b,d(3d− 2)) 6= 0.AcknowledgmentsMJ is partially supported by the CNPq grant number 302477/2010-1 and theFAPESP grant number 2011/01071-3. SM is supported by the FAPESP post-doctoral grant number 2012/07481-1. We started thinking about instantonsas p-Buchsbaum bundles after conversations with Joan Pons-Llopis; we thankhim for several fruitful discussions.Added in proof. Conjecture 3.3 was proposed by the second author inthe “Vector Bundles Days II”, held in Trieste in occasion of Emilia Mezzetti’sbirthday. We thank the editor for informing us that the conjecture has beenproved in [8].References[1] V. Beorchia and D. Franco, On the moduli space of ’t Hooft bundles, Ann.Univ. Ferrara Sez. VII (N.S.) 47 (2001), 253–268.[2] G. Bohnhorst and H. Spindler, The stability of certain vector bundles onPn, Lecture Notes in Math. 1507 (1992), 39–50.[3] L. Costa and R.M. Miró-Roig, Monads and regularity of vector bundles onprojective varieties, Michigan Math. J. 55 (2007), 417–439.[4] L. Costa and G. Ottaviani, Nondegenerate multidimensional matrices andinstanton bundles, Trans. Amer. Math. Soc. 355 (2003), 49–55.[5] L. Ein, Generalized null correlation bundles, Nagoya Math. J. 111 (1988), 13–24.[6] D. Eisenbud and F.-O. Schreyer, Betti numbers of graded modules and co-homology of vector bundles, J. Amer. Math. Soc. 22 (2009), 859–888.P -BUCHSBAUM RANK 2 BUNDLES 25[7] Ph. Ellia, Ordres et cohomologie des fibrés de rang deux sur l’espace projectif,Preprint (Nice), 1987.[8] Ph. Ellia and L. Gruson, On the Buchsbaum index of rank two vector bundleson P3, Rend. Istit. Mat. Univ. Trieste 47 (2015), 65–79.[9] Ph. Ellia and A. Sarti, On codimension two k-Buchsbaum subvarieties of Pn,Lecture Notes in Pure and Appl. Math., Dekker, New York 206 (1999), 81–92.[10] R. Hartshorne and A. Hirschowitz, Cohomology of a general instantonbundle, Ann. Sci. École Norm. Sup. 15 (1982), 365–390.[11] R. Hartshorne and I. Sols, Stable rank 2 vector bundles on P3 with c1 = −1,c2 = 2, J. Reine Angew. Math. 325 (1981), 145–152.[12] M. Jardim and R.V. Martins, Linear and Steiner bundles in projective vari-eties, Comm. Algebra 38 (2010), 2249–2270.[13] M. Jardim and M. Verbitsky, Trihyperkähler reduction and instanton bun-dles on CP3, Compositio Math. 150:11 (2014), 1836–1868.[14] N.M. Kumar and A.P. Rao, Buchsbaum bundles on Pn, J. Pure Appl. Algebra152 (2000), 195–199.[15] J.C. Migliore and R.M. Miró-Roig, On k-Buchsbaum curves in P3, Comm.Algebra 18 (1990), 2403–2422.[16] R.M. Miró-Roig, Non-obstructed subcanonical space curves, Publ. Mat. 36(1992), 761–772.[17] M. Roggero and P. Valabrega, Chern classes and cohomology for rank 2reflexive sheaves on P3, Pacific J. Math. 150 (1991), 383–395.[18] A.S. Tikhomirov, Moduli of mathematical instanton vector bundles with oddc2 on projective space, Izvestiya: Mathematics 76:5 (2012), 991–1073.[19] A.S. Tikhomirov, Moduli of mathematical instanton vector bundles with evenc2 on projective space, Izvestiya: Mathematics 77:6 (2013), 1331–1355.Authors’ addresses:Marcos JardimDepartamento de MatemáticaIMECC-UNICAMPRua Sérgio Buarque de Holanda 65113083-859 Campinas, SP, BrazilE-mail: jardim@ime.unicamp.brSimone MarchesiDepartamento de MatemáticaIMECC-UNICAMPRua Sérgio Buarque de Holanda 65113083-859 Campinas, SP, BrazilE-mail: marchesi@ime.unicamp.brReceived July 7, 2014Revised October 1, 2014

p-Buchsbaum rank 2 bundles on the projective space

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