Codimension one distributions of degree 2 on the three-dimensional projective space
aa r X i v : . [ m a t h . AG ] F e b CODIMENSION ONE DISTRIBUTIONS OF DEGREE ON THETHREE-DIMENSIONAL PROJECTIVE SPACE
HUGO GALEANO, MARCOS JARDIM, AND ALAN MUNIZ
Abstract.
We prove existence and non-existence results for codimension onedistributions D of degree 2 on P whose tangent sheaves T D have prescribedChern classes. These allow us to establish a full classification of these dis-tributions, leading to certain general conjectures regarding codimension onedistributions and foliations of arbitrary degree on P . Contents
1. Introduction 12. Preliminaries and notation 43. Basic constructions 104. Distributions with c ( T D ) ≤ c ( T D ) = 2 176. Distributions with c ( T D ) = 3 197. Distributions with c ( T D ) = 4 248. Distributions with c ( T D ) = 5 279. Distributions with c ( T D ) = 6 2810. Concluding Remarks 28References 291. Introduction
A codimension one holomorphic distribution D of degree d on the complex pro-jective space P n is given by an exact sequence of the form D : 0 −→ T D −→ T P n ω −→ I Z ( d + 2) −→ , induced by a 1-form ω ∈ H (Ω P n ( d + 2)) ≃ Hom(T P n , O P n ( d + 2)). The sheaf T D ≃ ker ω , called the tangent sheaf is a reflexive sheaf of rank n −
1, whileim ω ≃ I Z ( d + 2) is the ideal sheaf of a subscheme Z ⊂ P n called the singularscheme of D ; Z is frequently denoted by Sing( D ). In general, the scheme Z isnot pure dimensional, and we denote by Sing n − ( D ) the union of its irreduciblecomponents of pure codimension 2. (HG, MJ) IMECC - UNICAMP, Departamento de Matem´atica, Rua S´ergio Buarque deHolanda, 651, 13083-970 Campinas-SP, Brazil
E-mail addresses : [email protected], [email protected], [email protected] .2020 Mathematics Subject Classification.
Primary 58A17, 14D20, 14J60; Secondary 14D22,14F06, 13D02.
Key words and phrases.
Distributions, Singular scheme.
A codimension one distribution D is integrable , i.e. it defines a foliation , if T D ⊂ T P n is closed under the Lie bracket, [ T D , T D ] ⊂ T D ; this is equivalent to thecondition ω ∧ dω = 0 on the 1-form ω .The study of distributions, and especially foliations, on P n from the point of viewof Algebraic Geometry has been a very active theme of research in the past fewdecades. A central problem has been the classification of codimension one foliationson P n with a given degree d . In [19], Jouanolou classified codimension one foliationsof degrees 0 and 1 on P n ; in [7], Cerveau and Lins Neto classified codimension onefoliations of degree 2 on P n . These authors describe the irreducible components ofthe algebraic set F ( d, n ) := { [ ω ] ∈ P H (Ω P n ( d + 2)) | ω ∧ dω = 0 } for d = 0 , d on P n ; their classification is given in terms of a naive deformationtheory for the 1-form ω .If we remove the integrability condition then any two 1-forms in H (Ω P n ( d + 2))can be deformed into one another with no regard of the distributions they define.Thus a finer moduli theory was needed to study such objects. A novel approachto the study of flat deformations of codimension one distributions, regardless ofintegrability, was introduced in [3]. The authors propose a classification in terms ofnatural topological invariants associated to a codimension one distribution, namelythe Chern classes of tangent sheaf T D , and provide a full description for T D and Z when d = 0 or 1, see [3, Proposition 7.1] and [3, Section 8], respectively. In addition,a classification of codimension one distribution on P of degree 2 with locally freetangent sheaf and reduced singular scheme is also given, see [3, Theorem 9.5].The aim of this work is to present a complete picture for codimension one distri-bution of degree 2 on P . More precisely, we describe all possible tangent sheaves,addressing (semi)stability, Chern classes and spectra, if applicable. In addition, wealso describe all possible singular schemes. Our main result is the following. Main Theorem.
Let D be a codimension one distribution of degree 2 on P . Then T D is µ -semistable whenever it does not split as a sum of line bundles; it can bestrictly µ -semistable only when ( c ( T D ) , c ( T D )) = (1 , or (2 , . In addition, c ( T D ) = 0 , and the second and third Chern classes and spectra of T D are listed inTable 1, where Sing ( D ) is given. Finally, Sing( D ) is never contained in a quadricsurface. We recall that the spectrum of a rank 2 reflexive sheaf is a set of integers thatencodes some partial information on the cohomology of F , see Section 2.4. Exceptfor the cases (5 , ,
8) and (4 , c ( T D ) ≤ c ( T D ) , c ( T D )) = (2 ,
4) or (3 , c ( T D ) , c ( T D )) = (6 ,
20) corresponds to 1-forms ω ∈ H (Ω P (4) with only isolated singularities, which are known to form anopen subset. Furthermore, the following facts were also observed in [3, Section 8]:(1) if c ( T D ) = 5, then T D is µ -stable and c ( T D ) = 14;(2) if c ( T D ) = 4, then T D is µ -stable and 0 ≤ c ( T D ) ≤ c ( T D ) = 3, then T D is µ -semistable and 0 ≤ c ( T D ) ≤ ODIMENSION ONE DISTRIBUTIONS OF DEGREE 2 ON P c ( T D ) c ( T D ) Spectrum Sing ( D )6 20 { -3,-2,-1,-1,-1 } empty5 14 { -2,-2,-1,-1,-1 } line4 10 { -2.-1,-1,-1 } conic8 { -1,-1,-1,-1 } two skew lines6 { -1,-1,-1,0 } double line of genus −
23 8 { -2,-1,-1 } plane cubic curve6 { -1,-1,-1 } twisted cubic4 { -1,-1,0 } conic ⊔ line2 { -1,0,0 } three skew lines0 { } double line of genus − ⊔ line2 4 { -1,-1 } elliptic quartic curve2 { -1,0 } rational quartic curve0 { } twisted cubic ⊔ line1 2 { -1 } curve of degree 5, genus 20 { } elliptic curve of degree 50 0 split ACM curve of degree 6 genus 3-1 0 split ACM curve of degree 7 genus 5 Table 1.
Possible Chern classes and singular loci of codimensionone distribution of degree 2; the last column describes a genericpoint in the irreducible component of the Hilbert scheme whichcontains Sing ( D ).(4) if c ( T D ) = 2, then T D is µ -semistable and 0 ≤ c ( T D ) ≤ T D is either strictly µ -semistable or not µ -semistable is given in Theorem 3 below. For the sake of completeness, we revisethe cases with c ( T D ) ≤ c ( T D ) ≥ c ( T D ) , c ( T D )) is fixed, the degree and arithmeticgenus of Sing ( D ) can be computed via the expessions in display (17). When thedegree is less than 5, the curves with these given degree and arithmetic genusare explicit described in the literature, especially [21, 22]. So if we construct acodimension one distribution D with a given Sing ( D ), then T D will have thedesired properties.The first technique, described in Section 3.1, relies on codimension two distri-butions on P , i.e., foliations by curves : given a codimension two distribution, onecan find a codimension one distribution containing it and whose singular locus canbe explicitly described, see Proposition 6. This result is applied to prove the exis-tence of codimension one distributions D of degree 2 such that Sing ( D ) is a pairof disjoint lines, a smooth conic or a line, see Sections 7.2, 7.3 and 8, establishingthe existence of the cases ( c ( T D ) , c ( T D )) = (4 , , (4 ,
10) and (5 , ω vanishing along a given curve. This procedure provides H. GALEANO, M. JARDIM, AND A. MUNIZ
Foliation c ( T D ) c ( T D ) T D R (1 ,
3) 3 8 stable R (2 ,
2) 2 4 stable L (1 , ,
2) 1 2 str. µ -semistable L (1 , , ,
1) 0 0 O P ⊕ O P E (3) 0 0 O P ⊕ O P S (2 ,
3) -1 0 O P (1) ⊕ O P ( − Table 2.
The first column lists the 6 irreducible components of F (2 ,
3) according to Cervau and Lins Neto [7]. Each line providesa description of the tangent sheaves for a generic point lying ineach componentexplicit examples for the cases ( c ( T D ) , c ( T D )) = (3 , , (3 , , (3 ,
6) and (4 ,
6) (seeSections 6.1 and 7) and also allows to show, as a consequence of Corollary 9, thatthe cases ( c ( T D ) , c ( T D )) = (4 , , (4 ,
2) and (4 ,
4) cannot be realized.Finally, the third technique looks into properties of stable rank 2 reflexive sheaveswith desired second and third Chern classes. Exploring these properties, one canprove the existence of a monomorphism from a given sheaf F into T P whosecokernel is torsion free. This is applied to the case ( c ( T D ) , c ( T D )) = (3 , R ( a, b ) and L ( d , .., d ) arethe families of rational and logarithmic foliations ; E (3) is the family of exceptionalfoliations ; S (2 ,
3) are pullback foliations . Our thorough study of codimension onedistributions of degree 2 on P , together with previous study of distributions ofdegrees 0 and 1, allows us to propose certain conjectures, stated in Section 10, fordistributions and foliations of arbitrary degree. Acknowledgments.
We thank Maur´ıcio Corrˆea for enlightening discussions. HGwas supported by a PhD grant from CAPES, and the results of his PhD thesisare present here. MJ is supported by the CNPQ grant number 302889/2018-3and the FAPESP Thematic Project 2018/21391-1. The authors also acknowledgethe financial support from Coordena¸c˜ao de Aperfei¸coamento de Pessoal de N´ıvelSuperior - Brasil (CAPES) - Finance Code 001.2.
Preliminaries and notation
We begin by setting up the notation and nomenclature to be used in the rest ofthe paper.2.1.
Codimension one distributions on P . A codimension one distributionson P is given by an exact sequence(1) D : 0 −→ T D −→ T P −→ I Z ( d + 2) −→ , ODIMENSION ONE DISTRIBUTIONS OF DEGREE 2 ON P where d ≥ degree of D , and Z is its singular scheme . In general,dim Z ≤
1, and it may have both 0- and 1-dimensional components. Letting U bethe maximal 0-dimensional subsheaf of O Z , we obtain the exact sequence(2) 0 −→ U −→ O Z −→ O C −→ , where C is a pure 1-dimensional scheme, i.e., a Cohen-Macaulay curve. We alsodenote it by Sing ( D ) := C . One can show that [3, Theorem 3.1]: c ( T D ) = 2 − d ; c ( T D ) = d + 2 − deg( C ); c ( T D ) = length( U ) = d + 2 d + 2 d − deg( C ) · (3 d −
2) + 2 p a ( C ) − , (3)where p a ( C ) denotes the arithmetic genus of C .Since this paper is dedicated to the case d = 2, we specialize it further. Then c ( T D ) = 0 and T D ≃ T ∗ D , since T D is a rank two reflexive sheaf. The formulas indisplay (3) simplify to c ( T D ) = 6 − deg( C ); c ( T D ) = length( U ) = 18 − C ) + 2 p a ( C ) . (4)Dualizing the sequence in display (1) and using d = 2, we obtain(5) 0 −→ O P ( − −→ Ω P −→ T D ζ −→ ω C −→ . Computing the cohomology we get the following key lemma; a similar result holdsin any given degree.
Lemma 1.
Let D be a degree codimension one distribution on P then (1) h ( T D (1)) = h ( ω C (1)) ; (2) h ( T D ) ≤ ;and for p ≥ we have (3) h ( T D ( p )) = h ( ω C ( p )) ; (4) h ( T D ( p )) = 0 . Note that since C is a Cohen-Macaulay curve, Serre duality holds and we mayuse h i ( ω C ( p )) = h − i ( O C ( − p )) for i ∈ , p ∈ Z , see [14, III Corollary7.7] .An important invariant of C is its Rao module M C = L p H ( I C ( p )). From thesequence (1) and using that U = I C / I Z is zero-dimensional, we get h ( I C ( p )) ≤ h ( T D ( p − p ∈ Z . Thus we have a restriction on the curves that can appear in the singular loci ofdegree two distributions.2.2.
Foliations by curves on P . Consider now codimension 2 distributions on P ; they are given by an exact sequence(6) G : 0 −→ O P (1 − k ) ν −→ T P −→ N G −→ N G , called the normal sheaf, is torsion free. The integer k ≥ degree of the distribution determined by the vector field ν ∈ H (T P ( k − foliation by curves of degree k on P . For a detailed account on foliations by curveswe refer to [5, 10]. We will be mostly interested in the cases k = 0 and k = 1. H. GALEANO, M. JARDIM, AND A. MUNIZ
In order to make the role of the singular locus more explicit, we dualize thesequence in display (6) to obtain(7) 0 −→ N ∗ G −→ Ω P −→ I W ( k − −→ , where W is the scheme of zeros of the vector field ν . As for codimension onedistributions, W may not be pure dimensional, so we have an exact sequence(8) 0 −→ R −→ O W −→ O Y −→ , where R is a 0-dimensional sheaf and Y is a curve; we set Y := Sing ( G ) and R := Sing ( G ). The foliation by curves G is said to be generic if dim W = 0, thatis, ν only vanishes at points.When k = 0, the picture is very simple. The foliation G is given by a constantvector field ν vanishing at only one point, and N G is a stable rank 2 reflexive sheafgiven by a sequence of the following form0 −→ O P −→ O P (1) ⊕ −→ N G −→ N ∗ G ≃ N G ( − k = 1 and N ∗ G is locally free, then N ∗ G = O P ( − ⊕ O P ( −
2) and W consists of two skew lines, see [10, Theorem 5.1]. We describe below the cases where N ∗ G is not locally free; thus completing the classification of foliation by curves ofdegree one. First recall that a torsion free sheaf E on P n is µ -semistable if for everysaturated subsheaf F ⊂ E , c ( F )rk F ≤ c ( E )rk E . the sheaf E is µ -stable if it is µ -semistable and the inequalities are always strict. Theorem 2.
Let G be a foliation by curves of degree 1 on P . Then (1) N ∗ G is a µ -stable rank 2 reflexive sheaf with Chern classes c ( N ∗ G ) = 6 and c ( N ∗ G ) = 4 , and dim W = 0 ; (2) N ∗ G = O P ( − ⊕ O P ( − and W consists of 2 skew lines; (3) N ∗ G is a strictly µ -semistable reflexive sheaf given by an extension of theform (9) 0 −→ O P ( − −→ N ∗ G −→ I L ( − −→ , where L is a line in P , and W consists of a line disjoint from L plus 2points contained in L .Proof. The first item was proved in [10, Theorem 4.1]. So we may assume thatdim
W >
0. Let G be a foliation by curves of degree 1 as in sequence (7). We firstnote that N ∗ G must be µ -semistable. Indeed, c ( N ∗ G ) = −
4, and h (( N ∗ G ) η ( − h ( N ∗ G (1)) = 0 , where ( N ∗ G ) η denotes the normalization of N ∗ G .It then follows from the Bogomolov inequality that c ( N ∗ G ) ≥ c ( N ∗ G ) / c ( N ∗ G ) = 6 − deg( Y ) ≤ Y ) > G is not generic.If c ( N ∗ G ) = 4, then c (( N ∗ G ) η ) = 0 which implies, by µ -semistability, that( N ∗ G ) η = O P ⊕ O P , providing the first item in the classification.If c ( N ∗ G ) = 5, then c (( N ∗ G ) η ) = 1 and deg( Y ) = 1, so Y must be a line.Therefore, [10, Theorem 4.1 (iii)] forces c ( N ∗ G ) = c (( N ∗ G ) η ) = 2. The proof of [8, ODIMENSION ONE DISTRIBUTIONS OF DEGREE 2 ON P Lemma 2.1] guarantees that N ∗ G must fit into the exact sequence given in display(9).To complete the proof, we must show that the sheaves described in display (9)do occur as conormal sheaves of foliation by curves. Recall that a null correlationsheaf P is defined as the cokernel of a morphism O P ( − → Ω P (1) given by aglobal section τ ∈ H (Ω P (2)) that vanishes on a line L := ( τ ) . Dualizing theexact sequence 0 −→ O P ( − τ −→ Ω P (1) −→ P −→ P , we get0 −→ P ∗ −→ T P ( − −→ O P (1) −→ O L (1) −→ O P (1)):0 −→ P ∗ (1) −→ T P −→ I L (2) −→ . The previous sequence is a codimension one distribution of degree 0 whose singu-lar locus is precisely the line L , and we conclude, from the classification of suchdistributions in [10, Proposition 7.1], that P ∗ (1) ≃ O P (1) ⊕ . Using the canonicalmonomorphism P ֒ → P ∗∗ , we get0 −→ P −→ O ⊕ P −→ O L (1) −→ . Next, choose a line ℓ , disjoint from L , which is defined as the vanishing locusof a monomorphism ξ : O P ( − → O ⊕ P . The kernel of the composed morphism O P ( − → O ⊕ P → O L (1), must be of the ideal sheaf I L ( − Q of length 2 contained in the line L . Tobe precise, we get the commutative diagram0 (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / I L ( − (cid:15) (cid:15) / / O P ( − ξ (cid:15) (cid:15) / / O L ( − (cid:15) (cid:15) / / / / P (cid:15) (cid:15) / / O ⊕ P (cid:15) (cid:15) / / O L (1) / / (cid:15) (cid:15) / / I ℓ ∪ Q (1) (cid:15) (cid:15) / / I ℓ (1) (cid:15) (cid:15) / / O Q / / (cid:15) (cid:15)
00 0 0Therefore, we can consider the composed epimorphism ϕ : Ω P ։ P ( − ։ I ℓ ∪ Q ;ker ϕ must, by our previous arguments, be a µ -semistable sheaf as given in display(9). (cid:3) H. GALEANO, M. JARDIM, AND A. MUNIZ
Stability for distributions and sub-foliations.
In order to describe codi-mension one distributions, we analyse the stability of the possible tangent sheaves.If D is a degree d codimension one distribution on P such that T D is not µ -stable,there exists a line sub-bundle O P ( l ) ⊂ T D such that 2 l ≥ − d . On the other hand,we have O P ( l ) ֒ → T D ֒ → T P that can exist only if l ≤
1. For d = 2, we must have l = 0 or 1; the later only occurs if T D is not µ -semistable. These line sub-bundlesinduce sub-distributions of codimension two; next, we will describe them.If D is a codimension one distribution of degree two then − ≤ c ( T D ) ≤ c ( T D ) = 6 − deg Sing ( D ) and, on the other hand,the restriction of D to a general plane H is singular at Sing ( D ) ∩ H , whencedeg Sing ( D ) ≤
7; see [3, p. 28].If T D is not µ -semistable then h ( T D ( − = 0 and, owing to [3, Lemma 4.3],it must split as sum of line bundles and, as T D ֒ → T P , the only possibility is T D = O P (1) ⊕ O P ( − c ( T D ) = − T D is µ -semistable if c ( T D ) ≥ c ( T D ) = 0 then T D = O P ⊕ O P . If c ( T D ) = 1 then c ( T D ) = 0 or 2, owingto [16, Theorem 8.2]. For c ( T D ) = 0 we have that T D must be a null correlationbundle, which is stable; for c ( T D ) = 1 [8, Lemma 2.1] shows that T D is strictly µ -semistable.If c ( T D ) ≥
4, then it follows from [3, Proposition 6.3] that the tangent sheaf T D is stable. We will now complete this picture, and the first claim of Main Theorem1, with the following result. Theorem 3.
Let D be a codimension one distribution of degree 2. The stability of T D is described as follows: (1) T D is not µ -semistable if and only if c ( T D ) = − ; if that is the case T D = O P (1) ⊕ O P ( − ; (2) If T D is strictly µ -semistable then ( c ( T D ) , c ( T D )) = (0 , , (1 , or (2 , ;For all the other cases T D is µ -stable.Proof. The first item follows from the above discussion; we will prove now thesecond one. Consider a codimension 1 distribution of degree 2 D : 0 −→ T D φ −→ T P −→ I Z (4) −→ c ( T D ) ≥
2. If D is not µ -stablethen h ( T D ) = 0; so let σ ∈ H ( T D ) be a non trivial section, and let S := ( σ ) . Dueto µ -semistability, S is a curve of degree deg( S ) = c ( T D ) ≥
2; otherwise we couldfactor out the divisoral part of S , yielding a section in H ( T D ( − σ induces a (sub-)foliation by curves of degree 1 G : 0 −→ O P φ ◦ σ −→ T P −→ N G −→ . Since im( φ ◦ σ ) ∨ ⊂ im σ ∨ , we see that S ⊆ Y = Sing ( G ); in particular, we havedeg( Y ) ≥
2. According to Theorem 2, we must have N ∗ G = O P ( − ⊕ O P ( − G ) = Y consists of two skew lines; otherwise Y would have degree one.As S has degree at least two, we conclude that S = Y . Therefore, c ( T D ) = 2 and c ( T D ) = 2 p a ( S ) − (cid:3) One advantage of the µ -semistability to our classification is to bound the thirdChern class. For a µ -stable rank two reflexive sheaf F on P with c ( F ) = 0 we ODIMENSION ONE DISTRIBUTIONS OF DEGREE 2 ON P have, owing to [16, Theorem 8.2], that(10) c ( F ) ≤ c ( F ) − c ( F ) + 2 . Also recall that c ≡ c c (mod 2), hence c ( F ) is always even.2.4. The spectrum of a rank 2 reflexive sheaf.
For a normalized rank tworeflexive sheaf F on P , such that h ( F ( − { k , . . . , k c ( F ) } , called the spectrum of F , that encodes partial informationon the cohomology of F . It was first defined by Barth and Elencwajg for locallyfree sheaves and later extended for reflexive sheaves by Hartshorne; we will use [16,Section 7] as reference. Remark 4.
Let F be a µ -semistable rank 2 reflexive sheaf on P with c ( F ) = 0and set c ( F ) = n and c ( F ) = 2 l . If { k , . . . , k n } is the spectrum of F then: h ( F ( p )) = n X i =1 h ( P , O P ( k i + p + 1)) for each p ≤ − h ( F ( p )) = n X i =1 h ( P , O P ( k i + p + 1)) for each p ≥ − . (11)The possible spectra of a sheaf with given Chern classes can be determined via thefollowing properties, see [16, Propositions 7.2 and 7.3 and Theorem 7.5].(1) P ni =1 k i = − l ;(2) if k > , . . . , k ; if, in addition, F is µ -stable then 0 must also occur;(3) if k < k, k + 1 , . . . , − F is µ -stable then either 0 occurs or − F is locally free, then the spectrum is symmetric, i.e., if k i occurs, thenso does − k i .As we have seen in Lemma 1, being the tangent sheaf of a codimension onedistribution imposes certain restrictions on h ( F ( p )) and h ( F ( p )), which can inturn be used to rule out several possible spectra for sheaves with given Chern classes.Studying the spectrum of tangent sheaves also leads to a precise knowledge of theassociated singular schemes, and vice-versa, allowing us to collect the informationdisplayed in Table 1. Here is an immediate application. Lemma 5.
Let D be a codimension one distribution on P of degree and let { k , . . . , k n } be the spectrum of T D . Then k i ≥ − for every i and − occurs atmost once. Moreover, occurs in the spectrum if and only if Sing( D ) is containedin a quadric surface.Proof. From Lemma 1 we get h ( T D ) ≤
1; then from (11) we must have k i ≥ − i and − h ( T D ( − = 0 if and only if Sing( D ) is contained in a surface ofdegree at most 2. On the other hand, h ( T D ( − = 0 if and only if 1 occurs in thespectrum. (cid:3) Below, we will compute all possible spectra for tangent sheaves of distributionsof degree 2, and we check that 1 never appears. The second part of Lemma 5 thenguarantees that Sing( D ) is contained in a quadric surface, as stated in the MainTheorem. Basic constructions
In the remainder of this paper we will provide the existence or non-existenceof distributions according to the given Chern classes. To fulfill our purpose, wewill establish in this section the key results needed for the construction of exam-ples of codimension one distributions of degree 2; and some results to discard theimpossible cases.3.1.
Distributions from foliations by curves.
Let G be a degree k foliation bycurves as in display (7) and let σ ∈ H ( N ∗ G ( l )) be a section, for some l ∈ Z . Weassume that X := ( σ ) , the vanishing locus of σ , has codimension 2. We will use σ to produce a codimension one distribution.Now consider the composed monomorphism ρ : O P ( − l ) σ ֒ → N ∗ G ֒ → Ω P and let F := coker ρ . Using the Snake Lemma, we can see that F fits into the followingexact sequence(12) 0 −→ I X ( l − k − −→ F −→ I W ( k − −→ X and W have codimension at least two, F is torsion free.Therefore, we have a codimension one distribution of degree l − D : 0 −→ F ∗ −→ T P −→ I Z ( l ) −→ , called the codimension one distribution induced by the pair ( G , σ ). Note that E xt ( F, O P ) ≃ O Z ( l ); it is also clear that X ⊂ Z .Dualizing the exact sequence in display (12) yields the long exact sequence(13) 0 −→ O P (1 − k ) −→ F ∗ −→ O P ( k + 3 − l ) η −→ ω Y (5 − k ) −→−→ O Z ( l ) −→ ω X ( k − l + 7) −→ ω R −→ . where Y := Sing ( G ) and R := Sing ( G ), noting that E xt ( I W ( k − , O P ) ≃ ω Y (5 − k ) . In addition, the morphism η in display (13) can also be regarded as a section η ∈ H ( ω Y ( l − k + 2)). Proposition 6.
Let G be a foliation by curves with Y = Sing ( G ) , and take a nonzero section σ ∈ H ( N ∗ G ( l )) such that ( σ ) is a curve; let D be the codimensionone distribution induced by ( G , σ ) , and let η ∈ H ( ω Y ( l − k + 2)) be as above. If dim coker η = 0 , then Sing ( D ) = ( σ ) , Sing ( D ) = ( η ) ,and −→ O P (1 − k ) −→ T D −→ I W ( k + 3 − l ) −→ . Proof.
We will use the same notation as above. Breaking the exact sequence indisplay (13) into short exact sequences, we extract the following two:0 −→ coker η −→ O Z ( l ) −→ O Z ′ ( l ) −→ −→ O Z ′ ( l ) −→ ω X ( k − l + 7) −→ ω R −→ Z ′ ⊂ X ⊂ Z . As X is Cohen-Macaulay and R has dimension zero, (15)implies that Z ′ = X and it follows from (14) thatcoker η = I X/Z ( l ) . ODIMENSION ONE DISTRIBUTIONS OF DEGREE 2 ON P If dim coker η = 0 then X = Sing ( D ) which implies that ( η ) = Sing ( D ). Since T D ≃ F ∗ , the exact sequence in the statement comes from the dualization of theexact sequence in display (12), using the hypothesis dim coker η = 0. (cid:3) Remark 7.
We make two observations on the previous argument.(1) η = 0 if and only if F ∗ = O P (1 − k ) ⊕ O P ( k + 3 − l ), so that l − k ≥ η = 0 and Y := Sing ( G ) is irreducible and reduced, then the hypothesisdim coker η = 0 is automatically satisfied.Also note that the codimension one distribution D constructed above, and forwhich dim coker η = 0 holds, has the following invariants: c ( T D ) = 4 − l ; c ( T D ) = l ( k −
1) + 6 − c ( N ∗ G ); c ( T D ) = c ( N ∗ G ) + (1 − k − l ) c ( N ∗ G ) + ( k + 2 k + 3) l − . (16)They can be calculated from (3) using that Sing ( D ) = ( σ ) . Since we are interestedin degree two distributions, we specialize the above formulas to l = 4: c ( T D ) = 4 k + 2 − c ( N ∗ G ); c ( T D ) = c ( N ∗ G ) − ( k + 3) c ( N ∗ G ) + 4 k + 8 k + 8 . (17)3.2. Distributions from syzygies.
Another way to construct codimension onedistributions, maybe the most traditional one, is to give an explicit twisted 1-formthat defines it. We will see how to construct 1-forms with specified vanishing locus,so that the distribution has the desired invariants. We will proceed by studyingsome homogeneous ideals as in [11, Section 4].Recall that a degree d codimension one distribution D may be given by a homo-geneous 1-form ω D = A dx + A dy + A dz + A dw . The singular scheme Z of D is defined by the saturated ideal I Z = I sat = M l ∈ Z H ( I Z ( l )) , where I = ( A , A , A , A ) ⊂ C [ x, y, z, w ]. As the A j have degree d + 1, restrictionsare imposed on the possible subschemes of P that can fit into the singular loci ofdistributions. Lemma 8.
Let C ⊂ P be a subscheme with saturated ideal I C and let D ⊃ C be the subscheme defined by ( I C ) ≤ d +1 , the ideal generated by the elements of I C ofdegree ≤ d + 1 . If D is a distribution as above and C ⊂ Z , then D ⊂ Z .Proof. Since C ⊂ Z we have I Z ⊂ I C ; in particular ( A , A , A , A ) ⊂ I C . Infact, ( A , A , A , A ) ⊂ ( I C ) ≤ d +1 and the result follows from the inclusion of thesaturations. (cid:3) A direct consequence of this lemma is a bound on the genus of double and triplelines that can be included in the singular scheme of a distribution.
Corollary 9.
Let D be a degree d distribution on P and let C be a double lineof genus g ≤ . If g < − d then D is singular along the second infinitesimalneighborhood of C red , i.e., the curve defined by ( I C red ) . Proof.
Up to a linear change of coordinates, C red = { x = y = 0 } and the ideal of C is I C = ( x , xy, y , xp + yq ) where p and q have degree − g , see [21]. Note thatwe may assume that p and q depend only on z and w .If g < − d , then deg( xp + yq ) ≥ d + 2 and Lemma 8 implies that I Z ⊂ ( x, y ) ,hence Z contains the curve given by ( x , xy, y ) = ( x, y ) . (cid:3) The second infinitesimal neighborhood of a line has degree three. This will beuseful to bound c ( T D ) in some cases. Next we describe the restriction on triplelines.Fix C red = { x = y = 0 } . According to [21], a triple structure on C red is describedby a pair of numbers ( a, b ) and falls in one of two cases:(1) a = − C is a curve of genus p a ( C ) = 1 − b given by I C = ( x , xy, y , xq − y p ) , where deg p + 1 = deg q = b .(2) If a ≥ C is a curve of genus p a ( C ) = − − a − b given by I C = ( x, y ) ∩ ( x ( xf − yg ) , y ( xf − yg ) , p ( xg − yf ) − rx − sxy − ty ) , where deg f = deg g = a + 1 and deg p = b . Corollary 10.
Let D be a degree d distribution on P and let C be a triple lineof type ( a, b ) . If either a = − and b ≥ d ; or a ≥ and a + b ≥ d then Sing( D ) contains a multiple structure on C red of degree at least .Proof. If a = − b ≥ d then Sing( D ) must contain the curve given by the ideal( x , xy, y ), which has degree 4.If a ≥ a + b ≥ d then Sing( D ) must contain the curve given by the ideal( x, y ) ∩ ( x ( xf − yg ) , y ( xf − yg )), which has degree at least 4. (cid:3) Finally we recall the following result, [11, Proposition 4.4], that exhibits a cor-respondence between the 1-forms a closed subscheme and the linear syzygies of itshomogeneous ideal.
Proposition 11 ([11]) . Let Z ⊂ P n be a closed subscheme and let d ≥ be aninteger. Suppose that Z is not contained in a hypersurface of degree less than orequal to d . Then there exists a linear isomorphism between the spaces of degree d +2 twisted -forms singular at Z and linear first syzygies of the homogeneous ideal I Z . To produce the 1-forms we fix a minimal generating set { F , . . . , F r } for I Z andconsider a linear first syzygy { G , . . . , G r } . Then we define ω = F dG + · · · + F r dG r ;it is clear that ω is homogeneous and descends to P n since F G + · · · + F r G r = 0.Note that, for general ideals, one may also use higher degree syzygies and replace dG j by G j dG j in the definition of ω . But then ω may be non-homogeneous andthe syzygies must be chosen wisely.3.3. Morphisms to the tangent bundle.
Let F be a µ -stable rank 2 reflexivesheaf on P , with c ( F ) = 0. Assuming that Hom( F, T P ) = 0, we would like tofind conditions that guarantee the existence of a monomorphism F ֒ → T P withtorsion free cokernel. This is the case, for instance, if F (1) is globally generated,see [3, Corollary A.4]. ODIMENSION ONE DISTRIBUTIONS OF DEGREE 2 ON P If φ : F → T P is not injective, then im φ must be a torsion-free sheaf, whichimplies that ker φ is reflexive of rank 1, therefore ker φ ≃ O P ( − k ) and im φ ≃ I C ( k )for some curve C . The stability of F forces k ≥ φ ֒ → T P induces a nontrivial section in H (T P ( − k )), thus k = 1. Then φ decomposes as F s ∨ −→ O P (1) ν −→ T P where s ∈ H ( F (1)) and ν ∈ H (T P ( − Lemma 12.
Let F be a µ -stable rank 2 reflexive sheaf on P with c ( F ) = 0 . Thenthere exists an injective morphism φ : F ֒ → T P if and only if either (1) h ( F (1)) = 1 ; (2) h ( F (1)) = 1 and hom( F, T P ) > .Moreover, if h ( F (1)) = 0 then every morphism in Hom( F, T P ) is injective;Proof. Consider the map ξ : H ( F (1)) ⊗ H (T P ( − → Hom( F, T P ) defined by ξ ( s ⊗ ν ) = s ∨ ν ; clearly ξ is injective. Let Σ ⊂ H ( F (1)) ⊗ H (T P ( − s ⊗ ν . From our previousdiscussion, the locus of maps in Hom( F, T P ) that are not injective is precisely ξ (Σ).If h ( F (1)) = 0 then Σ = ∅ and we are done. If h ( F (1)) = 1 we have theother extremal case Σ = H ( F (1)) ⊗ H (T P ( − F, T P ) > dim Σ = 4. Finally, if h ( F (1)) ≥ F, T P ) ≥ h ( F (1)) > h ( F (1)) + 3 = dim Σ . Thus there exist injective morphisms. (cid:3)
Now we analyse the morphisms that are injective but have torsion in their cok-ernels. Let φ : F ֒ → T P be a monomorphism whose cokernel is not torsion free,and let(18) P := ker { coker φ −→ (coker φ ) ∨∨ } be the maximal torsion subsheaf of coker φ . The quotient (coker φ ) /P is a torsionfree sheaf of rank 1, so it must be of the form I Z ( d ′ + 2) for 1-dimensional scheme Z and some d ′ ≥
0. We end up with the following commutative diagram(19) 0 (cid:15) (cid:15) (cid:15) (cid:15) P (cid:15) (cid:15) / / F (cid:15) (cid:15) φ / / T P / / K / / (cid:15) (cid:15) / / F ′ φ ′ / / (cid:15) (cid:15) T P / / I Z ( d ′ + 2) / / (cid:15) (cid:15) P (cid:15) (cid:15) K := coker φ and F ′ := ker { T P ։ K ։ K/P } .The second row of the previous diagram defines a codimension one distribution D of degree d ′ , called the saturation of the monomorphism φ : F → T P ; note that T D = F ′ . Lemma 13.
Let F be a rank 2 reflexive sheaf on P , and let φ : F → T P be amonomorphism whose cokernel is not torsion free. If D is the saturation of φ , then: (1) The sheaf P defined in display (18) has pure dimension 2. (2) deg( D ) < − c ( F ) ; (3) hom( T D , T P ) ≤ hom( F, T P ) .Proof. For the first item it suffices to prove that E xt ( P, O P ) = 0 and E xt ( P, O P )has dimension 0. Consider the diagram in display (19). Dualizing the bottom rowand using that F ′ = T D is reflexive we get E xt ( I Z ( d ′ + 2) , O P ) = 0. Then,dualizing the rightmost column we get E xt ( K, O P ) −→ E xt ( P, O P ) −→ −→ E xt ( K, O P ) −→ E xt ( P, O P ) −→ . On the other hand, we dualize the top row to get dim E xt ( K, O P ) = 0 and E xt ( K, O P ) = 0; using this information in the sequences above we show that P is pure of dimension two. Therefore c ( P ) > D ) = 2 − c ( T D ) = 2 − ( c ( F ) + c ( P )) < − c ( F ) . Finally, we apply the functor Hom( · , T P ) to the leftmost column and get0 −→ Hom( T D , T P ) −→ Hom( F, T P ) , which proves the third item. (cid:3) In general, it is hard to describe the possible saturations and one may rely onad hoc methods. In our case, it will be useful to know the the dimensions ofHom( T D , T P ) for a distribution of degree 0 or 1. These dimensions were obtainedin [3] and we summarize them in Table 3. ODIMENSION ONE DISTRIBUTIONS OF DEGREE 2 ON P deg( D ) ( c ( T D ) , c ( T D )) hom( T D , T P )0 (2,0) 1(1,0) 81 (3,5) 1(2,2) 5(1,1) 12(0,0) 19 Table 3.
Possible values of hom( T D , T P ) for all possible D ofdegree ≤
1. 4.
Distributions with c ( T D ) ≤ c ( T D ) = − , c ( T D ) = − T D = O P (1) ⊕ O P ( − φ : T D ֒ → T P yieldsvector fields ν := φ (1 ,
0) and ν := φ (0 , D . Up to a linear change in coordinates, we may assume that ν = ∂∂w and ν ( w ) = 0; then [ ν , ν ] = 0, i.e. D is integrable, if and only if ν does notdepend on w . These are the so called linear pullback foliations S (3). Of course,one can easily produce a non-integrable distribution setting ν = ∂∂w and chossing ν depending on w .As T D splits we can use the surjection O P (1) ⊕ ։ T P to construct a freeresolution for C = Sing( D ).(20) 0 −→ O P ( − ⊕ O P ( − ⊕ O P ( − −→ O P ( − ⊕ −→ I C −→ . In particular, C is arithmetically Cohen-Macaulay (ACM). This is true whenever T D splits and in a more general setting, see [9, Theorem 1].If c ( T D ) ≥
0, we have µ -semistability, recall Theorem 3. In particular, for c ( T D ) = 0, this implies that c ( T D ) = 0, hence T D = O P ⊕ O P . In this case, weget from φ : T D ֒ → T P two linear vector fields ν and ν ′ . Clearly, the choice of anytwo (linearly independent) vector fields defines a distribution. The integrable cases,i.e. [ ν, ν ′ ] = aν + bν ′ with a, b ∈ C , come from representations of two-dimensionalcomplex Lie algebras. Among these we find logarithmic foliations L (1 , , , E (3)that come from representations of the affine Lie algebra aff ( C ).As above T D splits and we can build a free resolution(21) 0 −→ O P ( − ⊕ −→ O P ( − ⊕ −→ I C −→ C = Sing( D ).We summarize the previous discussion in the following proposition. Proposition 14.
Let D be a codimension one distribution on P of degree . If c ( T D ) ≤ then one of the following holds. (1) T D = O P (1) ⊕ O P ( − and Sing( D ) is an ACM curve as in (20) ; (2) T D = O P ⊕ O P and Sing( D ) is an ACM curve as in (21) . For c ( T D ) = 1 we have two possibilities: either c ( T D ) = 0 and T D is µ -stable;or c ( T D ) = 2 and T D must be strictly µ -semistable. Indeed, this holds for anyreflexive sheaf with such Chern classes, see [8, Lemma 2.1]. Moreover, if c ( T D ) = 0then T D is a null correlation bundle, see [23, Lemma 4.3.2]. We will show, in the next two results, that distributions with these invariants do exist. We recall that acurve is arithmetically Buchsbaum if its Rao module H ∗ ( I C ) is annihilated by theirrelevant ideal, see [1]; in particular, this is true if H ∗ ( I C ) is supported in onlyone degree. Proposition 15.
Let N be a null correlation bundle on P . Then there exists a dis-tribution D such that T D = N . Moreover, its singular scheme is an arithmeticallyBuchsbaum curve C of degree and arithmetic genus .Proof. As N (1) is a quotient of Ω P (2), it is globally generated. We then apply[3, Corollary A.4] to show that there exists a distribution D with tangent sheaf T D = N . Note that from (4) C = Sing( D ) is a curve of degree 5 and genus 1. Fromthe sequence 0 −→ N −→ T P −→ I C (4) −→ l ∈ Z ,0 −→ H ( I C (4 + l )) −→ H ( N ( l )) −→ H (T P ( l ));on the other hand, H ( N ( l )) = H (T P ( l − h ( I C ( l )) = 0 for l = 1and h ( I C (1)) = 1 and C is arithmetically Buchsbaum. (cid:3) We note that the spectrum of a null correlation bundle is { } ; it follows fromthe fact that h ( N ( l )) = 0 for l = − h ( N ( − Proposition 16.
Let F be µ -semistable rank 2 reflexive sheaf on P with Chernclasses c ( F ) = 0 , c ( F ) = 1 and c ( F ) = 2 . Then F can be realized as the tangentsheaf of a codimension one distribution of degree 2. Moreover, Sing ( D ) is an ACMcurve of degree 5 and genus 2 contained in a quadric.Proof. Let F be as in the statement. Consulting [8, Table 2.3.1] we check that: h ( F ( p )) = 0 for every p ∈ Z ; h ( F ( p )) = 0 for p ≥ −
3; and h ( F ( p )) = , p ≥ − , p = − , p ≤ − . Then the Castelnuovo–Mumford criterion implies that F (1) is globally generated.Owing to [3, Corollary A.4], there exists a codimension one distribution D of degree2 whose tangent sheaf is precisely F .For the second claim, let C := Sing ( D ); it follows from the equations in display(4) that C has degree 5 and arithmetic genus 2. We must show that h ( I C ( p )) = 0for every p , so that C is ACM.From the definition of C in the sequence in display (2), we have that(22) H ( I Z ( l )) → H ( I C ( l )) → H ( U ) → H ( I Z ( l )) → H ( I C ( l )) → h ( U ) = c ( F ) = 2. Also recall that h ( I Z ( l )) = h ( F ( l − l = 0 and h ( I Z ) ≤ h ( F ( − l ≥
3, we see that h ( I C ( l )) ≤ h ( I Z ( l )) = h ( F ( l − l ≤
1, we have h ( I C ( l )) = 0, since C cannot be planar. On the otherhand, h ( I Z ( l )) ≤ h ( F ( l − H ( U ) ≃ H ( I Z ( l )) hence h ( I C ( l )) = 0. ODIMENSION ONE DISTRIBUTIONS OF DEGREE 2 ON P For l = 2, we have h ( I Z (2)) = h ( F ( − h ( F ( − h ( I Z (2)) = 0. From (22) we only need to show that h ( I C (2)) = 1; the onlyother possibility is h ( I C (2)) = 2.Suppose that h ( I C (2)) = 2. Since deg C = 5, then H ( I C (2)) is spanned by f f and f f for some linear polynomials f, f , f ∈ H ( O P (1)). First we claim thatthis would imply that C contains a planar subcurve of degree 4 which is impossible,due to Lemma 8. Therefore h ( I C (2)) = 1 and C is ACM which concludes theproof.Now we prove the claim. We have two cases:(1) { f, f , f } is linearly independent then C is the union of a plane quartic C ′ and the line L = { f = f = 0 } ;(2) { f, f , f } is linearly dependent, so we may assume f = f ; then C is acurve in the double plane defined by f .In the second case, owing to [18, Proposition 2.1], we have0 −→ I Y ( − H ) −→ I C −→ I V/H ( − P ) −→ , where V ⊂ Y ⊂ P ⊂ H = { f = 0 } , for curves Y and P and a 0-dimensionalsubscheme V . Moreover, deg C = deg Y +deg P and P is the largest curve containedin C ∩ H . Also, Y is the residual intersection of C and H hence it must be the line { f = f = 0 } . Then P is a plane quartic contained in C . (cid:3) Among these distributions we find logarithmic foliations in L (1 , , µ -semistable sheaf F with c ( F ) = 0, c ( F ) = 1 and c ( F ) = 2 is {− } .5. Distributions with c ( T D ) = 2If D is a degree two distribution with c ( T D ) = 2 then the bound in (10) implies c ( T D ) = 0 , T D is µ -stable except for thecase c ( T D ) = 4 where it can be strictly µ -semistable.The existence of degree 2 codimension one distributions D such that c ( T D ) = 2and c ( T D ) = 0 or 4 was established in [3, Theorem 9.5] and [3, Section 11.5],respectively. We describe them here for the sake of completeness.If c ( T D ) = 0 then T D is an instanton bundle of charge two. Indeed, any suchbundle satisfies h ( T D ( − D )is a curve C of degree 4 and genus −
1. The Hilbert scheme H (4 , −
1) has threeirreducible components, described in [22, Theorem 6.2]. Since T D is an instanton,we know that h ( I C (2)) = h ( T D ( − C can only be the disjoint unionof a line and a twisted cubic (or some degeneration of that). Example 17.
Consider the curve C defined by I C = ( z − yw, yz − xw, y − xz ) ∩ ( x, w ) . Using its syzygies we could find the 1-form ω =( xz − xyw + y w − xzw + yzw + z w − xw − yw ) dx + ( − xyz + xz + x w − xyw + yzw + z w − xw − yw ) dy + ( xy − x z − xyz + x w − y w + xzw − yzw + xw ) dz + ( − xyz − xz + x w + 2 xyw + y w − xzw ) dw which is singular precisely at C . Thus T D is an instanton bundle of charge two.We note that [15, Lemma 9.4] says that h ( T D ( − { , } .If c ( T D ) = 4 then Sing( D ) is composed by 4 points and a curve C of degree4 and genus 1. Any such curve is (some degeneration of) a elliptic quartic curve,see [22]. In particular, h ( I C (2)) = 2 and there exists a unique pencil of quadricscontaining C . If C is general, this pencil defines a degree two rational foliation in R (2 , ( D ) is composed by the vertices of the cones appearing in thispencil. Moreover, T D is stable in this case. Conversely, any stable sheaf with theseChern classes can be realized as the tangent sheaf of a distribution that, in general,is not integrable. Proposition 18.
Let F be a µ -stable rank two reflexive sheaf on P with ( c ( F ) , c ( F ) , c ( F )) = (0 , , . Then there exists a distribution D such that T D = F . Moreover, Sing ( D ) is a, possibly degenerated, elliptic quartic.Proof. Owing to [8, Table 2.12.2] and Castelnuovo–Mumford criterion, F (1) is glob-ally generated. Then, due to [3, Corollary A.4], there exists a codimension onedistribution of degree 2 whose tangent sheaf is precisely F . The assertion aboutthe singular scheme follows from our previous discussion. (cid:3) We note that we can also find distributions with T D strictly µ -semistable. Example 19.
Consider the curve C given by the ideal I C = ( w − ( y − x ) z, z − xy )and the double point I p = ( x, y , w ). Using the syzigies we find the 1-form ω =( − xy xz − xyw + xzw − yzw + z w + zw w dx + ( x y − xz xzw − yzw + w dy + ( − x z + xyz − xw dz + ( x y − x z + y z − xz − xw − yw dw singular at the prescribed singular scheme and also at the double point given by I p = ( y + w, x − z − w, ( z + w ) ). Computing the kernel of ω we see that T D hasa free resolution0 −→ O P ( − −→ O P ( − ⊕ −→ O P ⊕ O P ( − ⊕ −→ T D −→ . In particular, h ( T D ) = 1 and T D is strictly µ -semistable. It is also not hard tofind a degree one foliation by curves tangent to this distribution.We note that, from Remark 4, the only possible spectrum for a µ -semistablesheaf F such that c ( F ) = 0, c ( F ) = 2 and c ( f ) = 4 is {− , − } .Now consider the case c ( T D ) = 2. Then the singular scheme is composed by 2points and a curve of degree 4 and genus 0. The Hilbert scheme H (4 ,
0) has twoirreducible components, whose generic points are given by either a rational quartic
ODIMENSION ONE DISTRIBUTIONS OF DEGREE 2 ON P curve, or a disjoint union of a plane cubic and a line. We now argue that the secondcase does not occur.The homogeneous ideal of a disjoint union of a plane cubic and a line is a product I C = ( H, F )( H , H ) where H, H and H have degree 1 and F has degree 3. Then( I C ) ≤ = ( HH , HH ) and, due to Lemma 8, Sing( D ) would have codimensionone.For C a rational quartic curve there exist distributions; we will now provide anexample. Example 20.
Fix the rational quartic curve given by I C = ( yz − xw, z − yw , xz − y w, y − x z ) , and the point (1 : 0 : 0 : 1). Using the syzygies we find ω = (2 xz − z − y w + yw ) dx + ( − y z + 2 xz + 2 yz + 2 xyw − y w − xzw ) dy + (2 y − x z − xyz − y z + xz + 2 x w + 2 xyw − y w + 2 yzw − xw ) dz + (2 y − x z + y z − yz − xyw + 2 xzw ) dw which is singular at the prescribed scheme and at the point (1 : 1 : 0 : 2). Therefore T D = ker ω has the desired Chern classes.Also, from Remark 4, we see that the only possible spectrum for a µ -semistablesheaf F such that c ( F ) = 0, c ( F ) = 2 and c ( f ) = 2 is {− , } .6. Distributions with c ( T D ) = 3In this case, we know that T D is µ -stable, so that c ( T D ) ∈ { , , , , } . Westart by recalling the case c ( T D ) = 8, which is discussed in [3, Section 11.6]. Nextwe will describe the cases c ( T D ) < F such that c ( F ) = 0, c ( F ) = 3 and c ( F ) = 8 arises as the tangent sheaf of degree two distribution. This is due to F (1)being globally generated, see [8, Lemma 3.8], and [3, Corollary A.4]. The singularscheme is composed by 8 points and a curve of degree 3 and genus 1, which canonly be a plane cubic, see [21, Proposition 3.1]. Furthermore, {− , − , − } is theonly possible spectrum for such sheaves.Among these distributions we find rational foliations in R (1 , C lying on a plane H , there exists a unique cubic surface that contains C but not H ; thus we have aunique rational foliation for a general C . The 0-dimensional part of the singularscheme is the vertex of the unique cone in this pencil.6.1. Case ≤ c ( T D ) ≤ . Consider R (0 , , l ) the moduli space of stable ranktwo reflexive sheaves F on P satisfying c ( F ) = 0, c ( F ) = 3 and c ( F ) = 2 l . Webegin by noting that when l >
0, the moduli space R (0 , , l ) is irreducible andcontains a nonempty open subset R (0 , , l ) = (cid:8) [ F ] | h ( F (1)) = 0 (cid:9) , see [8]. It follows that, for each [ F ] ∈ R (0 , , l ), a representative F admits amonomorphism to T P . Proposition 21.
Let l ∈ { , , } . Then every stable rank 2 reflexive sheaf F withChern classes ( c ( F ) , c ( F ) , c ( F )) = (0 , , l ) satisfying h ( F (1)) = 0 admits amonomorphism F ֒ → T P .Proof. Applying Hom( F, · ) to the Euler sequence and using that h ( F (1)) = 0 weget 0 −→ H ( F (1)) ⊕ −→ Hom( F, T P ) −→ H ( F ) −→ , so that hom( F, T P ) = 4 h ( F (1)) + h ( F ). Consulting the cohomology tables in[8] we get that h ( F (1)) = l − h ( F ) = 3 − l . Therefore, the result followsfrom a direct application of Lemma 12. (cid:3) It is not clear if every such sheaf can be realized as the tangent sheaf of a distribu-tion; i.e. it is not clear if for any given [ F ] ∈ R (0 , , l ) there exist monomorphisms F ֒ → T P with torsion free cokernel. For the moment, we can show that this is truegenerically. Indeed, we only need to provide an example to show that an every sheafin an open subset of R (0 , , l ) can be realized as tangent sheaves of distributions. Example 22. If c ( T D ) = 6, any such distribution must be singular at a curve C of degree 3 and genus 0, plus six points. According to [25, Lemma 1], any suchcurve is a, possibly degenerated, twisted cubic. Then let C be given by I C = ( − y + xz, − yz + xw, − z + yw )and add the points (0 : 1 : 0 : 0), (0 : 0 : 1 : 0) and (1 : 1 : − ω = ( − y z + yz + xyw − y w + yzw − z w − xw + yw ) dx + ( xyz + yz − x w − xzw + z w − yw ) dy + ( − xyz − y z + x w + xyw + 2 y w − xzw + 4 yzw − xw ) dz + ( xy − x z − xyz − y z + 3 xz − yz + x w − xyw + y w + 4 xzw ) dw which is singular at the prescribed scheme and also at the points (3 : 0 : 1 : 0),(1 : − −
7) and (9 : −
31 : 24 : − T D = ker ω has the desiredChern classes. Example 23.
For c ( T D ) = 4 we must have C = Sing ( D ) a degree 3 curve ofgenus −
1. Then C must be extremal, owing to [20, Theorem 4.1]; this means that C must contain a conic, possibly degenerated. So let us fix C a disjoint union of aconic and a line, given by I C = ( x, y ) ∩ ( w, z − xy );then we choose the points (1 : 1 : 1 : 1), (4 : 1 : 2 : 1) and (1 : 4 : 2 : 1). We get a1-form ω = ( − xy + yz − xyw + 45 y w + xzw − yzw + 70 xw + 11 yw ) dx + ( x y − xz + 45 x w − xyw − xzw + 53 yzw + 115 xw − yw ) dy + ( − x w + 216 xyw − y w − xw + 49 yw ) dz + ( − x w − xyw + 34 y w + 211 xzw − yzw ) dw that also vanishes at the point (289 : 1225 : −
595 : − T D = ker ω has the desired Chern classes. ODIMENSION ONE DISTRIBUTIONS OF DEGREE 2 ON P Example 24. If c ( T D ) = 2 then C = Sing ( D ) is a degree 3 curve of genus − C is composed by three skew lines, possibly degenerated;(2) C is the union of a double line of genus − D ) cannot contain a double line of genus − C falls in the first case. So fix C given by I C = ( x, y ) ∩ ( z, w ) ∩ ( x − z, y − w )and add the point P = (1 : − ω = ( xyz + y z + yz + xyw − y w − xzw − yzw + 2 xw + yw ) dx + ( − x z − xyz + xz + yz − x w + xyw + xzw − yzw ) dy + ( − xyz − y z + x w + xyw − y w + xzw + yw ) dz + ( x z + 2 xyz + 2 y z − xz − x w − xyw − yzw ) dw that is singular at C ∪ { P, Q } where Q = (1 : − T D = ker ω has the desired Chern classes.Now we will show that the special sheaves in R (0 , , l ) \ R (0 , , l ) cannot berealized as tangent sheaves of distributions. Proposition 25.
Fix an integer l ∈ { , , } and let D be a distribution such that [ T D ] ∈ R (0 , , l ) then h ( T D (1)) = 0 .Proof. Recall that Lemma 1 implies h ( T D (1)) = h ( ω C (1)) = h ( I C ( − C = Sing ( D ), for any distribution of degree 2. Then we only need to show thatone of these other cohomologies vanish.If l = 3 then C is a, possibly degenerated, twisted cubic. In particular, we havea resolution 0 −→ O P ( − ⊕ −→ O P ( − ⊕ −→ I C −→ , and it follows that h ( I C ( − l = 2 then C is extremal. As deg C = 3 and p a ( C ) = −
1, we have that h ( I C ( l )) = 0 for l ≤ −
1, see [20].If l = 1, we know from Example 24 that C is the disjoint union of threelines, possibly degenerated. From the proof of [21, Proposition 3.4] we have that h ( I C ( − (cid:3) Corollary 26.
Let l ∈ { , , } . If [ F ] ∈ R (0 , , l ) \ R (0 , , l ) then there doesnot exist distribution with F as the tangent sheaf. Despite that we do not have distributions, many of these special sheaves admitmonomorphisms to T P . For instance, using [8, Table 3.5.1] we see that for any[ F ] ∈ R (0 , , \ R (0 , ,
6) we get h ( F (1)) = 3 hence, due to Lemma 12, there ex-ists a monomorphism F ֒ → T P . Therefore the cokernel of any such monomorphismhas torsion.6.2. Case c ( T D ) = 0 . Let us finally consider codimension one distributions suchthat T D is a locally free sheaf with c ( T D ) = 3; hence C = Sing( D ) is a curve ofdegree 3 and genus −
3. From [21, Proposition 3.5] we know that either(1) C is the union of a double line of genus − C is a disjoint union of a double line of genus − Corollary 9 implies that the first case cannot occur, so C belongs to the second one.Owing to [21, Proposition 3.3], in fact to its proof, any such curve C admits a freeresolution0 −→ O P ( − ⊕ −→ O P ( − ⊕ −→ O P ( − ⊕ −→ I C −→ . and satisfies h ( I C ( l )) = 2 h ( P , O P ( l ))+ h ( P , O P ( l +1)). With this informationone can compute h i ( I C ( l )) for every i and l . Therefore, using Lemma 1 and theexact sequence 0 −→ T D −→ T P −→ I C (4) −→ T D as in Table 4. Note that the complementof this table can be computed via Serre duality. In particular, T D is described bythe next result. − − l ≥ h ( T D ( l )) 0 0 0 0 l + 2 l + l − h ( T D ( l )) 0 3 4 1 0 h ( T D ( l )) 0 0 0 0 0 h ( T D ( l )) 0 0 0 0 0 Table 4.
Cohomology table for T D with c ( T D ) = 3 and c ( T D ) =0. Theorem 27.
Let E be stable rank two vector bundle on P with c ( E ) = 0 and c ( E ) = 3 . Then E is the tangent sheaf of a distribution if and only if it is a genericinstanton bundle E of charge with a unique jumping line of order 3. Recall that a line Y ⊂ P is said to be a jumping line of order k for E if E | Y = O Y ( − k ) ⊕ O Y ( k ). According to [13, Section 1], generic instanton bundlesof charge 3 admit at most one jumping line of order 3. Proof.
First suppose that there exists D such that E = T D . From Table 4 we havethat E is an instanton bundle with natural cohomology, i.e. for every l at most one h ( E ( l )) does not vanish, see [17]. Due to [13, Lemme 1.1], either E has a uniquejumping line of order three or the map ξ : H ( E ) ⊗ H ( O P (1)) → H ( E (1)), givenby multiplication, is a nondegenerate pairing. We will prove that the later cannotoccur.Tensorizing the Euler sequence with E we get0 −→ H ( E ⊗ T P ) −→ H ( E ) ζ −→ H ( E (1)) ⊗ H ( O P (1)) ∨ −→ H ( E ⊗ T P )and it follows that ζ is an isomorphism if and only if ξ is nondegenerate. Byhypothesis, h ( E ⊗ T P ) = hom( E, T P ) = 0 and ζ cannot be an isomorphism,whence E has a unique jumping line of order 3.Now we prove the converse. Suppose that E is a generic instanton bundle ofcharge 3, its cohomology being given by Table 4, and suppose that E has a uniquejumping line of order 3. From our previous argument, there exists φ : E → T P which is injective, due to Lemma 12. In fact, owing to the proof of [13, Lemme 1.1], ODIMENSION ONE DISTRIBUTIONS OF DEGREE 2 ON P the map ζ has rank two, hence hom( E, T P ) = 2. To conclude we need to provethat coker φ is a torsion free sheaf.Suppose, by contradiction, that coker φ has torsion and let F be the saturateddistribution; then(23) 0 −→ E β −→ T F −→ P −→ . According to Lemma 13, P is a pure sheaf of dimension two, deg( F ) ≤ T F , T P ) ≤ hom( E, T P ) = 2. From Table 3 we see that F must fall in oneof two cases that we will analyze now.First, deg( F ) = 1 and c ( T F ) = 3 and c ( T F ) = 5. In this case, Sing( F )is a non-planar zero-dimensional subscheme of length 5; and P is supported on a(reduced) plane. Dualizing the sequence (23) we get E xt ( P, O P ) −→ E xt ( T F , O P ) −→ . Thus, Sing( F ) must be contained in Supp( P ), which is absurd.The second case is: deg( F ) = 0 and c ( T F ) = 2 and c ( T F ) = 0. In this case, T F = N (1), where N is a null correlation bundle.Consider G (1 , ⊂ P the Grassmannian of lines in P and let T ⊂ G (1 ,
3) bethe curve of jumping lines of order at least 2 for E . Let H ⊂ P a hyperplane suchthat H ∩ G (1 ,
3) is the set of jumping lines for N . Also consider X ⊂ G (1 ,
3) theFano variety of lines in Supp P .Owing to [13, 2.2], the moduli space of instanton bundles of charge 3 with ajumping line of order 3 is birational to the Hilbert scheme of rational quintic curvesin ˇ P . Assume that E is general so that it corresponds to a smooth rational quinticΓ ⊂ ˇ P not contained in a quadric. Under this assumption Γ has a unique 4-secantline which is precisely ˇ L , for L the jumping line of order 3 of E . According to[13, 3.4.2], T parameterizes the trisecant lines for Γ hence T is an integral curvenot contained in X . Also T is the intersection of 7 quadrics hence it cannot becontained in H . We conclude that T \ ( H ∪ X ) = ∅ , which we will use this to geta contradiction.Let L be a line in P corresponding to [ L ] ∈ T \ ( H ∪ X ); as L Supp P wehave that Tor ( P, O L ) = 0. Restricting (23) to L we get0 −→ O L (2) ⊕ O L ( − −→ O L (1) ⊕ −→ P | L −→ , which is an absurd, since Hom( O L (2) , O L (1)) = 0. (cid:3) Remark 28.
We complete this section by observing that for distributions D asabove, the two lines of C red = Sing ( D ) red are jumping lines for T D : the double lineof C is supported on the unique jumping line of order 3 for T D , while the reducedline of C is a jumping line of order 2. Indeed, let L denote the support of thedouble line of C and L its the reduced line; dualizing the sequences0 −→ O L (1) ⊕ O L −→ O C −→ O L −→ , −→ T D −→ T P −→ I C (4) −→ , we obtain the composition of epimorphisms T D − ։ ω C − ։ O L ( − ⊕ O L ( − . which can only exist if L and L are jumping lines for T D of orders 3 and 2,respectively. Distributions with c ( T D ) = 4Let D be a degree two distribution such that c ( T D ) = 4. Then Theorem 3implies that T D is stable and, due to (10), c ( T D ) ∈ { , , , , , , , } . In fact,we can prove that c ( T D ) ∈ { , , } .First note that C = Sing ( D ) is a curve of degree 2 and, in particular, p a ( C ) ≤ c ( T D ) = 10 + 2 p a ( C ) ≤ . On the other hand, [21, Corollary 1.6] also says that any degree two curve of genusless than − p a ( C ) ≥ − c ( T D ) ≥ . Before we show the existence of such distributions, we make a few remarks onthe cohomology of the possible tangent sheaves.
Lemma 29. If T D is the tangent sheaf of a distribution D of degree 2 such that c ( T D ) = 4 and c ( T D )) = 2 l , then h ( T D (1)) = ( , l = 3 , , l = 5 . Proof.
From Lemma 1, we have that h ( T D (1)) = h ( ω C (1)) = h ( O C ( − C = Sing ( D ); we will compute h ( O C ( − C ) = 2 and p a ( C ) = l − l ≤
4, it follows that0 −→ O L (3 − l ) −→ O C ( − −→ O L ( − −→ L and L are reduced lines and L = L only if l = 4, see [21]. It is thenclear that h ( O C ( − l = 5 the curve C is a conic (not necessarily irreducible nor reduced) whence0 −→ O P ( − −→ O P ( − ⊕ O P ( − −→ I C ( − −→ h ( O C ( − h ( I C ( − (cid:3) Case c ( T D ) = 6 . In this case we have that Sing( D ) is the union of a doubleline C of genus − Example 30.
Consider the double structure C on the line { x = y = 0 } given bythe ideal I C = ( x , xy, y , x ( z − w ) − yzw )and fix the points (1 : 0 : 0 : 0), (0 : 1 : 0 : 0), (1 : 1 : 1 : 1) and ( − − ω =( − xy + xyz + y z + 2 xz + 2 xyw − y w − yzw − xw ) dx + ( x y + x z + xyz + xz − x w − yzw − xw ) dy + ( − x y − xy − x z − xyz + x w + 4 xyw + 2 y w ) dz + ( xy − x z − xyz − y z + 2 x w + xyw ) dw, which is singular at the prescribed scheme and also at (38 : −
19 : 14 : −
4) and(9 : −
27 : 17 : 13).
ODIMENSION ONE DISTRIBUTIONS OF DEGREE 2 ON P Next we compute the possible spectra. Let { k , k , k , k } be the spectrum of T D for a distribution D with c ( T D ) = 4 and c ( T D ) = 6. Then using Remark 4and k + k + k + k = − −
3. It follows that the spectrum is either(24) {− , − , , } or {− , − , − , } . For our example, we get h ( T D ( − h ( I Z (3)) − T D must be {− , − , − , } .7.2. Case c ( T D ) = 8 . For D a distribution of degree two with c ( T D ) = 4 and c ( T D ) = 8 we have that Sing( D ) is a curve C of degree 2 and genus − C is either a pair of skew lines or a double line. We will showthat such distributions exist using an auxiliary foliation.Let N be a null correlation bundle, and let σ ∈ H ( N (1)) such that C := ( σ ) consists of the union of two disjoint lines. Let G be a generic foliation by curves ofdegree 3 given by the exact sequence G : 0 −→ N ( − −→ Ω P −→ I W (2) −→ , so that W is smooth and connected, and thus irreducible; such foliations exist by [10,Proposition 8.3]. As deg( G ) = 3 and W is irreducible, we may apply Proposition6 to conclude that the distribution D induced by ( G , σ ) satisfies Sing ( D ) = C .Moreover, it follows from (17) that c ( T D ) = 4 and c ( T D ) = 8.The next step is to determine the possible spectra of the tangent sheaf T D , andwe proceed as in the previous subsection. Let { k , k , k , k } be the spectrum of T D . As k + k + k + k = − − {− , − , − , } or {− , − , − , − } . For a distribution constructed as above, we have0 −→ O P ( − −→ T D −→ I W (2) −→ h ( T D ( − h ( I W (1)) = 0. Thus, as 0 cannot occur, thespectrum must be {− , − , − , − } . Remark 31.
It is not clear to us whether the other spectra in displays (24) and(25), namely {− , − , , } and {− , − , − , } , are realized by the tangent sheafof a codimension one distribution on P .7.3. Case c ( T D ) = 10 . A distribution D in this case is singular at a conic plus 10points. We may show that such distributions exist using an auxiliary foliation bycurves.Start with a generic foliation by curves of degree 2 of the form(26) G : 0 −→ O P ( − ⊕ O P ( − −→ Ω P −→ I W (1) −→ , with W being a smooth and irreducible curve of degree 5 and genus 1. Since N ∗ G (4) = O P (2) ⊕O P (1), we can find σ ∈ H ( N ∗ G (4)) such that C = ( σ ) is a conicdisjoint from W . Therefore, Proposition 6 guarantees that the induced codimensionone distribution D is such that Sing ( D ) = C provided η ∈ H ( ω Z (2)) is non zero;the formulas in display (4) yield c ( T D ) = 4 and c ( T D )) = 10. We only need toshow that η = 0. Assume by contradiction that η = 0, so that the long exact sequence in display(13) breaks into two parts0 −→ O P ( − −→ F ∗ −→ O P (1) −→ −→ ω W (3) −→ O Z (4) −→ ω C (5) −→ . Using that ω W = O W and ω C = O C ( −
1) we conclude that0 −→ O W ( − −→ O Z −→ O C −→ Z = W ∪ C and W ∩ C = ∅ , which is absurd.The next step is to determine the spectrum of the tangent sheaf. Let D be a dis-tribution of degree 2 satisfying c ( T D ) = 4 and c ( T D )) = 10 and let { k , k , k , k } be the spectrum of T D . First we claim that h ( T D ( − k ≤ − k i ≥ −
2; indeed as k + k + k + k = − {− , − , − } contained in the spectrum. Therefore, the only possible spectrum is {− , − , − , − } . Now we prove our claim.
Lemma 32. If T D is the tangent sheaf of a codimension one distribution D ofdegree 2 with ( c ( T D ) , c ( T D )) = (4 , , then h ( T D ( − .Proof. By Lemma 29, h ( T D (1)) = 1, so let σ ∈ H ( T D (1)) be a nontrivial sectionand let X := ( σ ) be its vanishing locus. We have then the following commutativediagram: 0 (cid:15) (cid:15) (cid:15) (cid:15) O P ( − σ (cid:15) (cid:15) O P ( − (cid:15) (cid:15) / / T D (cid:15) (cid:15) φ / / T P (cid:15) (cid:15) / / I Z (4) / / / / I X (1) (cid:15) (cid:15) / / G (cid:15) (cid:15) / / I Z (4) / /
00 0where Z = Sing( D ) is a conic C plus 10 points and G is the cokernel of φ ◦ σ . Fromthe first column from the left we have h ( T D ( − h ( I X ); we will compute h ( I X ).Dualizing the second column from the left we get a foliation by curves(27) 0 −→ G ∗ −→ Ω P −→ I W (1) −→ W ⊃ X ; and dualizing the bottom row we get0 −→ O P ( − −→ G ∗ −→ O P ( − ζ −→ ω C −→ · · · where ζ ∈ H ( ω C (1)). Note that as C is a conic, ω C = O C ( −
1) and also note that ζ is surjective; otherwise ζ = 0 and G ∗ would not admit an injective map to Ω P . ODIMENSION ONE DISTRIBUTIONS OF DEGREE 2 ON P In particular, we get0 −→ O P ( − −→ G ∗ −→ I C ( − −→ G ∗ = O P ( − ⊕ O P ( − W has puredimension one and by degree reasons X = W . Thus (27) gives as a resolution for I X from which we get h ( T D ( − h ( I X ) = 0 (cid:3) Distributions with c ( T D ) = 5If D is a distribution of degree two such that c ( T D ) = 5, then Sing ( D ) is acurve of degree 1, i.e., a reduced line. Thus c ( T D ) = 14, owing to (4). Thesedistributions do exist and, in fact, we can prove a more general result. Proposition 33.
For each d ≥ and any line L ⊂ P , there exists a codimensionone distribution D of degree d whose singular scheme consists of L plus d + 2 d − d points, so that ( c ( T D ) , c ( T D )) = ( d + 1 , d + 2 d − d ) . In particular, taking d = 2 in the statement above implies that there exists acodimension one distribution D of degree 2 such that ( c ( T D ) , c ( T D )) = (5 , Proof.
The starting point is a foliation by curves of the form(28) G : 0 −→ O P ( − d − ⊕ φ −→ Ω P −→ I W (2 d − −→ G ) = 2 d − ≥
3, so W must be connected by [10, Proposition 4.3].By taking a generic morphism φ , Ottaviani’s Bertini-type Theorem [24, Teorema2.8] implies that we can assume W to be smooth, so W is an irreducible curve.Since N ∗ G = O P ( − d − ⊕ , we can find σ ∈ H ( N ∗ G ( d + 2)) so that ( σ ) = L , forany given line L . We then apply Proposition 6 to the pair ( G , σ ) we just described.Note that for d ≥ G ) ≥ d + 1 and Remark 7 implies η = 0; since W is irreducible, dim coker η = 0.The induced distribution D will have degree d and Sing ( D ) = L ; the Chernclasses for T D follow directly from (16). (cid:3) Remark 34.
There exist distributions as in the statement of Proposition 33 alsofor d = 0 or 1, but the proof above does not apply to these cases: when d = 0,there is no morphism O P ( − ⊕ → Ω P ; when d = 1, the singular scheme W isnot connected.To conclude this section we compute the possible spectra for the tangent sheafof a codimension one distribution of degree 2 with c ( T D ) = 5. Let { k , . . . , k } be the spectrum of T D ; note that k + · · · + k = − k i ≥ − h ( T D ( − ≤ h ( ω L ( − k i ≤
0. Then the spectrum must be one of the following possibilities:(29) {− , − , − , − , − } , {− , − , − , − , } or {− , − , − , − , } . For the distributions constructed via Proposition 33 we have, as a consequence ofthe exact sequence in the statement of Proposition (6),0 −→ O P ( − −→ T D −→ I W (2) −→ d = 2, we get h ( T D ( − h ( I W (1)) = 0. Therefore 0cannot occur and the spectrum is {− , − , − , − , − } . Remark 35.
It is not clear if there exist distributions whose tangent sheaves realizethe two other spectra in display (29).9.
Distributions with c ( T D ) = 6The first line of Table 1 refers to the distributions with c ( T D ) = 6. These arethe generic distributions of degree 2, that is, those distributions whose singularscheme is 0-dimensional. In particular, c ( T D ) = 20. The subset of P H ( O P (4))parameterizing isomorphism classes of generic distributions of degree two is Zariskiopen. One can easily find an example of such a distribution. Example 36.
Let D be the distribution given by the following 1-form, inspired byJouanolou’s examples [19, p. 160], ω = ( x y − w ) dx + ( y z − x ) dy + ( z w − y ) dz + ( w x − z ) dw. Its singular scheme is the reduced set of points { ( ξ : ξ − : ξ : 1) | ξ = 1 } .Generic distributions of degree two are completely determined by its singularscheme, see [11, Corollary 4.7]; and they are never integrable, due to the dimensionof the singular scheme.It only remains for us to determine the spectrum of the tangent sheaf. Let D be ageneric distribution and let { k , . . . , k } be the spectrum of T D . Since Sing( D ) hascodimension three, E xt ( I Z , O P ) = 0 and dualizing the exact sequence in display(1) we obtain 0 −→ O P ( − −→ Ω P −→ T D −→ h ( T D ( − h ( T D (1)) = 0 and h ( T D ) = 1. Dueto (11), we have − ≤ k i ≤ −
1, for every i , and − k + · · · + k = −
10, an easy computation of integer partitions shows that thespectrum is {− , − , − , − , − , − } .10. Concluding Remarks
Recall that isomorphism classes of codimension one distributions D of degree d on P whose tangent sheaves have Chern classes c ( T D ) = c and c ( T D )) = c areparameterized by a quasi-projective variety denoted by D ( d, c , c ), see [3]. Someof these moduli spaces have been explicitly described in [3, Section 11] and [11],using different techniques.From this point of view, the main goal of this paper was simply to determine forwhich values of c and c we have D (2 , c , c ) = ∅ . However, the results in Section6.2 allow us a detailed description of D (2 , , Proposition 37.
The moduli space D (2 , , of codimension one distributions D of degree 2 such that c ( T D ) = 3 and c ( T D ) = 0 is irreducible of dimension 21.Proof. Let B (3) denote the moduli space of stable rank 2 locally free sheaves E with c ( E ) = 0 and c ( E ) = 3. If [ D ] ∈ D (2 , , T D ] ∈ B (3), thus [3, Lemma 2.5] yields a forgetful morphism ̟ : D (2 , , → B (3).Let I (3) denote the space of instanton bundles with a unique jumping line oforder 3, which was show to be an irreducible quasi-projective variety of dimension20 in [13, 1.4], and let J denote the open subset of I (3) consisting of instantonbundles corresponding to a smooth rational quintic as in [13, 2.2]. Following theproof of Theorem 27, we note that J ⊆ im( ̟ ) ⊆ I (3). ODIMENSION ONE DISTRIBUTIONS OF DEGREE 2 ON P Furthermore, the fibre of ̟ over [ E ] ∈ im( ̟ ) is precisely the set of monomor-phisms φ : E → T P whose cokernel is torsion free, so it is an open subset ofHom( E, T P ). However, as we saw in the proof of Theorem 27, if [ E ] ∈ J , thenevery φ ∈ Hom( E, T P ) is injective and has torsion free cokernel, thus in fact ̟ − ([ E ]) = P Hom( E, T P ) ≃ P whenever [ E ] ∈ J . Since hom( E, T P ) = 2 forevery [ E ] ∈ I (3) it follows that dim ̟ − ([ E ]) = 1 for every E ∈ im( ̟ ). Theconclusion is then an immediate consequence of the Theorem on the Dimension ofFibres. (cid:3) More generally, we observe that once we find a point [ D ] ∈ D (2 , c , c ), then thetangent space T [ D ] D (2 , c , c ) of D (2 , c , c ) at [ D ] is isomorphic to Hom( T D , I Z (4)).This means that D (2 , c , c ) has an irreducible component of dimension at mosthom( T D , I Z (4)) (equality occurs when D (2 , c , c ) is smooth at [ D ]), a quantitythat can be explicitly computed in several examples. A description of D (2 , c , c )along the lines of [3, Section 11] and [11] is a natural next step.From the classification of distributions of degrees 0 and 1, presented in [3], and2, presented here, we see evidence that for the following set of conjectures regardingthe topological, geometrical and algebraic properties of codimension one distribu-tions and especially foliations.First, if D is a codimension one distribution of degree d ≤
2, then its singularscheme Sing( D ) is never contained in a surface of degree d . This claim also holdsfor codimension one distributions D of arbitrary degree such that Sing( D ) hasdimension zero, see [11, Lemma 4.2]. We wonder if this is true in general. In par-ticular, this would provide an equivalence between degree d distributions singularat Sing( D ) and syzygies for the saturated homogeneous ideal of Sing( D ).In order to understand the properties imposed by integrability condition, weobserve that if D is a codimension one foliation of degree d ≥ c ( T D ) ≤ d + 2 and this bound is sharp;(2) Sing ( D ) is arithmetically Cohen–Macaulay (ACM), hence connected;(3) H ∗ ( T D ) = 0 or, equivalently, the ideal I generated by the coefficients of ahomogeneous 1-form defining D is saturated.We wonder if these properties hold in general, for d ≥
3. Indeed, (2) and (3) arerefinements of questions that have appeared earlier in the literature. The connect-edness of the 1-dimensional part of the singular scheme of a foliation was asked byCerveau in [6] and it was well known to hold in degrees d ≤
2; our remark here isthat the 1-dimensional part of the singular scheme should be ACM. Recently, theauthors of [4] proved that the ACM property holds, under mild assumptions, forthe Kupka subcheme
K ⊂
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