Daniele Faenzi

Marcos Jardim

Jean Vall Ès

Logarithmic sheaves of complete intersections

Keywords: Mathematics Subject Classification. 14F05, 14J60, 14M10, 32S65 Logarithmic sheaves, freeness and local freeness. Complete intersections. Stable syzygy sheaves. Pencils of quadrics. Algebraic Foliations

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Introduction

Let κ be an algebraically closed field of characteristic zero and consider a regular sequence σ " pf 1 , . . . , f k q of homogeneous polynomials f i P R " κrx 0 , . . . , x n s of degree d i `1, for some 0 ď d 1 ď ¨¨¨ď d k and k ď n. Let I σ :" pf 1 , . . . , f k q denote the ideal generated by the sequence σ, and V pσq be the associated scheme-theoretic complete intersection in P n . Consider the Jacobian matrix of σ, namely:

J σ :" ¨∇f 1 . . . ∇f k ‹ '.
This can be viewed as a morphism of sheaves:

J σ : O 'n`1 P n ÝÑ ' k i"1 O P n pd i q.
The main focus of this paper concerns the sheaf: T σ :" kerpJ σ q.

We call T σ the logarithmic tangent sheaf associated to σ. This nomenclature for T σ is motivated by the following observation. Set X " V pσq and recall that the Zariski tangent sheaf TX and the sheaf T 1 X supported at SingpXq fit into:

0 Ñ TX Ñ TP n | X Ñ k à i"1 O X pd i `1q Ñ T 1 X Ñ 0.
The sheaf T P n xXy of vector fields on P n tangent to X is the kernel of the natural morphism TP n Ñ À k i"1 O X pd i `1q, see [10, Chapter 3]. It turns out that T σ p1q is a subsheaf of rank n ´k `1 of T P n xXy. More precisely (see Lemma 2.4), writing V σ " À k i"1 I X pd i `1q{O P n and denoting by Q σ the cokernel of J σ , we have: 0 Ñ T σ p1q Ñ T P n xXy Ñ V σ Ñ Q σ p1q Ñ T 1 X Ñ 0. When the sequence σ consists of a single polynomial f (so that k " 1q, then V σ " 0 so T σ p1q » T P n xXy is precisely the logarithmic tangent sheaf associated to the divisor V pf q, see for instance [11] or the celebrated [9].

Note that, for k ě 2, the sheaf T P n xXy cannot be locally free. On the other hand, as we shall see, T σ may be locally free or even completely decomposable. Hence we propose the following three definitions, whose goal is to generalize the usual concept of a free divisor introduced in [9].

Definition. A regular sequence σ is said to be:

(1) locally free if the associated logarithmic tangent sheaf T σ is locally free.

(2) free if the logarithmic tangent sheaf T σ splits as a sum of line bundles.

(3) strongly free if every sequence σ 1 such that I σ 1 " I σ is free.

Clearly, every free regular sequence of length k " 1 is also strongly free. On the other extreme case, we observe that if σ is a regular sequence of length k " n, then T σ " O P n peq for some negative integer e, since every rank 1 reflexive sheaf on P n is a line bundle. Therefore, every regular sequence σ in κrx 0 , . . . , x n s of length n is strongly free. We provide explicit examples of free and strongly free regular sequences of length 2 in κrx 0 , x 1 , x 2 , x 3 s, see Example 2.7 and Example 2.8 below.

Regarding the middle range 1 ă k ă n, recall that it is notoriously hard to construct indecomposable locally free sheaves of rank r on P n when 2 ď r ď n ´2. In fact, only two examples are known, the Horrocks-Mumford rank 2 bundle on P [START_REF] Calvo-Andrade | Codimension one holomorphic distributions on projective threespaces[END_REF] and Horrocks' rank 3 parent bundle on P [START_REF] Eisenbud | Commutative algebra. With a view toward algebraic geometry[END_REF] . Furthermore, Hartshorne's conjecture predicts that every locally free sheaf of rank r on P n with 3r ă n splits as a sum of line bundles, which would imply that locally free regular sequences of length k in κrx 0 , . . . , x n s are free whenever 3k ą 2n `3.

With these facts in mind, it seems natural to investigate regular sequences of length 2 in R " κrx 0 , . . . , x n s. Following two directions usually pursued in the literature concerning logarithmic sheaves for hypersurfaces, our goal is to find criteria to determine when, on the one hand, a regular sequence σ " pf 1 , f 2 q is (locally) free, and, on the other hand, when the associated logarithmic tangent sheaf T σ is slope-stable.

We start in Section 2 by setting up basic definitions and examples. Most importantly, we provide criteria to determine when a regular sequence in R is locally free (see Lemma 2.2), as well as examples of free regular sequences that are not strongly free (Example 2.7), and of a strongly free regular sequence (Example 2.8). In Section 2.2 we show that regular sequences of length k induces codimension k ´1 distributions on P n , see Lemma 2.3. Yet another important notion in this paper is the compressibility of a k-web σ (that is, a regular sequence of length k consisting only of forms of equal degree): σ is said to be compressible if there is a variable that occurs in none of the forms contained in σ; a regular sequence that is not compressible is said to be incompressible (see Definition 2.12).

Sections 3 through 6 are dedicated to a thorough study of regular sequences of consisting of 2 polynomials of degree 2, also referred to as pencils of quadrics.

First, recall that the slope of a torsion-free sheaf F of rank p ą 0 on P n of determinant `Źp F ˘__ » O P n peq is defined as µpF q " e{p. The sheaf F is said to be slope-(semi)stable if any proper subsheaf K of E has slope µpKq ă pďq µpEq; F is slope-polystable if it is the direct sum of slope-stable sheaves with the same slope, and F is slope-unstable if it is not slope-semistable. The following result is proved in Section 3.

Theorem A. Let σ be a pencil of quadrics in P n and let r 0 be the maximal corank of the Hessian matrix for each quadric of the pencil.

(1) If σ contains two double hyperplanes, then T σ " O 'pn´1q P n .

(2) If σ contains only one double hyperplane, then T σ is slope-stable if and only if σ is incompressible. (3) If σ is compressible and contains no double hyperplanes, then T σ is slopeunstable. (4) If σ is incompressible and contains no double hyperplanes, then i) T σ is slope-stable when 2r 0 ă n `1; ii) T σ is strictly slope-semistable when 2r 0 " n `1; iii) T σ is slope-unstable when 2r 0 ą n `1.

The upshot is that, for the most interesting case (namely that of incompressible pencils without double hyperplanes), stability depends only of the maximal corank r 0 of the quadrics in the pencil. By [1], semistability of a pencil of quadrics in the sense of geometric invariant theory is equivalent to the fact that the discriminant of the pencils is non-zero (i.e. the pencil is regular) and has no root of multiplicity greater than pn`1q{2. So there are many GIT-unstable pencils σ whose logarithmic sheaf T σ is still slope-semistable or even slope-stable see Remark 3.14.

Next, we look at freeness and local freeness of pencils of quadrics and, more generally, at projective dimension of T σ , both in the local and in the graded senses. This turns out to depend on more subtle invariants of the pencil. To review them, note that the pencil of quadrics defined by σ gives a symmetric matrix ρ σ of linear forms on P 1 , whose generic corank r 1 is the corank of the Hessian matrix of a generic quadric in the pencil. Note that r 1 " 0 if and only if σ contains smooth quadrics, we call σ regular in this case and irregular otherwise. When σ is irregular, there are integers c 1 ď ¨¨¨ď c r1 such that the torsion free part of C σ " cokerpρ σ q is À r1

i"1 O P 1 pc i q. We call c " pc 1 , . . . , c r1 q the degree vector of σ. If Λ " tλ 1 , . . . , λ ℓ u Ă P 1 is the support of the torsion part C t of C σ , then, for each j P t1, . . . , ℓu, denoting by λ paq j the a-tuple structure over λ j , the localization at λ j of C t is

À sj i"1 O 'pj,i λ pa j,i q j ,
for some s j and pa j,i , p j,i | i P t1, . . . , s j uq. These data are arranged into the Segre symbol Σ " rΣ 1 , . . . , Σ ℓ s, defined for all j P t1, . . . , ℓu by: Σ j " pa j,1 , . . . , a j,1 loooooomoooooon pj,1

, . . . , a j,sj , . . . , a j,sj looooooomooooooon pj,s j q, with a j,1 ą ¨¨¨ą a j,sj .

It turns out that the data pr 1 , Λ, Σq completely characterize an incompressible pencil of quadrics up to homography, thus generalizing a classical result attributed to Segre and Weierstrass for the case of regular pencils, see Section 4.1 for further details. With these data, we describe the scheme-theoretic structure of the Jacobian scheme, when σ is regular, as a union of nilpotent structures on pairwise disjoint linear spaces whose dimension and degree of nilpotency depend on Σ and whose position depends on Λ. If σ is irregular, the Jacobian scheme contains an additional component which is a rational normal scroll of dimension r 1 and degree c 1 `¨¨¨`c r1 that connects all the linear spaces, with a prescribed intersection along each space.

The upshot is that these invariants also characterize the projective and global projective dimensions of T σ , as it is described in the following two results, proved in Section 4.2 and Section 4.3, respectively Theorem B. Let σ have Segre symbol Σ. For q ą 0, Ext q P n pT σ , O P n q ‰ 0 if and only if there are j P t1, . . . , ℓu and k P t1, . . . , s j u such that: q `pj,1 `. . . `pj,k " n ´r1 ´1, or r 1 ą 0 and q `r1 " n ´2. In particular we have: i) if σ is regular and p " mintp j,1 | j P t1, . . . , ℓuu, then pdimpT σ q " n ´p ´1; ii) if σ is irregular, then pdimpT σ q " n ´r1 ´2.

We have a rather different situation for the graded projective dimension gpdimpT σ q -namely, the projective dimension of the module of global sections of T σ . This is summarized in the following result.

Theorem C. For a regular pencil of quadrics σ in P n we have gpdimpT σ q " n ´2 except if σ has Segre symbol r1 p , 1 q s for p ě q ě 1 or rp2 q , 1 p qs with q ě 1. In both these cases gpdimpT σ q " n ´q ´1.

For an irregular pencil of quadrics σ of generic corank r 1 we have gpdimpT σ q " n ´1 except if σ has degree vector p1, . . . , 1q, in which case gpdimpT σ q " n ´r1 ´2.

With this in mind, after a careful analysis of pencils of quadrics in P 3 , performed in Section 5, we come to the conclusion that freeness and local freeness are equivalent conditions for pencils of quadrics in P n and we completely classify pencils satisfying such condition.

Theorem D. A pencil of quadrics σ in P n , n ě 3, is free if and only if T σ is locally free. More precisely, the only free pencils of quadrics are displayed in Table 52.

By contrast, we provide in Section 7 a series of examples of locally free pencils of degree k ě 3 that are not free. This indicates that potentially interesting vector bundles may arise as logarithmic sheaves associated to regular sequences of higher degree having deep singularities. To understand our following result, recall that a null correlation bundle is defined as the cokernel of a non vanishing morphism O P 3 p´1q Ñ Ω 1 P 3 p1q; every slope-stable rank 2 locally free sheaf N on P 3 with c 1 pN q " 0 and c 2 pN q " 1 arises in this way.

Theorem E. Fix k ě 0, and consider the pencil σ " pf, gq of degree k `3 with:

f " x 0 x k`2 1 `xk`3 2 `xk`2 2 x 3 and g " x 2 x 3 px k`1 2 ´xk`1 1 q.
Then T σ » N p´k ´2q, where N is a null correlation bundle.

We complete this paper with an application of our results to the study of rational codimension one foliations on P n , see Section 8. To be precise, recall that a rational 1-form is a twisted 1-form given by the expression ω " pd 1 `1qf 1 ¨df 2 ´pd 2 `1qf 2 ¨df 1 P H 0 pΩ 1 P n pd 1 `d2 `2qq, where f i P H 0 pO P n pd i `1qq for i " 1, 2 and f 1 , f 2 have no common factors. Regarding ω as a morphism TP n Ñ O P n pd 1 `d1 `2q, we consider the kernel sheaf K ω :" ker ω. We show in Section 8 that the natural 1-1 correspondence pf 1 , f 2 q Ø ω between regular sequences of length 2 and rational 1-forms is such that K ω " T σ p1q, see Lemma 8.1. This fact has two important consequences. First, we can invoke a result from the general theory of codimension one distributions on P 3 , presented in [4], to obtain simple criteria to establish when T σ is slope-(semi)stable, see Corollary 8.2.

Second, we provide a negative answer to a problem posed by Calvo-Andrade, Cerveau, Giraldo and Lins Neto, see [3, Problem 2]; namely, these authors asked whether the tangent sheaf of a codimension one foliation must split as a sum of line bundles whenever it is locally free. While Theorem D implies that this claim is true for rational foliations of type p2, 2q, Theorem E says that for each k ě 0 there are rational foliations of type pk `3, k `3q on P 3 whose tangent sheaf is a slope-stable locally free sheaf.

All things considered, we believe that the results presented in this paper point to a rich, interesting general theory of (local/strong) freeness for complete intersection subschemes that in some sense parallels the widely known theory of freeness for divisors in P n .

Acknowledgments. We thank Eduardo Esteves for pointing us out to the results of [1]. This research is part of the CAPES/COFECUB project Moduli spaces in algebraic geometry and applications, Capes reference number 88887.191919/2018-00. D.F. is partially supported by EIPHI Graduate School ANR-17-EURE-0002 and ANR Fano-HK ANR-20-CE40-0023. M. J. is partially supported by the CNPQ grant number 302889/2018-3 and the FAPESP Thematic Project 2018/21391-1. The authors also acknowledge the financial support from Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -Brasil (CAPES) -Finance Code 001.

Basic definitions and examples

Let κ be a field of characteristic 0, n P N and put R " κrx 0 , . . . , x n s. In this section we will give some preliminary properties of logarithmic tangent sheaves associated with complete intersections. Many properties remain valid for κ of characteristic different from 2.

2.1. General framework. Let σ :" pf 1 , . . . , f k q be a regular sequence of R. Consider the associated complete intersection variety X " V pσq and set J σ for the associated Jacobian matrix. Let us denote by B i the partial derivative B Bxi and ∇f " pB 0 f, . . . , B n f q the gradient of a homogeneous polynomial f P R.

We put T σ for the associated logarithmic tangent sheaf as defined in the introduction, namely T σ is the kernel of J σ . In addition, we define the sheaves M σ :" impJ σ q and Q σ :" cokerpJ σ q. The sheaf M σ is torsion free, it can be thought of as the natural extension to P n of the equisingular normal sheaf of X " V pσq and that T σ is reflexive. We have the fundamental exact sequences:

(1) 0 ÝÑ T σ ÝÑ O 'n`1

P n Jσ ÝÑ M σ ÝÑ 0 (2) 0 ÝÑ M σ ÝÑ ' k i"1 O P n pd i q ÝÑ Q σ ÝÑ 0 We define the Jacobian scheme Ξ σ as the degeneracy locus of J σ , Ξ σ :" V ˜k ľ J σ ¸.
This is the subscheme of P n defined by the common zeros of the k ˆk minors of J σ . The reduced structure pΞ σ q red coincides with the support of the sheaf Q σ . Note that pΞ σ q red may contain a hypersurface.

More precisely, the image of the exterior power morphism:

(3)

k ľ J σ : O 'p n`1 k q P n Ñ O P 3 pd 1 `¨¨¨`d k q;
is of the form I Wσ pd 1 `¨¨¨`d k ´lq, where l :" c 1 pQ σ q is the degree of the hypersurface contained in Ξ σ , and W σ Ă P n is a subscheme of codimension at least 2, possibly not pure. Let us illustrate this discussion with an explicit example.

Example 2.1. Let f " x 0 x 1 `x2 x 3 and g " x 0 x 1 x 2 x 3 , so that:

J σ :" ˆx1 x 0 x 3 x 2 x 1 x 2 x 3 x 0 x 2 x 3 x 0 x 1 x 3 x 0 x 1 x 2 ˙.
The curve C " V pf, gq is the union of the four lines V px i , x j q with i " 0, 1 and j " 2, 3. Two of the 2 ˆ2 minors of J σ vanish identically, and we have that

2 ľ J σ " px 0 x 1 ´x2 x 3 q ¨p0 x 0 x 3 x 1 x 2 x 1 x 3 x 0 x 2 0q.
It follows that Ξ σ consists of the quadric V px 0 x 1 ´x2 x 3 q, so that l " c 1 pQ σ q " 2 in this case, plus two lines V px 0 , x 1 q and V px 2 , x 3 q. So W σ is the union of these two skew lines. Note that neither f and g have common factors, nor do ∇f and ∇g.

Lemma 2.2. Let σ be a regular sequence. Then:

i) σ is locally free if and only if Q σ has no subsheaf of codimension ě 3; ii) if σ is locally free, Ξ σ has no irreducible component of codimension ě 3.
Note that pΞ σ q red may have no irreducible component of codimension at least 3 even when Q σ admits a subsheaf of codimension at least 3, see Section 7. This means that the converse of item (ii) above does not hold in general.

Proof. Taking duals of ( 1) and ( 2) we obtain Ext n´1 P n pT σ , O P n q " 0 and, for j ď n´1:

Ext j P n pT σ , O P n q » Ext j`1 P n pM σ , O P n q » Ext j`2 P n pQ σ , O P n q.
The sheaf T σ is locally free if and only if Ext j P n pT σ , O P n q " 0 for j ě 1, which is equivalent to requiring that Ext j P n pQ σ , O P n q " 0 for j ě 3. This gives the equivalence in the first claim.

If pΞ σ q red has an irreducible component Y of codimension j ě 3, then, since pΞ σ q red is the support of Q σ , it follows that Q σ has a non trivial subsheaf V ãÑ Q σ supported on Y , hence codim V " j. The previous item then implies that T σ is not locally free.

2.2.

Regular sequences and distributions. Recall that a codimension r distribution on P n is a short exact sequence of the form

D : 0 ÝÑ T D ÝÑ TP n ÝÑ N D ÝÑ 0
where N D is a torsion free sheaf of rank r and T D is a reflexive sheaf of rank n ´r, respectively called the normal and tangent sheaves of D. We refer to [4, Section 2.1] for further details on the general theory of distributions.

Let us point out how distributions are related to regular sequences. First, thinking of the Koszul complex attached to σ we consider σ " ppd 1 `1qf 1 , . . . , pd k `1qf k q t and the Koszul syzygy sheaf S σ :" cokerpσq, so:

(4) 0 Ñ O P n p´1q Ñ k à i"1 O P n pd i q Ñ S σ Ñ 0. Let η : O P n p´1q ÝÑ O 'n`1
P n be the Euler morphism, namely η " px 0 , . . . , x n q t . The Euler relation gives η ¨∇f i " pd i `1qf i for all i P t1, . . . , ku. This allows us to construct the following commutative diagram:

0 0 O P n p´1q η O P n p´1q σ 0 / / T σ / / O 'n`1 P n Jσ / / À k i"1 O P n pd i q 0 / / T σ / / TP n p´1q / / S σ 0 0 (5)
Here we used that κ is of characteristic zero, or rather that the characteristic of κ does not divide d i `1 for all i P t1, . . . , ku.

Note that the image of σ is contained in M σ and set N σ for the cokernel of σ, corestricted to M σ . The previous diagram gives:

0 0 O P n p´1q η O P n p´1q σ 0 / / T σ / / O 'n`1 P n Jσ / / M σ / / 0 0 / / T σ / / TP n p´1q / / N σ / / 0 0 0 (6)
Furthermore, we have a second diagram featuring the cokernel sheaf Q σ :

0 0 O P n p´1q O P n p´1q σ 0 / / M σ / / ' k i"1 O P n pd i q / / Q σ / / 0 0 / / N σ / / S σ / / Q σ / / 0 0 0 (7)
It follows that the bottom line in diagram in display (6) defines, for k ě 2, a codimension k ´1 distribution D σ on P n , given by the exact sequence (8) D σ : 0 ÝÑ T σ p1q ÝÑ TP n ÝÑ N σ p1q ÝÑ 0.

Summing up, we have proved the following statement.

Lemma 2.3. Every regular sequence σ of length k on n `1 variables induces a codimension k ´1 distribution D σ on P n such that T Dσ " T σ p1q.

However, not every codimension k ´1 distribution on P n comes from a regular sequence via the construction above. For instance, given a codimension k ´1 distribution D on P n , the monomorphism T D ãÑ TP n may not factor through O P n p´1q 'n`1 .

2.3. Logarithmic tangent sheaf and deformations. Let us point out the relationship between our sheaf and classical sheaves of tangent vector fields, in connection with locally trivial deformations of embeddings.

2.3.1. Tangent vector fields along a complete intersection. Given the regular sequence σ we have a complete intersection X " V pσq Ă P n , whose ideal sheaf I X is generated by σ_ : ' k i"1 O P n p´d i q Ñ I X p1q. In addition, we have the equisingular normal sheaf N [START_REF] Avritzer | Pencils of quadrics, binary forms and hyperelliptic curves[END_REF] X{P n , see [10, § 3.4.4], which is defined as the quotient sheaf TP n | X {TX, and therefore satisfies the following exact sequence

0 Ñ TX Ñ TP n | X Ñ N 1
X{P n Ñ 0; here, TX denotes the Zariski tangent sheaf of X. Note also that N 1

X{P n is a subsheaf of the normal bundle N X{P n » À k i"1 O X pd i `1q, so:

N 1 X{P n ãÑ k à i"1 O X pd i `1q;
the quotient of this monomorphism is denoted by T 1 X , see [10, § 1.1.3]; it is supported at the singular locus of X. For further reference, we write its defining exact sequence:

(9) 0 Ñ N 1 X{P n Ñ k à i"1 O X pd i `1q Ñ T 1 X Ñ 0.
The sheaf of vector fields on P n tangent to X, denoted by T P n xXy, is defined as the kernel of the composed epimorphism TP n ։ TP n | X ։ N 1 X{P n , yielding the exact sequence (10) 0 Ñ T P n xXy Ñ TP n Ñ N 1 X{P n Ñ 0. The main motivation for introducing T P n xXy is given by [10, Proposition 3.4.17]; namely, H 1 pT P n xXyq and H 2 pT P n xXyq are the tangent space and the obstruction space of the semiuniversal space of locally trivial deformations of the embedding X ãÑ P n . Here we show that T σ p1q is a subsheaf of T P n xXy and, to a certain extent, describe the quotient T P n xXy{T σ p1q.

Since the forms f 1 , . . . , f k generate the homogeneous ideal of X in P n , we may view σ " ppd 1 `1qf 1 , . . . , pd k `1qf k q t as a morphism O P n Ñ À k i"1 I X pd i `1q. We define a torsion free sheaf V σ " cokerpσq fitting into:

0 Ñ O P n Ñ k à i"1 I X pd i `1q Ñ V σ Ñ 0.
Note that, when k " 1, we have V σ " 0, as I X pd 1 `1q » O P n in this case, so:

T σ p1q » T P n xXy, for k " 1.
For k ě 2 the relationship between the two sheaves T σ p1q and T P n xXy is expressed by the following lemma. Lemma 2.4. We have an exact sequence:

(11) 0 Ñ T σ p1q Ñ T P n xXy Ñ V σ Ñ Q σ p1q Ñ T 1 X Ñ 0.
Proof. We use the Koszul syzygy sheaf S σ of Subsection 2.2 to write the following exact sequence relating S σ and V σ :

0 Ñ V σ Ñ S σ p1q Ñ k à i"1 O X pd i `1q Ñ 0.
We get a commutative diagram:

0 0 0 / / O P n σ / / η À k i"1 I X pd i `1q 0 / / T σ p1q / / O 'pn`1q P n p1q Jσ / / À k i"1 O P n pd i `1q 0 / / T P n xXy / / TP n ̟ / / À k i"1 O X pd i `1q 0 0
where ̟ is given by the composition TP n ։ N 1 X{P n ãÑ

À k i"1 O X pd i `1q.
The exact sequence in display (11) is then obtained via the snake lemma, since V σ :" cokerpσq, Q σ :" cokerpJ σ q and T 1 X :" cokerp̟q.

2.3.2. Tangent vector fields along hypersurfaces. We look at the relationship between T σ and the tangent vector field to one of the hypersurfaces defining σ.

Lemma 2.5. We have:

(12) T σ " k č j"1 T fj .
Further, for each j P t1, . . . , ku, set Z j " Sing `V pf j q ˘. Then there is an exact sequence:

(13) 0 Ñ T σ Ñ T fj Ñ à iPt1,...,kuztju O P n pd i q Ñ Q σ Ñ O Zj pd j q Ñ 0.
Proof. For any j P t1, . . . , nu, we have:

T fj " ker ´∇f j : O 'pn`1q P n Ñ O P n pd j q ¯.
Therefore, since T σ is defined as kernel of the matrix obtained by stacking ∇pf 1 q, . . . , ∇pf k q, we get (12). Next, for any j P t1, . . . , ku, we have the following commutative diagram:

0 0 0 / / T σ / / T fj / / À iPt1,...,kuztju O P n pd i q 0 / / T σ / / O 'n`1 P n Jσ / / ∇fi À k i"1 O P n pd i q I Zj pd j q / / O P n pd j q 0 0 (14)
Since Z j is the Jacobian scheme of f j , the completion of this diagram via the snake lemma leads to (13).

These observations will play an important role in the proof of Theorem 7.1 below.

2.4.

Syzygies and global sections. Let us point out the relationship between the Jacobian syzygies and the global sections of T σ . Let ν : O P n p´aq Ñ O 'n`1 P n be a sygyzy of degree a for the Jacobian matrix J σ of a regular sequence σ, that is, J σ ˝ν " 0; assume that the entries of ν have no common factors of positive degree, so that N ν :" cokerpνq is a torsion free sheaf. We have the commutative diagram:

0 0 O P n p´aq O P n p´aq ν 0 / / T σ / / O 'pn`1q P n Jσ / / M σ / / 0 0 / / M σ,ν / / N ν / / M σ / / 0 0 0 (15)
It follows that every syzygy of degree a for J σ induces a section in H 0 pT σ paqq.

Conversely, every non trivial section in H 0 pT σ paqq induces a syzygy of degree a for J σ , thus we obtain an isomorphism of vector spaces (16)

H 0 pT σ paqq » Syz a pJ σ q,
where Syz a pJ σ q is the vector space of syzygies of degree a for the matrix J σ .

Example 2.6. Let us consider the case n " k " 1 as a toy model for the theory we are proposing; Take g P H 0 pO P 1 pd `1qq and the associated morphism

J g " ∇g : O '2 P 1 Ñ O P 1 pdq;
If B 0 g and B 1 g have no common factors, then J g is surjective and T g " O P 1 p´dq. The observation (16) implies that ∇g has no syzygies of degree less than d. More generally, if B 0 g and B 1 g have a common factor of degree e (for instance, if g " x 2 0 x 2 1 , then x 0 x 1 is a common factor for B 0 g and B 1 g), then M g » O P 1 pd ´eq, and T g » O P 1 p´d `eq. Example 2.7. Here is an example of a free regular sequence that is not strongly free. Consider the following regular sequences in R " κrx 0 , x 1 , x 2 , x 3 s σ :" px 0 x 1 , gq and σ 1 :" px 0 x 1 , x 2 0 x 1 `gq; where g is a polynomial of degree 3 depending only on x 2 and x 3 . Assume that B 2 g and B 3 g have no common factors, so that, as it was observed in Example 2.6 above, ∇g has no syzygies of degree ă 2.

Clearly, I σ " I σ 1 . We argue that σ is free, while σ 1 is not. Indeed, their Jacobian matrices are given by:

J σ " ˆx1 x 0 0 0 0 0 B 2 g B 3 g ˙and J σ 1 " ˆx1 x 0 0 0 2x 0 x 1 x 2 0 B 2 g B 3 g ˙.
Note that J σ has two independent syzygies, given by ν 1 " p´x 0 , x 1 , 0, 0q and ν 2 " p0, 0, B 3 g, ´B2 gq of degrees 1 and 2, respectively. Therefore, we have a monomorphism ν : O P 3 p´1q ' O P 3 p´2q ãÑ T σ whose cokernel, being a subsheaf of I L p1q ' I C p2q with L " V px 0 , x 1 q and C " V pB 2 g, B 3 gq, must be torsion free. It follows that ν must be an isomorphism, thus T σ » O P 3 p´1q ' O P 3 p´2q.

To see that T σ 1 does not split as a sum of line bundles, note that ν 2 is also a syzygy for J σ 1 , thus h 0 pT σ 1 p2qq ą 0. On the other hand, since J σ 1 has no syzygy of degree ď 1, we have that h 0 pT σ 1 p1qq " 0. In addition, the minors appearing in Ź 2 J σ 1 have no common factor, thus c 1 pQ σ 1 q " 0 and c 1 pT σ 1 q " ´c1 pM σ 1 q " ´3. Thus if T σ 1 " O P 3 paq ' O P 3 pbq with a ď b, then a `b " ´3, and a, b ď ´2, which is impossible.

In fact, note that Ξ σ 1 consists of the line V px 0 , x 1 q together with the following 0-dimensional schemes:

V px 0 , B 2 g, B 3 gq and V px 1 , B 2 g, B 3 gq, each of length equal to 4. Therefore, pΞ σ 1 q red contains at least two irreducible components of codimension 3; the second item of Lemma 2.2 implies that T σ 1 is not locally free.

Example 2.8. We show that the regular sequence σ " px 0 , x 2 3 q in R " κrx 0 , x 1 , x 2 , x 3 s is a strongly free sequence consisting of polynomials of different degrees. Any regular sequence σ 1 such that I σ " I σ 1 must be of the form σ 1 " pαx 0 , x 0 l `βx [START_REF] Bruce | On the classification of cubic surfaces[END_REF] 3 q for some linear form h P H 0 pO P 3 p1qq and α, β P κ ˚. Setting h " ax 0 `bx 1 `cx 2 `dx 3 , that Jacobian matrix for σ 1 is given by

J σ 1 " ˆα 0 0 0 2ax 0 bx 0 cx 0 dx 0 `2βx 3 ˙.
If c ‰ 0, then ν 1 " p0, ´c, b, 0q and ν 2 " p0, 0, dx 0 `2βx 3 , ´cx 0 q are independent syzygies of degree 0 and 1, respectively. Following the argument in Example 2.7, so we can conclude that T σ 1 " O P 3 ' O P 3 p´1q. When c " 0 and b ‰ 0 then ν 1 " p0, 0, 1, 0q and ν 2 " p0, dx 0 `2βx 3 , 0, ´bx 0 q are independent syzygies of degree 0 and 1, respectively, so again we conclude that T σ 1 " O P 3 ' O P 3 p´1q.

Finally, if b " c " 0, then ν 1 " p0, 0, 1, 0q and ν 2 " p0, 1, 0, 0q are independent syzygies of degree 0, thus T σ 1 " O '2 P 3 . It is worth noticing that, in general, detpT σ q is not fixed and may change with the choice of generators for I σ .

2.5. Webs. Fix integers d ě 0 and k ě 1 and let σ " pf 1 , . . . , f k q be a regular sequence of forms of degree d `1 in R " κrx 0 , . . . , x n s; we call σ a k-web in P n ; a 2-web is usually called a pencil. In this section, we establish some basic properties of logarithmic tangent sheaves associated to k-webs, which will be useful later on.

Freeness of webs.

Here is the first fundamental fact. Lemma 2.9. Let σ be a k-web. If σ is free, then it is strongly free.

Proof. Let σ 1 " pf [START_REF] Avritzer | Pencils of quadrics, binary forms and hyperelliptic curves[END_REF] 1 , . . . , f 1 k q be another regular sequence such that I σ 1 " I σ ; one can check that there is a matrix P P GL k pκq such that

¨f 1 1 . . . f 1 k ‹ '" P ¨f1 . . . f k ‹ '.
It follows that J σ 1 " P J σ , thus in fact T σ 1 » T σ , from which the desired statement follows immediately.

A particular case of the previous result leads to the simplest example of a strongly free regular sequence.

Example 2.10. Take a regular sequence σ " pf 1 , . . . , f k q such that each f i is a linear polynomial; note that V pσq is a linear subspace of codimension k. The Jacobian matrix is then a constant matrix of maximal rank, inducing a surjective morphism O 'n`1

P n Ñ O 'k P n . It follows that T σ " O 'n`1´k P n , M σ " O 'k P n , and Q σ " 0. 2.5.2.
Webs versus regular sequence. Let us point out how to associate a web to any regular sequence, keeping the logarithmic sheaf unchanged. Let σ " pf 1 , . . . , f k q be a regular sequence, with degpf i q " d i `1, for some d 1 `1, . . . , d k `1 P N. Let e be the least common multiple of d 1 `1, . . . , d k `1. For i P t1, . . . , ku, put ℓ i " e{pd i `1q. Set:

τ " pf ℓ1 1 , . . . , f ℓ k k q. Note that τ is a web of degree e.
Lemma 2.11. We have:

T τ " T σ .
Proof. For i P t1, . . . , ku, set g i " f ℓi i , so that τ " pg 1 , . . . , g k q. By the chain rule, for each i P t1, . . . , ku we have:

∇pg i q " ℓ i f ℓi´1 i ∇pf i q.
In other words, considering the morphism defined by the diagonal matrix:

P " diagpℓ 1 f ℓ1´1 1 , . . . , ℓ k f ℓ k ´1 k q : k à i"1 O P n pd i q Ñ O P n peq 'k ,
we get that:

J τ " P ˝Jσ . Since κ is of characteristic zero, P is injective, so T τ " kerpJ τ q " kerpJ σ q " T σ .

2.5.3. Compressibility of webs. Next, we introduce the following definitions. Definition 2.12. We say that the k-web σ is:

(1) regular if there is pz 1 , . . . , z k q P κ k such that the hypersurface V p ř k i"1 z i f i q is non singular;

(2) compressible if, up to a linear coordinate change, there is a variable that occurs in none of the forms f 1 , . . . , f k . A k-web that is not regular is called irregular ; a k-web that is not compressible is called incompressible.

Note that regular k-webs are incompressible. Furthermore, as it was observed in the proof of Lemma 2.9 above, if σ is a k-web, then the logarithmic tangent sheaf T σ is independent from the choice of generators of the ideal I σ generated by σ. Lemma 2.13. A k-web σ is compressible if and only if H 0 pT σ q ‰ 0.

Proof. The condition H 0 pT σ q ‰ 0 does not depend on the given choice of a system of coordinates, and if none of the forms f 1 , . . . , f k depends on a given variable, then all partial derivatives of f 1 , . . . , f k with respect to this variable are zero. This means that J σ contains a column containing only 0 and thus the kernel sheaf T σ contains a copy of O P n .

Conversely, assume H 0 pT σ q ‰ 0. For all pi, jq P N 2 with 1 ď i ď k and 0 ď j ď n, we set:

f i,j " Bf i Bx j P R.
Since the sheaf T σ satisfies H 0 pT σ q ‰ 0, there is a non-zero vector pb 0 , . . . , b n q P κ n`1 such that:

b 0 f i,0 `¨¨¨`b n f i,n " 0, for all 1 ď i ď k.
Then we define new coordinates px 1 0 , . . . , x 1 n q by choosing an invertible matrix pa i,j q of size n `1 with the condition a j,0 " b j for all 0 ď j ď n and putting :

x j " n ÿ ℓ"0 a j,ℓ x 1
ℓ , for all 0 ď j ď n.

Then, for all for all 1 ď i ď k, we have:

Bf i Bx 1 0 " n ÿ j"0 Bx j Bx 1 0 f i,j " n ÿ j"0 b j f i,j " 0.
Therefore, in the new coordinates px 1 0 , . . . , x 1 n q, none of the forms appearing in σ depends on x 1 0 .

The compressibility of a k-web σ " pf 1 , . . . , f k q is defined as the number of independent variables that can be removed from the polynomials f i , in a suitable coordinate system. In other words, σ has compressibility m if and only if h 0 pT σ q " m; note that 0 ď m ď n ´1. We set n :" n ´m; it indicates the minimal number of variables where the web is defined.

Lemma 2.14. If σ is a compressible k-web in R, then there is an incompressible k-web σ in κrx 0 , . . . , x ns and a n-dimensional linear space L Ă P n such that T σ " O

'pn´nq P n ' E where E| L » T σ .
Proof. Assume that σ " pf 1 , . . . , f k q is compressible, so that m :" h 0 pT σ q ą 0; set n :" n ´h0 pT σ q. We get a monomorphism O 'm P n ãÑ T σ so that the following composition

O 'm P n ãÑ T σ ãÑ O 'pn`1q P n ։ O 'm P n is the identity morphism; it follows that T σ " O 'pn´nq P n
' E, where the sheaf E fits in the exact sequence

(17) 0 Ñ E Ñ O 'pn`1q P n µ Ñ O P n pd ´1q 'k .
As we have seen in the proof of Lemma (2.13), there are new coordinates px 1 0 : ¨¨¨:

x 1
n q such that the variables x 1 0 , . . . , x 1 m´1 do not appear in the polynomials f i P σ. This means that the first m columns of the Jacobian matrix consist only of zeros, and that the matrix µ in display (17) is precisely the submatrix of trivial columns of J σ .

In addition, the k-web σ can be regarded as a k-web in κrx 1 m`1 , . . . , x 1 ns, which we rename σ. Setting L " V px 1 0 , ¨¨¨, x 1 m´1 q, we have that µ| L " J σ , thus E| L » T σ .

As an immediate consequence, we have:

Corollary 2.15. The logarithmic tangent sheaf of a compressible k-web is never slope-stable.

Recall that a coherent subsheaf F of a coherent sheaf E is saturated if E{F is torsion free. The following technical observation will be useful later on.

Lemma 2.16. If σ is an incompressible k-web, then every saturated subsheaf K Ă T σ satisfies c 1 pKq ă 0. Proof. Any saturated subsheaf K of T σ is also a saturated subsheaf of O 'n`1 P n
, which is a slope polystable sheaf. It follows that c 1 pKq ď 0, and if c 1 pKq " 0, then [8, Corollary 1.6.11] implies that K " O 'm P n , so σ is compressible.

We complete this section with a characterization of the degeneracy locus of kwebs as the set of points which are singular for some hypersurface of the web.

Lemma 2.17. Let σ be a k-web. The reduced degeneracy locus pΞ σ q red of the Jacobian matrix J σ coincides with the union of the singular loci of the singular hypersurfaces contained in the web.

Proof. Set σ " pf 1 , . . . , f k q. A point x P P n belongs to pΞ σ q red if and only if the gradients of f i are linearly dependent, that is, there is pz 1 , . . . , z k q P κ n`1 zt0u such that:

∇ ˜k ÿ i"1 f i ¸pxq " k ÿ i"1 z i ∇f i pxq " 0.
But this is the same as saying that x lies in the singular locus of V p ř k i"1 z i f i q.

Stability for pencils of quadrics

In this section κ is algebraically closed of characteristic different from 2. Given a quadric hypersurface Q in P n , we denote by rkpQq the rank of the Hessian matrix of an equation of Q, that is, the rank of a quadratic form associated with Q. We set corkpQq " n `1 ´rkpQq. When non empty, the singular locus of Q is a linear subspace of P n of dimension corkpQq ´1. The quadric Q is a double plane if and only if rkpQq " 1.

In this section, we focus on regular sequences σ " pf 1 , f 2 q such that degpf 1 q " degpf 2 q " 2, to which we can associate the pencil of quadrics

Q λ :" V `z1 f 1 `z2 f 2
˘, where λ " rz 1 : z 2 s P P 1 . Let r 0 :" max corkpQ λ q | λ P P 1 ( .

Our goal is to present the proof of Theorem A as follows. We will start by considering the easiest case, namely pencils that contain at least one double hyperplane, and prove the first two items of Theorem A in Section 3.1. The third item is an immediate consequence of Lemma 2.13; the most involved part of Theorem A is the last item, and its proof will take the bulk of Section 3.2.

3.1.

Stability of pencils with a double hyperplane. If Q λ contains two double hyperplanes, then we can take f 1 " x 2 0 and f 2 " x 2 1 , so that

J σ " ˆ2x 0 0 0 ¨¨¨0 0 2x 1 0 ¨¨¨0 ˙.
It is then easy to see that T σ " O 'pn´1q P n , as desired. Assume now that Q λ contains only one double plane. We can take f 1 " x 2 0 and put g " f 2 , so that

J σ " ˆ2x 0 0 ¨¨¨0 g 0 g 1 ¨¨¨g n ˙,
with g i " Bg{Bx i , for i P t0, . . . , nu. The hypothesis that Q λ contains only one double plane implies that at least two of the partial derivatives g i are non trivial for i P t1, . . . , nu. Since

2 ľ J σ " 2x 0 ¨pg 1 ¨¨¨g n q,
it follows that Ξ σ " V px 0 q Y V pg 1 , . . . , g n q, thus c 1 pT σ q " ´1.

Lemma 3.1. Let σ be a pencil of quadrics containing only one double plane. Then T σ is slope-stable if and only if σ is incompressible.

Proof. If σ is compressible, then h 0 pT σ q ‰ 0 by Lemma 2.13, thus T σ cannot be slope-stable since µpT σ q " ´1{pn ´1q.

Conversely, assume that T σ is not slope-stable, and let K ãÑ T σ be a destabilizing subsheaf; set r " rkpKq. If c 1 pKq ď ´1, then

´1 n ´1 ď c 1 pKq r ď ´1 r ðñ r ě n ´1,
which is a contradiction. It follows that c 1 pKq " 0, and Lemma 2.16 implies that σ must be compressible.

3.2.

The stability criterion for pencils of quadrics. If a pencil of quadrics σ contains no double hyperplanes, then c 1 pT σ q " ´2. Lemma 2.13 then implies that compressible pencils of quadrics containing no double hyperplanes have slopeunstable logarithmic sheaves, thus proving the third item of Theorem A. We can finally address incompressible pencils of quadrics σ " pf 1 , f 2 q containing no double hyperplanes. The result depends on the maximal corank of the quadrics Q λ , where for each λ " pz 1 : z 2 q P P 1 we write Q λ " V pz 1 f 1 `z2 f 2 q. To be precise, we prove the following result. Theorem 3.2. Let n ě 3 and let σ be an incompressible pencil of quadrics in P n containing no double hyperplane. Put r 0 " maxpcorkpQ λ q | λ P P 1 q.

i) If 2r 0 ă n `1, then T σ is slope-stable. ii) If 2r 0 " n `1, then T σ is strictly slope-semistable. iii) If 2r 0 ą n `1, then T σ is slope-unstable.
Since σ contains no double plane, we have codimpQ σ q ě 2 so the slope of T σ is:

µpT σ q " 2 1 ´n .
The proof of Theorem 3.2 will be divided in three parts. We start by establishing items ii) and iii) in Section 3.3. For the proof of item i), we first consider regular pencils in Section 3.4, leaving the case of irregular pencils for Section 3.6. We start with the following observation. Lemma 3.3. Let σ be an incompressible pencil of quadrics and let λ, µ P P 1 be distinct points such that Q λ and Q µ are singular. Then the singular loci of Q λ and Q µ are disjoint linear spaces of dimension corkpQ λ q ´1 and corkpQ µ q ´1. Proof. The singular locy of Qλ and Q µ are defined by linear equations and the corank of Qλ and Q µ is precisely the number of independent equations. In addition, these two linear spaces are disjoint, as the coordinates of a point of P n lying in the singular locus of two distinct quadrics f 1 , f 2 of the pencil would annihilate the derivatives of f 1 and f 2 , so such derivatives would fail to span H 0 pO P n p1qq. Thus we could choose coordinates so that one of the variables x 0 , . . . , x n occurs neither in f 1 nor in f 2 . However, this is excluded by the hypothesis that σ is incompressible.

3.3. Koszul subsheaves. As a preliminary step towards the proof of items ii) and iii), let R M be the first Koszul syzygy sheaf of a linear subspace M Ă P n of codimension r ą 1, namely the sheaf fitting into the following short exact sequence:

0 Ñ R M Ñ O 'r
P n p´1q Ñ I M Ñ 0. We claim that R M is slope-stable. Indeed, note that µpR M p1qq " ´1{pr ´1q; moreover, any saturated subsheaf F ãÑ R M p1q must, by the argument in the proof of Lemma 2.16, have c 1 pFq ď ´1, since h 0 pR M p1qq " 0. If F destabilizes R M p1q, then

c 1 pFq rkpFq ą ´1 r ´1 ùñ c 1 pFq ą ´rkpFq r ´1 ą ´1,
providing a contradiction.

Koszul subsheaves from singular quadrics.

A linear subspace of Ξ σ is called maximal if it is not strictly contained in another linear subspace of Ξ σ . The following technical lemma is quite useful.

Lemma 3.4. Let σ be a incompressible pencil, ̺ ě 3 be an integer and L Ă Ξ σ be a maximal linear subspace of dimension ̺ ´1. Then there is a linear subspace M Ă P n of codimension ̺ and a subscheme W Ă M such that T σ fits into

(18) 0 Ñ R M p1q Ñ T σ Ñ R L p1q Ñ I W {M p1q Ñ 0.
Proof. Since the pencil σ is incompressible, the linear forms appearing in the Jacobian matrix of σ span H 0 pO P n p1qq, hence the sheaf Q σ has rank 1 at each point of its support, in particular this happens at each point of L, so

Q σ | L is a line bundle on L, namely there is e P Z such that Q σ | L » O L peq. Since O L p1q '2 surjects onto O L peq and ̺ ě 3, we conclude that e " 1.
The surjection Q σ Ñ O L p1q allows to write the following commutative exact diagram:

0 / / M σ / / O P n p1q '2 / / Q σ / / 0 0 / / I L p1q / / O P n p1q / / O L p1q / / 0 0 0
Put F and G for the kernel and cokernel of the induced morphism M σ Ñ I L p1q, respectively; in addition, let Q 1 denote the kernel of the epimorphism

Q σ ։ O L p1q.
The snake lemma provides the following exact sequence

0 Ñ F Ñ O P n p1q Ñ Q 1 Ñ G Ñ 0,
thus there is a subscheme W Ă P n such that F » I W p1q and SupppGq Ă SupppQ 1 q Ă SupppQ σ q " pΞ σ q red . Since M σ is the image of the the Jacobian matrix O 'pn`1q

P n Ñ O P n p1q '2 ,
O '̺ P n / / I W p1q 0 / / T / / O 'pn`1q P n / / M σ / / 0 0 / / R L p1q / / O 'pn`1´̺q P n / / I L p1q / / 0 0 0 (19)
Note that the image of the morphism O '̺ P n Ñ I W p1q is the ideal sheaf of a linear subspace M Ă P n of codimension ̺ containing W , twisted by O P 3 p1q; its cokernel is the ideal of W in M , also twisted by O P 3 p1q. The snake lemma then yields the exact sequence (18).

3.3.2.

Destabilizing Kozsul subsheaves. Now we can prove items ii) and iii). Indeed, we set ̺ " r 0 and consider a quadric Q λ in the pencil σ having corkpQ λ q " r 0 . The assumption r 0 ě pn `1q{2 forces r 0 ě 3 or pn, r 0 q " p3, 2q. The latter case follows from the full classification of pencils of quadrics in P 3 and their logarithmic tangent sheaves provided in Subsection 5. Hence we can assume r 0 ě 3, so the linear space L Ă P n of dimension r 0 ´1 appearing as the singular locus of Q λ satisfies the hypotheses of Lemma 3.4, thus T σ contains the Koszul subsheaf R M p1q which has slope 1{p1 ´r0 q. The condition r 0 ą pn `1q{2 implies:

µpR M p1qq " 1 1 ´r0 ą 2 1 ´n " µpTq.
Finally, for item ii), we use the exact sequence in display (18), which yields:

(20) 0 Ñ R M p1q Ñ T σ Ñ E Ñ 0,
where E is the kernel of R L p1q ։ I W {M p1q. Since M has codimension r 0 ě 2, the sheaves E and R L p1q share the same slope, namely 1{p1 ´r0 q. This implies that any destabilizing subsheaf of E would also destabilize R L p1q, thus E is slope-stable. But if r 0 " pn `1q{2, then R L p1q also has slope equal to 1{p1 ´r0 q. Therefore, the exact sequence in display (20) shows that T is strictly slope-semistable; in addition, R M p1q and E are the factors of the Jordan-Holder filtration of T σ .

3.4. Proof of stability for regular pencils. Let σ be a regular pencil of quadrics containing no double hyperplane, so that there are only finitely many points λ P P 1 such that Q λ is singular and at each such point the singular locus of Q λ is a linear space of dimension corkpQ λ q ´1. A regular pencil is incompressible so these spaces are disjoint by Lemma 3.3, so pΞ σ q red is the union of finitely linear spaces of dimension at most r 0 ´1. In order to prove i) we assume, by contradiction, that T has a saturated destabilizing subsheaf K of rank p, with 1 ď p ď n ´2 with `Źp K ˘__ » O P n p´eq. Since σ is incompressible, Lemma 2.16 implies that e ą 0. The condition that K destabilizes T amounts to: pn ´1qe ď 2p. Since p ď n ´2, this gives e " 1. Also, we get p ě pn ´1q{2.

Choose a sufficiently general linear subspace M Ă P n of dimension n ´r0 . Since dimpM q `dimpΞ σ q " n ´1, we may assume that M is disjoint from the degeneracy locus Ξ σ and that M meets transversely the locus where T{K is not locally free. The second assumption implies that Tor 1 pT{K, O M q " 0, so we get a subsheaf K| M ãÑ T| M which still destabilizes T| M . The first assumption yields Q σ | M " 0 " Tor 1 pM σ , O M q, so the restricted Jacobian matrix gives an exact sequence:

(21) 0 Ñ T| M Ñ O 'pn`1q M Ñ O '2 M p1q Ñ 0.
The sheaf T| M is locally free and, setting q " n ´1 ´p we get:

´p ľ T| M ¯p1q » ´q ľ T _ | M ¯p´1q.
Since K| M is a subsheaf of T| M , we obtain a monomorphism

O M p´1q » `p ľ K| M ˘__ ãÑ `p ľ T| M ˘__ » `p ľ T| M which
in turn gives H 0 ´´Ź q T _ | M ¯p´1q ¯‰ 0. We need to prove that this is absurd.

In order to check this, we dualize the exact sequence in display (21) and take exterior powers to get a long exact sequence:

0 Ñ O M p´q ´1q 'pq`1q Ñ O M p´qq 'qpn`1q Ñ ¨¨¨Ñ ´q ľ T _ | M ¯p´1q Ñ 0.
All of the terms in the sequence above, except for the rightmost one, are copies of O M p´tq for some integer t with 1 ď t ď q `1. In the range p ě pn´1q{2, we have q " n ´1 ´p ď pn ´1q{2 so q `1 ď pn `1q{2. Now, the assumption r 0 ă pn `1q{2 guarantees pn `1q{2 ă n ´r0 `1 " dimpM q `1, thus q ă dimpM q. Therefore H ˚pO M p´tqq " 0 for all 1 ď t ď q `1 and hence H 0 ´´Ź q T _ | M ¯p´1q ¯" 0. This is the contradiction we were looking for, thus proving i).

This finishes the proof of Theorem 3.2 for regular pencils.

3.5. Irregular pencils of quadrics. The goal of this section is to set up some basic analysis of irregular pencils that is necessary for the proof of item i) for irregular pencils of quadrics. In particular, we study in detail a special kind of pencil that we call completely irregular, which is actually the only kind of irregular pencils where the proof given in Subsection 3.4 fails.

3.5.1.

The regular part of an irregular pencil. Considering the polarization (or Hessian) matrix of the quadrics in the pencil σ we obtain a pencil of symmetric matrices of size n `1:

ρ σ : O P 1 p´1q 'pn`1q Ñ O 'pn`1q P 1 .
Definition 3.5. The splitting type of a pencil of quadrics σ is the unique pair of integers pu, vq, such that impρ σ q » O P 1 p´1q 'u ' O 'v P 1 . Note that σ is regular if and only if ρ σ is injective, if and only the splitting type of σ is pn `1, 0q. In addition, if σ is incompressible, then cokerpρ σ q contains no trivial direct summand. Set r 1 :" min corkpQ λ q | λ P P 1 ( . We call r 1 the generic corank of P 1 . We have:

r 1 " n `1 ´rkpρ σ q " n `1 ´u ´v.
Set K :" kerpρ σ q, F :" impρ σ q, C σ :" cokerpρ σ q and write the long exact sequence:

(22) 0 Ñ K Ñ O P 1 p´1q 'pn`1q Ñ O 'pn`1q P 1 Ñ C σ Ñ 0.
The cokernel sheaf C σ decomposes as C σ » C tf ' C t , where C t is its torsion part and C tf is its torsion free part. Lemma 3.6. If σ has splitting type pu, vq then u ě v and there is a regular pencil of symmetric matrices ρσ with size u ´v satisfying cokerpρ σ q » C t .

Proof. Using that ρ σ is symmetric and that C t is zero-dimensional, we get:

C tf » C __ σ » K _ p´1q, C t » Ext 1 P 1 pC σ , O P 1 p´1qq
. Therefore, dualizing the above sequence we get:

0 Ñ C _ σ p´1q Ñ O P 1 p´1q 'pn`1q α Ñ F _ p´1q Ñ C t Ñ 0, 0 Ñ F _ p´1q β Ñ O 'pn`1q P 1 Ñ K _ p´1q Ñ 0. (23)
Again by the symmetry of ρ σ , the following composition of morphisms

O P 1 p´1q 'pn`1q α Ñ F _ p´1q β Ñ O 'pn`1q P 1
coincides with ρ σ , thus the image of the morphism α in display (23) is precisely F.

We get an injective twisted endomorphism ρ σ of the vector bundle F and an exact sequence:

0 Ñ F ρ σ ÝÑ F _ p´1q Ñ C t Ñ 0.
This endomorphism is also symmetric. Let us rewrite this in terms of the splitting type:

0 Ñ O P 1 p´1q 'u ' O 'v P 1 ρ σ ÝÑ O P 1 p´1q 'v ' O 'u P 1 Ñ C t Ñ 0. The morphism O 'v P 1 Ñ O 'u P 1 extracted from ρσ is thus injective. Therefore its transpose O P 1 p´1q 'u Ñ O P 1 p´1q 'v is surjective -in particular we must have v ď u.
These constant maps of maximal rank can be expunged from ρσ so that ρ σ reduces to a symmetric morphism, still denoted by ρ σ , which takes the form:

ρ σ : O P 1 p´1q 'pu´vq ãÑ O 'pu´vq P 1 .
The torsion part C t is a sheaf of length h 0 pC t q " u ´v which is supported at the points λ P P 1 such that corkpQ λ q ą r 1 . We obtain the exact sequence:

(24) 0 Ñ O P 1 p´1q 'pu´vq ρ σ ÝÑ O 'pu´vq P 1 Ñ C t Ñ 0.
Definition 3.7. The pencil of quadrics associated with the pencil of matrices ρσ given by the previous lemma is called the regular part of σ.

Note that the regular part of a pencil of quadrics may be empty: this happens for u " v. We will shall call these pencils completely irregular and treat them in detail a bit further on. Note also that the regular part of an irregular pencil may fail to be a pencil, namely when u " v `1. By convention, a pencil of symmetric matrices of size 0 (the empty pencil) and a non-zero pencil of symmetric matrices of size 1 are regular.

3.5.2. Recovering the pencil. The torsion-free part C tf of C σ decomposes as a sum of ample line bundles. Namely, there are integers c 1 , . . . , c r1 such that:

C tf » r1 à i"1 O P 1 pc i q, 1 ď c 1 ď ¨¨¨ď c r1 , r1 ÿ i"1 c i " v.
We call c " pc 1 , . . . , c r1 q the degree vector of σ. The following result allows to recover a pencil from its regular part and the degree vector.

Proposition 3.8. Let n 1 P N and let ρ be a regular pencil of symmetric matrix of size n 1 . Fix integers r 1 and 1 ď c 1 ď ¨¨¨ď c r1 . Then, up homography, there is a unique incompressible pencil σ such that C σ » C t ' C tf with:

(25) C tf » r1 à i"1 O P 1 pc i q, C t » cokerpρq, ρσ " ρ.
Proof. According to the previous subsection, we put:

(26) v " r1 ÿ i"1 c i , u " n 1 `v, n " r 1 `u `v ´1.
For all m P N ˚we consider the exact sequence:

(27) 0 Ñ O P 1 p´1q 'm τm Ý Ý Ñ O 'pm`1q P 1 Ñ O P 1 pmq Ñ 0,
given by the mth symmetric power of the Euler sequence on P 1 . The matrix τ m is unique up to a coordinate change on the source and target and up to homography of P 1 . We consider the block-diagonal matrix formed by the morphisms τ ci for all i P t1, . . . , r 1 u. In view of the definition of v this gives:

r1 à i"1 τ ci : O P 1 p´1q 'v Ñ O 'pv`r1q P 1 .
Stacking this morphism together with its transpose and with ρ, in view of our definition of n we get a symmetric matrix pencil of the form:

(28) r1 à i"1 pτ ci ' τ t ci q ' ρ : O P 1 p´1q 'pn`1q Ñ O 'pn`1q P 1
.

We obtain a pencil of quadrics σ on P n . By construction σ is incompressible and satisfies the conditions in display (25). About uniqueness of σ, we argue as follows. First note that, under the assumption of incompressibility, the dimension n and the splitting type pu, vq are determined by r 1 , c 1 , . . . , c r1 through the equalities in display (26). Next, we observe that for any pencil σ satisfying (25), the direct sum decomposition of C σ gives: obtained by concatenating minimal presentation matrices of O P 1 pc 1 q, . . . , O P 1 pc r1 q. In other words, this second block is τ c1 ' ¨¨¨' τ cr 1 . Up to choosing coordinates of P n adapted to this decomposition, the above pencil appears in a block decomposition of ρ σ and by symmetry of ρ σ the residual block must be τ t c1 ' ¨¨¨' τ t cr 1 . So σ is obtained as in (28). 3.5.3. The Jacobian scheme of an irregular pencil. We continue with the assumption that σ is an incompressible irregular pencil and seek a set-theoretic description of degeneracy scheme Ξ σ .

0 Ñ O 'pu´vq P 1 p´1q ' O 'v P 1 p´1q Ñ O 'pu´vq P 1 ' O 'pv`r1q P 1 Ñ C t ' C tf Ñ 0.
The torsion free part C tf defines a projective bundle Y " PpC tf q and, since the vector bundle C tf is very ample, Y " PpC tf q embeds via the linear system of the tautological relatively ample divisor h as a rational normal scroll of degree v, spanning a linear space L Ă P n of dimension n ´u. Lemma 3.9. Let σ be an incompressible pencil of quadrics. Then Ξ σ satisfies:

(29) pΞ σ q red " Y Y ď λPSupppCtq P r λ ´1,
where the linear subspaces tP r λ ´1 | λ P SupppC t qu are disjoint. In particular:

dimpΞ σ q " maxpr 0 ´1, r 1 q, r 1 " n `1 ´u ´v.
Proof. We look at the projectivization of the vector bundle C tf and of the coherent sheaves C t , C σ The epimorphisms O

'pn`1q

P 1 ։ C σ , O 'pn`1q P 1 ։ C t ,

and O

'pn`1q P 1

։ C tf induce embeddings PpC σ q ãÑ P 1 ˆPn , PpC t q ãÑ P 1 ˆPn and Y ãÑ P 1 ˆPn . Similarly, the epimorphism O '2 P n ։ Qp´1q induces an embedding PpQ σ p´1qq ãÑ P 1 ˆPn . The two subschemes PpC σ q and PpQ σ p´1qq of P 1 ˆPn are defined by the same bihomogeneous equations. Indeed, denoting by λ " pz 1 : z 2 q and x " px 0 : . . . : x n q the points of P 1 and P n and recalling the notation f i,j " Bfipxq Bxj , we have: PpQ σ p´1qq " PpC σ q " tppx, λq P P 1 ˆPn | f 1,j z 1 `f2,j z 2 " 0, @j " 0, . . . , nu, which in turn gives a Kozsul complex (in the obvious notation):

(30) ¨¨¨Ñ O P 1 ˆPn p´1, ´1q 'pn`1q Ñ O P 1 ˆPn Ñ O PpCσq Ñ 0.
We get thus a correspondence:

PpC σ q ϕ { { ① ① ① ① ① ① ψ # # • • • • • • P 1 P n (31)
where the map ϕ : PpC σ q Ñ P 1 is generically a P r1´1 -bundle and ψ : PpC σ q Ñ Ξ σ Ă P n is an isomorphism at the points where Q σ has rank 1.

At each point λ of the support of the torsion part C t we have a skyscraper sheaf supported at λ, whose rank we denote by r λ . The surjection O 'pn`1q P 1 Ñ O 'r λ λ induces an embedding P r λ ´1 Ă Ξ σ Ă P n . We noticed in Lemma 3.3 that the linear spaces appearing as singular loci of distinct points of SupppC t q are disjoint. This achieves the proof.

In addition, we will also use the following statement later on. Lemma 3.10. An incompressible pencil of quadrics satisfies r 1 ď n`1 3 .

Proof. We assume that σ is an incompressible pencil of quadrics. Recall that σ has a splitting type pu, vq with u ě v. Since σ is incompressible, the integers c 1 , . . . , c r1 appearing in the degree vector are strictly positive, hence:

v " r1 ÿ i"1 c i ě r 1 .
Therefore we have n `1 " r 1 `u `v ě r 1 `2v ě 3r 1 .

3.5.4. Completely irregular pencils. We say that the incompressible pencil σ is completely irregular if has no regular part, which is to say, if u " v. This is equivalent to the condition C t " 0, which in turn is tantamount to r 0 " r 1 .

We take a closer look at the degeneration scheme in this case. Denote by F the divisor class of a fibre of Y Ñ P 1 , so O Y pF q » ϕ ˚pO P 1 p1qq, in the notation of the diagram in display (31); write also H " c 1 pψ ˚pO P n p1qqq. Note that ψ ˚pO Y pF qq » Q σ and that the Koszul complex (30) is exact at O P 1 ˆPn . Tensoring it with O P 1 ˆPn p1, 0q and applying ψ ˚, we get an exact sequence:

O 'pn`1q P n p´1q Ñ O '2
P n Ñ O Y pF q Ñ 0. The rightmost morphism above agrees with the Jacobian matrix, so we have an exact sequence:

(32) 0 Ñ T σ Ñ O 'pn`1q P n Jσ Ñ O P n p1q '2 Ñ O Y pH `F q Ñ 0.
Example 3.11. Let us list all the possible splitting types pu, vq for n " 5 together with the data of the scroll, compressibility and so forth. Regular pencils give C σ " C t , a finite-length scheme with h 0 pC σ q " 6. For irregular pencils we have the following possibilities. Observe that there is only one incompressible, completely irregular pencil of quadrics in P 5 ; moreover, n " 5 is the lowest dimension in which such pencils can occur.

Lemma 3.12. Let σ be an incompressible completely irregular pencil. Then there is an action of SL 2 " SL 2 pκq on P n for which the exact sequence in display (32) is equivariant.

Proof. Let the group SL 2 act by homographies on P 1 regarded as the base of the pencil. For any integer m P N we let V m be the irreducible representation SL 2 of weight m, so V 0 is the trivial representation of rank 1, V 1 is the standard representation of rank 2, while V m " S m V 1 has rank m `1. By convention we set V ´1 " 0. For all m P N ˚we rewrite (27) as in SL 2 -equivariant form:

0 Ñ V m´1 b O P 1 p´1q τm Ý Ý Ñ V m b O P 1 Ñ O P 1 pmq Ñ 0.
Given an incompressible, completely irregular pencil σ, the matrix pencil σ is formed by stacking the morphisms τ ci with their transpose for all i P t1, . . . , n 1 ´2uu whereby obtaining a symmetric block matrix. Thus we rewrite the exact sequence in display (22) in equivariant form:

0 Ñ n`1´2u à i"1 O P 1 p´1 ´ci q Ñ n`1´2u à i"1 pV ci´1 ' V ci q b O P 1 p´1q ρ Ý Ñ ρ Ý Ñ n`1´2u à i"1 pV ci´1 ' V ci q b O P 1 Ñ n`1´2u à i"1 O P 1 pc i q Ñ 0.
Since the matrix pencil ρ is SL 2 -equivariant, so is the pencil σ obtained by ρ. Hence the Jacobian matrix of σ is also SL 2 -equivariant and this induces an SL 2 -action on its kernel and cokernel sheaves. Since the pencil σ is determined by the degree vector pc 1 , . . . , c n`1´2u q by Proposition 3.8, we get that this construction holds for any incompressible completely irreducible pencil.

Note that the SL 2 -action on O Y pH `f q is simply the action by homographies on the basis of the scroll Y . Also, the SL 2 -action on P n induces the isomorphism of SL 2 -modules:

(33) H 0 pP n , O P n p1qq " n`1´2u à i"1 pV ci´1 ' V ci q.
The following lemma says that, if σ is a completely irregular pencil of quadrics, then the logarithmic sheaf T σ is simple. Lemma 3.13. Let σ be a completely irregular incompressible pencil of quadrics with u ě 2. Then End P n pT σ q » κ.

Proof. We use the exact sequence in display (32). We first use its rightmost part, namely:

(34) 0 Ñ M σ Ñ O P n p1q '2 Ñ O Y pH `F q Ñ 0.
Since O Y is a line bundle on the smooth irreducible variety Y , we have End X pO Y q » κ. Since the morphism O '2 P n Ñ O Y pF q induces an isomorphism on global sections and H p pO Y pF qq " 0 for p ą 0, we get H ˚pM σ p´1qq " 0. Also, Ext p P n pO Y pF `hq, O P n p1qq " 0 for p " 0, 1 by Serre duality since dimpY q " n `1 ´2u ă n ´1. Therefore applying Hom P n p´, M σ q and Hom P n pO Y pH `F q, ´q to the exact sequence in display (34) we get: End P n pM σ q » Ext 1 P n pO Y pF `hq, M σ q » End P n pO Y q » κ. Also, note that Serre duality gives Ext p P n pO Y pF `hq, O P n q " 0 for p " 1, 2 as dimpY q " n `1 ´2u ă n ´2. We deduce that Ext p P n pM σ , O P n q " 0 for p " 0, 1. Next, we write the exact sequence:

(35) 0 Ñ T σ Ñ O 'pn`1q P n Ñ M σ Ñ 0.
Applying Hom P n pM σ , ´q to (35) and using that Ext p P n pM σ , O P n q " 0 for p " 0, 1, we get:

Ext 1 P n pM σ , T σ q » End P n pM σ q » κ.
Finally, we apply Hom P n p´, T σ q to (35) and use that, since σ is incompressible, we have H 0 pT σ q " H 1 pT σ q " 0. This concludes the proof, since:

End P n pT σ q » Ext 1 P n pM σ , T σ q » κ.
3.6. Proof of stability for irregular pencils of quadrics. It only remains for us to prove item i) for irregular incompressible pencils σ containing no double hyperplane.

By hypothesis, we have 2r 0 ă n `1. As in Subsection 3.4, we assume by contradiction that T admits a saturated subsheaf K of rank p and again we get p ě pn ´1q{2 and p Ź p Kq __ » O P n p´1q. Next, observe that, if r 1 ă r 0 , then by Lemma 3.9 we have dimpΞ σ q ď r 0 ´1. In this case, the proof of i) given in Subsection 3.4 goes through as again a sufficiently general linear subspace M Ă P n of dimension n ´r0 does not meet Ξ σ , while the rest of the argument is still valid for irregular pencils.

Therefore, we may assume until the end of the subsection that r 0 " r 1 , which is to say, that the pencil is completely irregular, so u " v, C σ " C tf and dimpΞ σ q " r 0 .

This time, with a slight difference with respect to the proof of item i) given in Subsection 3.4, we choose a general linear subspace M Ă P n of dimension n ŕ0 ´1. In particular, we may assume that M does not meet Ξ σ and that the p th exterior power of K| M ãÑ T| M gives a non-zero element of H 0 p Ź q T _ | M p´1qq, with q " n ´1 ´p; this equality q " n ´1 ´p comes from duality of the sheaf T, which is of rank n ´1.

If n is odd, we write n " 2n 0 `1 and 2r 0 ă n `1 gives r 0 ď n 0 , while p ě pn ´1q{2 " n 0 gives q " 2n 0 ´p ď n 0 . Since dimpM q " 2n 0 ´r0 ě n 0 we get q ă dimpM q unless dimpM q " n 0 . If q ă dimpM q, again the argument given in Subsection 3.4 remains valid, so we may assume, without loss of generality, that dimpM q " n 0 . It then follows that n 0 " dimpM q " 2n 0 ´r0 , thus n 0 " r 0 ; since r 0 " n `1 ´2u, n 0 is even, say n 0 " 2n 1 , and u " n 1 `1. Summing up, if n is odd, then:

(36)

n " 4u ´3, dimpM q " p " q " r 0 " 2pu ´1q, u ě 2.

Similarly, if n " 2n 0 , then we get that r 0 ď n 0 ď p and thus q ď n 0 ´1 ď dimpM q, so we may assume that dimpM q " n 0 ´1. It follows that n 0 " r 0 " n `1 ´2u so n 0 is odd; setting n 0 " 2n 1 `1, we obtain u " n 1 `1. Summarizing, if n is even, then:

(37) n " 4u ´2, dimpM q " q " 2pu ´1q, r 0 " p " 2u ´1, u ě 2.

In any case, the sheaves K and L " T{K are slope-stable.

Having established these numerical constraints, we proceed with the next step of the proof, which requires looking at the exact sequence in display (32). Working in the linear span L " P n`1´u Ă P n of Y , we write an exact commutative diagram

0 0 I Y {L p1q '2 I Y {L p1q '2 0 / / M L / / O L p1q '2 / / O Y pH `F q / / 0 0 / / O Y ph ´F q / / O Y phq '2 / / O Y pH `F q / / 0 0 0
Here, the sheaf M L , defined by the middle row, can be though of as the normal sheaf associated to the pencil of quadrics restricted to the smaller space L. Using the leftmost column of the previous diagram and the fact that the morphism O L p1q '2 Ñ O Y pH`F q in the exact sequence in display (32) factors through O L p1q '2 Ñ O Y pHF q, we get an exact sequence:

0 Ñ I L{P n p1q '2 Ñ M σ Ñ M L Ñ 0.
Using this exact sequence and the one in display (32) we get a second exact commutative diagram:

0 0 0 0 / / R L p1q '2 / / O '2u P n / / I L{P n p1q '2n / / 0 0 / / T / / O 'pn`1q P n / / M σ / / 0 0 / / G / / O 'pn`1´2uq P n / / M L / / 0 0 0 0
Here the sheaf G, defined with the bottom row, has c 1 pGq " 0 and rkpGq " n`1´2u, while R L p1q is the Koszul syzygy of L which we already proved to be stable of slope 1{p1 ´uq.

In view of the leftmost column of the previous diagram and of the slopesemistability of R L p1q '2 , the inclusion K ãÑ T must descend to an inclusion K ãÑ G. We get thus two injections with isomorphic cokernel:

(38) K ãÑ G, R L p1q '2 ãÑ L.
Denote by P this common cokernel sheaf, so P » G{K » L{R L p1q '2 . Then, note that independently on whether n is even or odd, we get:

rkpKq " p " n `1 ´2u " rkpGq.

Therefore, since p Ź p Kq __ » O P n p´1q, we get a hyperplane H Ă P n as support P. Note that the hyperplane H is determined uniquely by T. Indeed, if K 1 ãÑ T is an embedding of any saturated destabilizing subsheaf of T, then the induced morphism K 1 Ñ L is either zero or an isomorphism, since K 1 and L are stable of the same slope. If K 1 Ñ L is zero then K 1 Ñ T factors through K so it determines the same hyperplane H. If K 1 Ñ L is an isomorphism, then T is decomposable, which is absurd by Lemma 3.13. Now, recall from Lemma 3.12 that T σ is equivariant for a natural SL 2 -action on P n . So the hyperplane H must be fixed by this action, in other words, it must correspond to a trivial summand V 0 in the decomposition in display (33).

Set t for the number of indices i P t1, . . . , n `1 ´2uu such that c i " 1, so:

(39) u " t `n`1´2u ÿ i"t`1 c i ě 2pn ´2u ´t `1q `t.
If t ě 3, then we can equip Y (and consequently T σ ) with a further SL 2 -action by letting SL 2 operate as V t´1 b O P 1 p1q on the summands of C σ of the form O P 1 p1q. Again we obtain that H 0 pO P n p1qq contains no copy of V 0 . In all these cases the SL 2 -fixed hyperplane H cannot exist and we conclude that T σ is stable.

Finally if t ď 2 then using (39), depending on whether n is odd or even, we get from (36) or (37) that u ď 2 or u ď 1, which leaves the only case n " 5, u " 2, c 1 " c 2 " 1. This last case corresponds to the pencil of quadrics σ " px 1 x 5 `x3 x 4 , x 2 x 4 `x0 x 5 q. For this explicit pencil, direct computation shows that H 0 ´`Ź 2 T σ ˘_p´1q ¯" 0 so that T σ is stable. Remark 3.14. One may check that GIT-semistability of σ implies slopesemistability of T σ . Indeed, [1, Theorem 3.1] says that GIT-semistability of σ amounts to σ being regular with C σ supported at λ 1 , . . . , λ ℓ with the condition that for all j P t1, . . . , ℓu, λ j has multiplicity at most pn `1q{2 as a root of detpρ σ q. In terms of the Segre symbol (see the introduction or the next paragraph), this means that ř sj i"1 a j,i p j,i ď pn `1q{2. This implies ř sj i"1 p j,i ď pn `1q{2, which amounts to r λj ď pn `1q{2, so T σ is slope-semistable, as r 0 " maxt ř sj i"1 p j,i | j P t1, . . . , ℓuu. However, the converse implication fails as one can see reverting the argument or considering that there are irregular pencils σ having a slope-semistable sheaf T σ .

Projective dimension for pencils of quadrics

Also in this section, κ is an algebraically closed field of characteristic different from 2.

4.1. Segre symbols. Let σ be a pencil of quadrics. Following the notation introduced in Section 3.5.1, let pu, vq be the splitting type of σ, so that its generic corank r 1 satisfies n `1 " u `v `r1 . Let tλ 1 , . . . , λ ℓ u Ă P 1 be the support of the torsion sheaf C t . Its localization at λ j of C t can be written in the following way:

pC t q λj » sj à i"1 O 'pj,i λ pa j,i q j .
Here, we denoted by λ paj,iq j the subscheme defined by the ideal pg aj,i j q where g j is a linear form vanishing at λ j for each j P t1, . . . , ℓu.

The integers a j,i can then be arranged into the so called Segre symbol. We write, for each j P t1, . . . , ℓu: Σ j " pa j,1 , . . . , a j,1 loooooomoooooon pj,1 , . . . , a j,sj , . . . , a j,sj looooooomooooooon pj,s j q, with a j,1 ą ¨¨¨ą a j,sj .

The Segre symbol Σ for a pencil of quadrics σ is defined to be the multi-set rΣ 1 , . . . , Σ ℓ s. We will use exponential notation for repeated entries; for instance, the Segre symbol rp1, 1, 1q, p3, 3, 1q, 2s in exponential notation reads r1 3 , p3 2 , 1q, 2s.

Note that, as we are dealing with potentially irregular pencils σ, we always refer to the Segre pencil of the regular part σ of σ. In case σ is completely irregular, its Segre symbol is H by convention. Note that, in contrast to the behavior for regular pencils, the Segre symbol of an irregular pencil may be of the form r1 p s, for some integer p.

The Segre symbol is a the key invariant of regular pencils. Indeed, the content of the Segre-Weierstrass theorem is that the set of singular quadrics of a regular pencil together with its Segre symbol classifies the regular pencil up to homography of P 1 and P n . For a reference of this theorem going back to Corrado Segre's thesis, the reader may look at the classical textbook [7, §XIII.10] or at the more recent paper [6]. This classical result is extended to irregular pencils as follows.

Proposition 4.1. Given ℓ, r 1 P N and integers 1 ď c 1 ď . . . ď c r1 , fix distinct points tλ 1 , . . . , λ ℓ u Ă P 1 , and a multiset Σ " rΣ 1 , . . . , Σ ℓ s. Then there is an incompressible pencil of quadrics σ, unique up to homography, such that:

C tf » r1 à i"1 O P 1 pc i q,
SupppC t q " tλ 1 , . . . , λ ℓ u, Σpρ σ q " Σ.

Proof. In view of the Segre-Weierstrass theorem, there is a regular pencil, and thus a regular symmetric pencil of matrices, uniquely defined by the datum of the settheoretic support of C t together with the Segre symbol. Therefore the result follows from Proposition 3.8.

Note that p j,i ď n for all indices pi, jq. The splitting type pu, vq satisfies:

(40)

ℓ ÿ j"1 sj ÿ i"1
a j,i p j,i " h 0 pC t q " u ´v, according to the exact sequence in display (24).

Ext sheaves.

The main result of this section provides necessary and sufficient conditions in terms of Σ for the Ext sheaves Ext q P n pT σ , O P n q to be non trivial. Theorem 4.2. For a pencil of quadrics σ with Segre symbol Σ and for q ą 0, we have Ext q P n pT σ , O P n q ‰ 0 if and only if there are j P t1, . . . , ℓu and k P t1, . . . , s j u such that:

(41) q `pj,1 `. . . `pj,k " n ´r1 ´1,
or r 1 ą 0 and q `r1 " n ´2.

Proof. We prove the theorem under the assumption that σ is incompressible, see the end of the proof for compressible pencils. Since the question is local and T σ is free of rank n ´1 locally around any point outside Ξ σ , it is enough to prove the claim on charts containing a single primary component of Ξ σ . In view of Lemma 3.9, these components are supported either at disjoint linear subspaces associated with distinct points in the support of C t , or at the rational normal scroll Y . Note that the points of the support of C t correspond to the parenthesized pieces of the Segre symbol. Also, the proof at the points of Y is similar if the support of C t contains one point or many. So we may assume, without loss of generality, that ℓ " 1 and simplify the notation to Σ " pa p1 1 , . . . , a ps s q with a 1 ą ¨¨¨ą a s ą 0. As we did in the proof of item i) of Theorem 3.2 (see Section 3.4), we observe that the sheaf Q σ is a line bundle supported at Ξ σ . Therefore, given q ą 0, we have Ext q`2 P n pQ σ , O P n q ‰ 0 if and only if Ext q`2 P n pO Ξσ , O P n q ‰ 0, so this in turn is equivalent to Ext q P n pT σ , O P n q ‰ 0. Let us analyze Ξ σ more in detail and recall the notation of Subsection 3.5.3. Since we are working under the assumption ℓ " 1 the sheaf C t , if nonzero, is supported at a single point of P 1 and we may fix coordinates so that this point is λ " p0 : 1q. In such coordinates and up to the action of GL n`1 pκq, the pencil of matrices ρ " ρσ of the regular part of σ is a block-diagonal matrix with blocks of sizes a 1 , . . . , a s , repeated p 1 , . . . , p s times, where a block of size a P N ˚takes the form (below, pz 1 , z 2 q is a basis for H 0 pO P 1 p1qq):

ρ a " ¨0 0 ¨¨¨0 z 2 z 1 0 ¨¨¨0 z 2 z 1 0 . . . . . . . . . . . . 0 . . . 0 z 2 z 1 0 . . . . . . z 2 z 1 0 . . . . . . z 1 0 ¨¨¨. . . ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ '
.

The cokernel of this matrix is the structure sheaf of O λ paq , where λ paq is the a-tuple point of P 1 defined by the ideal pz a 1 q. Therefore, we have:

C t » s à i"1 O 'pi λ pa i q .
For each i P t1, . . . , su, we consider the injection λ paiq Ă λ pasq . Concatenating the surjections O λ pas q Ñ O λ pa i q we get an epimorphism:

C t ։ s à i"1 O 'pi λ pas q .
For k P t1, . . . , su, put q k " ř k i"1 p i . From the above epimorphism we get the exact sequence:

0 Ñ s´1 à i"1 O 'pi λ pa i ´as q Ñ C t Ñ O 'qs λ pas q Ñ 0.
Iterating this procedure we obtain a natural filtration:

0 " D p0q Ă D p1q Ă ¨¨¨Ă D psq " C t ,
where, for all k P t1, . . . , su, we have (with the convention a s`1 " 0):

D pkq " k à i"1 O 'pi λ pa i ´ak`1 q , C pkq :" D pkq {D pk´1q " O 'q k λ pa k ´ak`1 q .
To make the proof more transparent, we carry it out first under the assumption that σ is regular, hence ρ " ρ σ , r 1 " 0, pu, vq " pn `1, 0q, C tf " 0 and C t " C σ . For any k P t1, . . . , su, projectivizing the surjection D pkq ։ C pkq we get a closed embedding PpC pkq q ãÑ PpD pkq q of schemes of the same dimension, with a residual subscheme of PpC pkq q in PpD pkq q which is isomorphic to PpD pk´1q q and has thus strictly smaller dimension -by convention, PpD p0q q " H. Note that, for each k P t1, . . . , su, the exact sequence 0 Ñ I PpC pkq q{PpD pkq q Ñ O PpD pkq q Ñ O PpC pkq q Ñ 0 induces an exact sequence : 0 Ñ H 0 pI PpC pkq q{PpD pkq q q Ñ H 0 pO PpD pkq q q Ñ H 0 pO PpC pkq q q Ñ 0. Thus, since I PpC pkq q{PpD pkq q is supported at PpD pk´1q q, we have an isomorphism H 0 pI PpC pkq q{PpD pkq q q » H 0 pO PpD pk´1q q q, which in turn implies that I PpC pkq q{PpD pkq q is isomorphic to O PpD pk´1q q .

Recalling the correspondence (31), we send this filtration to P n and define, for each k P t1, . . . , su the subschemes Ξ pkq " ψpPpD pkq q ˆP1 PpC σ qq Ă P n and Υ pkq " ψpPpC pkq q ˆP1 PpC σ qq Ă P n . Since ψ is an embedding on the fibres of ϕ, this gives Υ pkq Ă Ξ pkq for each k P t1, . . . , su and finally a stratification:

0 " O Ξ p0q Ă O Ξ p1q Ă ¨¨¨Ă O Ξ ps´1q Ă O Ξ psq " O Ξσ , with : (42) O Ξ pkq {O Ξ pk´1q » O Υ pkq , (43)
for all each k P t1, . . . , su, with Ξ p1q " Υ p1q . Each component Υ pkq is the projectivization of a trivial bundle of rank q k over a subscheme of length a k in P 1 . As such, it is equidimensional and Cohen-Macaulay of codimension n ´qk `1 and therefore satisfies:

(44) Ext q`2 P n pO Υ pkq , O P n q » " 0, if q `2 ‰ n ´qk `1, ω Υ pkq pn `1q, if q `2 " n ´qk `1.
So this sheaf is non zero if and only if q " n ´p1 ´¨¨¨´p k ´1, which in turn agrees with (41).

We apply Ext Pn p´, O P n q to the filtration (42). To compute this we use (43) and induction on k P t0, . . . , su. Since for all k P t1, . . . , su the sheaves (44) are line bundles supported on subschemes sharing no common component, the boundary morphisms induced by applying Ext Pn p´, O P n q to (42) are all zero. We deduce that Ext q`2 P n pO Ξσ , O P n q ‰ 0 if and only if there is k P t1, . . . , su such that (41) is satisfied. This concludes the proof when σ is regular. Now let us assume that σ is irregular. Restricting C tf to each of the subschemes λ pa1q , . . . , λ pasq , we obtain the sheaves:

(45) Dpkq " D pkq ' C tf p´a k`1 q, Ĉpkq " Dpkq { Dpk´1q " O 'pq k `r1q
λ pa k ´ak`1 q , with the filtration:

(46) C tf p´a 1 q " Dp0q Ă Dp1q Ă ¨¨¨Ă Dpsq " C σ .
Again, for any k P t1, . . . , su, we get an embedding Pp Ĉpkq q ãÑ Pp Dpkq q of schemes of the same dimension. The residual subscheme is Pp Dpk´1q q and has strictly smaller dimension -this time Pp Dp0q q " Y . The component Y of Ξ σ is a rational normal scroll over P 1 . We denote by F the divisor class of a fibre of the scroll map Y Ñ P 1 .

Using the diagram (31) we define, for each k P t1, . . . , su the subschemes Ξpkq " ψpPp Dpkq q ˆP1 PpC σ qq Ă P n and Υpkq " ψpPp Ĉpkq q ˆP1 PpC σ qq Ă P n . We get Υpkq Ă Ξpkq . Note that ψ ˚pϕ ˚pO P 1 p1qqq » O Y pF q and that F | Υ pkq " 0. Hence, in view of (45) we obtain for each k P t1, . . . , su an exact sequence:

(47) 0 Ñ O Ξpk´1q p´a k F q Ñ O Ξpkq Ñ O Υpkq Ñ 0.
With our convention, Y " Ξp0q " Υp0q so for k " 1 the leftmost term of the above sequence is O Ξp0q p´a 1 F q » O Y p´a 1 F q.

We have obtained a stratification of Ξ σ that allows to compute the desired Ext sheaves. Indeed, to compute Ext q`2 P n pO Ξσ , O P n q for q ą 0 we apply Ext Pn p´, O P n q to (47) and use induction on k P t0, . . . , su together with twists by O Y ptF q for suitable t P Z. For k " 0 we observe that, since Y Ă P n is smooth of codimension n ´r1 , for any t P Z we have Ext q`2 P n pO Y ptF q, O P n q ‰ 0 if and only if q " n ´r1 ´2. For k ě 1, Υpkq is the projectivization of a trivial bundle of rank q k `r1 over a subscheme of length a k in P 1 , so:

(48)

Ext q`2 P n pO Υpkq , O P n q ‰ 0 if and only if q " n ´r1 ´qk ´1, and this sheaf is ω Υpkq pn`1q if q " n´r 1 ´qk ´1. Again, since these sheaves are line bundles on Υp0q , . . . , Υpsq and since these subschemes have no common component, we have the vanishing of all the boundary morphisms of the long exact sequence obtained by applying Ext Pn p´, O P n q to (47). Therefore, Ext q`2 P n pO Ξσ , O P n q ‰ 0 if and only if q " n ´r1 ´p1 ´¨¨¨´p k ´1 for some k P t1, . . . , su or q " n ´r1 ´2. This concludes the proof if σ is incompressible.

Finally, if σ has compressibility m with 1 ď m ď n, then we set n " n ´m as in Lemma 2.14 and work with the incompressible pencil of quadrics σ in P n associated with σ. We obtained already a stratification of Ξ σ Cohen-Macaulay closed subschemes of P n which are projective bundles over subschemes of P 1 , or the scroll Y . The equations of these subschemes, viewed in P n define cones over such subschemes, which are still Cohen-Macaulay of the same codimension. Therefore, for all q ą 0, we have Ext q P n pT σ , O P n q ‰ 0 if and only if Ext q P n pT σ , O P n q ‰ 0. This concludes the proof.

Let us give a couple of explicit examples to show the stratification appearing in the proof of the theorem.

Example 4.3. Consider a regular pencil with Segre symbol rp6 3 , 3 4 , 2 3 qs, so that ℓ " 1, s " 3, and pa 1 , a 2 , a 3 q " p6, 3, 2q, pp 1 , p 2 , p 3 q " p3, 4, 3q. We have a torsion sheaf C σ " cokerpρq with h 0 pC σ q " 18 `12 `6 " 36 " n `1, so n " 35 and ρ " ρ '3 6 ' ρ '4 4 ' ρ '3 2 . We have:

C σ " D p3q " O '3 λ p6q ' O '4 λ p3q ' O '3 λ p2q .
The Jacobian subscheme Ξ σ " Ξ p3q is set-theoretically a linear subspace of P 35 of dimension 3 `4 `3 ´1 " 9. We have C psq " C p3q " O '10 λ p2q . The scheme Ξ σ contains Υ p3q " ψ ˚pϕ ˚pC p3q qq which is a double structure over P 9 Ă P 35 . We have:

D p2q " O '3 λ p4q ' O '4 λ p1q , C p2q " O '7
λ p1q , Note that Υ p2q " ψ ˚pϕ ˚pPpC p2q qqq and Ξ p2q " ψ ˚pϕ ˚pPpD p2q qqq have dimension 6. The residual subscheme of Υ p3q in Ξ p3q " Ξ σ is Ξ p2q . This is set-theoretically a P 6 and contains Υ p2q which is a reduced P 6 . We have:

D p1q " C p1q " O '3
λ p3q . The residual subscheme of Υ p2q in Ξ p2q is Ξ p1q " Υ p1q . This is a triple structure on P 2 . For q ą 0, we have Ext q P n pT σ , O P n q ‰ 0 if and only q P t24, 27, 31u.

Example 4.4. An incompressible pencil with r 1 " 3, pc 1 , c 2 , c 3 q " p1, 2, 2q and Segre symbol Σ " rp3 2 , 1 4 q, p4 5 , 3 2 , 2 3 qs, hence with pu, vq " p47, 5q, lives in P 54 . Its Jacobian scheme consists of a rational normal scroll Y of dimension 3 and degree 5 spanning a P 7 and of two linear spaces pΞ 1 q red and pΞ 2 q red of dimension 5 and 9 meeting Y along two disjoint projective planes appearing as fibres of the scroll. The subscheme pΞ 1 q contains a simple P 5 with a double line as residual subscheme. On the other hand the subscheme pΞ 2 q contains a double P 9 , whose residual subscheme still contains a simple P 6 with a simple P 4 as residual subscheme.

We have ℓ " 2 and Ext q P n pT σ , O P n q ‰ 0 if and only q " 54 ´2 ´r1 or q " 54 ´1 ´r1 ´p1,1 or q " 54 ´1 ´r1 ´p1,1 ´p1,2 or q " 54 ´1 ´r1 ´p2,1 or q " 54 ´1 ´r1 ´p2,1 ´p2,2 or q " 54 ´1 ´r1 ´p2,1 ´p2,2 ´p2,3 which gives q P t40, 43, 44, 45, 48, 49u.

From the proof of the previous theorem, we extract some precise information on the primary components of Ξ σ . Assume σ is an incompressible pencil of quadrics having Segre symbol Σ and degree vector c, with: Σ " rΣ 1 , . . . , Σ ℓ s, Σ j " pa pj,1 j,1 , . . . , a pj,s j j,sj q, c " pc 1 , . . . , c r1 q, for some integers r 1 , ℓ, s 1 , . . . , s ℓ , tpa j,i , p j,i q | j P t1, . . . , ℓu, i P t1, . . . , s j u with a j,1 ą ¨¨¨ą a j,sj for all j P t1, . . . , ℓu and c 1 ď ¨¨¨ď c r1 . Recall the convention a j,i " 0 for i ą s j and for each j P t1, . . . , ℓu set q j,k ´1 " ř k i"1 p j,i ´1. Corollary 4.5. Let σ be an incompressible pencil of quadrics.

i) If σ is regular, then the Jacobian scheme Ξ σ has primary components: Υ pkq j , for j P t1, . . . , ℓu and k P t1, . . . , s j u,

where the components Υ pkq j are projective spaces of dimension q j,k ´1 over subschemes of length a j,i ´aj,k`1 of P 1 . We have:

h 0 pO Ξσ q " ℓ ÿ j"1 a j,1 , Υ pkq j X Υ pk 1 q j 1 " H, if j ‰ j 1 .
ii) If σ is irregular, then the Jacobian scheme Ξ σ consists of a smooth scroll Y of dimension r 1 and degree v " ř r1 i"1 c i and of the primary components: Υpkq j , for j P t1, . . . , ℓu and k P t1, . . . , s j u, where the components Υpkq j are projective spaces of dimension r 1 `qj,k ´1 over subschemes of length a j,i ´aj,k`1 of P 1 . Also:

h 0 pO Ξσ q " 1, Υpkq j X Υpk 1 q j 1 " H, if j ‰ j 1 .
Finally, setting Ξ for the residual scheme of Y in Ξ σ , we have:

h 0 pO Ξq " ℓ ÿ j"1 a j,1 .
Proof. We gave in Lemma 3.9 a set-theoretic description of the Jacobian scheme Ξ σ which shows that Ξ σ consists of ℓ pairwise disjoint linear spaces, together with the scroll Y in case σ is irregular, and in this case we also noticed that Y has dimension r 1 and degree v. Also, in the proof of Theorem 4.2 we gave the structure of each primary component supported at any of the linear spaces mentioned above. Taking the union over all such spaces we get precisely the set tΥ pkq j

| j P t1, . . . , ℓu, k P t1, . . . , s j uu, or t Υpkq j | j P t1, . . . , ℓu, k P t1, . . . , s j uu depending on whether σ is regular or not. Note that in the proof of Theorem 4.2 we also had the component Υp0q , but this is just the scroll Y which is already accounted for.

To compute h 0 pO Ξσ q, note that taking h 0 of the structure sheaf is an additive operation on disjoint primary components, which is invariant under taking projective bundles and takes value a at λ paq j Ă P 1 for any a P N ˚. So for regular pencils we get:

h 0 pO Ξσ q " ℓ ÿ j"1 sj ÿ k"1 h 0 pO Υ k j q " ℓ ÿ j"1 sj ÿ k"1 pa j,i ´aj,k`1 q " ℓ ÿ j"1 a j,1 .
For irregular pencils, the Jacobian scheme is connected as Y meets all the components t Υpkq j | j P t1, . . . , ℓu, k P t1, . . . , s j uu, hence we have h 0 pO Ξσ q " 1. Finally, the primary components of Ξ are precisely the t Υpkq j | j P t1, . . . , ℓu, k P t1, . . . , s j uu, so the last formula follows as in the regular case.

4.3.

Applications to projective dimension. Theorem 4.2 allows to compute the projective dimension pdimpT σ q of the logarithmic tangent sheaf associated to a pencil o quadrics, namely, the minimal length of a locally free resolution of T σ . Proposition 4.6. Let σ be an incompressible pencil of quadrics. i) Assume σ is irregular. Then pdimpT σ q " n ´r1 ´2.

ii) Assume σ is regular and put p " mintp j,1 | j P t1, . . . , ℓuu. Then:

pdimpT σ q " n ´p ´1.
Proof. By Theorem 4.2, we can compute for which values of q ě 1 one has Ext q P n pT σ , O P n q ‰ 0. On the other hand, we have: pdimpT σ q " maxtq P N | Ext q P n pT σ , O P n q ‰ 0u. Fixing j P t1, . . . , ℓu and letting k vary in t1, . . . , s j u the maximal value for n ´r1 ´pj,1 ´¨¨¨´p j,k ´1 is attained by choosing k " 1. Such value is thus n ´r1 ´pj,1 ´1. Letting j vary in t1, . . . , ℓu, the maximal value of n ´r1 ´pj,1 ´1 is n ´r1 ´p ´1. The maximum between n ´r1 ´p ´1 and n ´r1 ´2 is n ´r1 ´2 because p ě 1. This gives the result.

Example 4.7. A pencil σ as in Example 4.3 has pdimpT σ q " 31. For Example 4.4, we get pdimpT σ q " 49.

Example 4.8. The completely irregular pencil σ " px 1 x 5 `x3 x 4 , x 2 x 4 `x0 x 5 q showing up at the end of the proof of Theorem 3.2 has n " 5, u " v " r 1 " 2. So for q ą 0 we have Ext q P n pT σ , O P n q ‰ 0 if and only if q " 1. We get pdimpT σ q " 1. Corollary 4.9. A regular pencil of quadrics is locally free if and only if n P t1, 2u or its Segre symbol is rp1, 1q, p1, 1qs or rp2, 2qs.

Proof. A regular pencil σ is locally free if and only if all the integers q satisfying (41) are non positive. This is always the case for n ď 2 and holds true for the Segre symbols rp1, 1q, p1, 1qs or rp2, 2qs so one implication is proved.

Conversely, assume n ě 3 and σ locally free. Set a " minta j,1 | j P t1, . . . , ℓuu. For all j P t1, . . . , ℓu, take k " 1 and define q by (41). The inequality q ď 0 gives p j,1 ě n ´1, which implies, in view of (40), that:

n `1 ě ℓ ÿ j"1 ˜aj,1 pn ´1q `sj ÿ i"2 a j,i p j,i ¸.
This gives npaℓ ´1q ď aℓ `1 and therefore either a " ℓ " 1, or aℓ " 2 and n " 3.

In the former case s 1 " 1 so (40) gives p 1,1 " n `1, which is impossible. In the latter, either pa, ℓq " p1, 2q and the Segre symbol is rp1 2 q, p1 2 qs " rp1, 1q, p1, 1qs or pa, ℓq " p2, 1q and the Segre symbol is rp2 2 qs " rp2, 2qs.

In the same spirit we have, more generally, the following.

Corollary 4.10. Let σ be an incompressible pencil of quadrics. Then: i) If σ is regular, then pdimpT σ q ě n´3 2 . ii) If σ is irregular, then pdimpT σ q ě 2n´7 3 . Proof. Assume σ is irregular. Then by Lemma 3.10 we get:

pdimpT σ q " n ´r1 ´2 ě n ´2 ´n `1 3 " 2n ´7 3 .
Next, suppose σ is regular. Again put a " minta j,1 | j P t1, . . . , ℓuu, p " mintp j,1 | j P t1, . . . , ℓuu and use (40) to get n `1 ě ℓap. By Corollary (4.6), we obtain:

n `1 ě ℓapn ´pdimpT σ q ´1q. Rearranging the terms, this yields:

pdimpT σ q ě n ´n `1 ℓa ´1.
We saw in the previous proof that aℓ ě 2 so this gives pdimpT σ q ě n´3 2 . Remark 4.11. The previous bounds are sharp. Indeed, if σ is a completely irregular incompressible pencil, then 3r 1 " n `1 so pdimpT σ q " n´3 2 . Also, if σ is a regular pencil with n ě 3 odd, say n `1 " 2m, then we may take σ to have Segre symbol rp1 m q, p1 m qs or rp2 m qs and we get pdimpT σ q " n´3 2 . Example 4.12. Let us list all the possible cases for irregular pencils of quadrics in P 3 . We give the possible Segre symbols of the regular part. The pencil with empty Segre symbol is completely irregular. We see that only one case gives an incompressible pencil. This one has pdimpT σ q " 1. 4.4. Graded projective dimension. Let n ě 2. We call graded projective dimension of a torsion free sheaf E on P n : gpdimpEq " maxtq P t0, . . . , n ´1u | Ext q P n pE, O P n q ˚‰ 0u. Here, Ext q P n pE, O P n q ˚is a shortcut for ' tPZ Ext q P n pE, O P n ptqq. The graded projective dimension is the length of a sheafified minimal graded free resolution of the module of global sections of E. Theorem 4.13. Let σ be a pencil of quadrics in P n . i) Assume σ is regular. Then:

gpdimpT σ q " n ´2,
unless the Segre symbol Σ of σ r1 p , 1 q s for some p ě q ě 1, or rp2 q , 1 p qs for some p ě 0 and q ě 1, in which case:

gpdimpT σ q " n ´q ´1.
ii) Assume σ is irregular of generic corank r 1 . Then:

gpdimpT σ q " n ´1,
unless σ has degree vector p1, . . . , 1q, in which case:

gpdimpT σ q " n ´r1 ´2.
We underline that the graded projective dimension of T σ depends on the Segre symbol only if σ is regular; otherwise, gpdimpT σ q only depends on whether or not the degree vector c " pc 1 , . . . , c r1 q contains a value strictly greater than 1.

Proof. First of all we observe that, without loss of generality, we can assume that the pencil σ is incompressible. Indeed, if σ has compressibility m ą 0, then we may work in a projective space of dimension n " n ´m whose coordinates do occur in the quadrics of σ. The minimal resolution obtained over the coordinate ring of this space is a minimal resolution of T σ {O 'm P n and thus computes gpdimpT σ q. Next we note that, according to the proof of Theorem 3.2, the sheaf Q σ p´1q of an incompressible pencil σ is isomorphic to O Ξσ pF q, where F is the class of a fibre of the scroll map Y Ñ P 1 . The divisor F is trivial on the components t Υpkq j | j P t1, . . . , ℓu, k P t1, . . . , s j uu. Set a " ř ℓ j"1 a j,1 . By Corollary 4.5, we get: (49) h 0 pQp´1qq " a, if σ is regular.

Also, if σ is irregular, denoting again by Ξ the residual scheme of Y in Ξ σ , we get an exact sequence:

(50) 0 Ñ O Y pF q Ñ Qp´1q Ñ O Ξ Ñ 0.
In order to prove the result, we will need two more ingredients, namely two equivalent definitions of the graded projective dimension. For the first one, for any coherent sheaf E on P n and q P N, put H q ˚pEq " À tPZ H q pEptqq. Set: q 0 " mintq P N ˚| H q ˚pT σ q ‰ 0u. We have, by Serre duality: gpdimpT σ q " n ´q0 .

The second one is worked out in the framework of graded modules over the polynomial ring R " κrx 0 , . . . , x n s. Consider the matrix J σ as a map of graded modules R n`1 Ñ Rp1q 2 and define the R-modules Q σ , M σ and T σ as the cokernel, image and kernel of this map, so that sheafifying the graded modules Q σ , M σ and T σ we get back Q σ , M σ and T σ . We write down the exact sequence of graded R-modules:

0 Ñ T σ Ñ R n`1 Jσ Ý Ý Ñ Rp1q 2 Ñ Q σ Ñ 0.
The Auslander-Buchsbaum formula gives: gpdimpT σ q " n `1 ´depthpT σ q.

We compute the depth of T σ by the relation: depthpT σ q " mintq P N | Ext q R pκ, T σ q ‰ 0u, where κ is the residual field, namely κ " R{px 0 , . . . , x 1 q.

Having set up all this, we are in position to prove i). So assume σ is regular. First we compute gpdimpT σ q when Σ " r1 p , 1 q s or Σ " rp2 q , 1 p qs. If Σ " r1 p , 1 q s with p ě q ě 1 then p `q " n `1 and the generators pf, gq of σ can be chosen to be f " x 2 0 `¨¨¨`x 2 p´1 and g " x 2 p `¨¨¨`x 2 n . Set L " V px 0 , . . . , x p´1 q and M " V px p , . . . , x n q. Looking at J σ , we see that:

T σ » R L p1q ' R M p1q,
where R L and R M are the Koszul syzygies of L and M , see §3.3.2. Now gpdimpR L p1qq " p ´2 ě q ´2 " gpdimpR M p1qq, so: gpdimpT σ q " p ´2 " n ´q ´1.

Next, we deal with Σ " rp2 q , 1 p qs, for q ě 1 and p ě 0. Note that p `2q " n `1, so n ´q ´1 ě q. When p " 0 we have Q σ p´1q » O Υ p1q , where Υ p1q is a projective space P n´q over a length-2 subscheme of P 1 , while for p ą 0 we have a filtration:

0 Ñ O Υ p1q Ñ Q σ p´1q Ñ O Υ p2q Ñ 0,
where Υ p1q is a reduced P n´q and Υ p2q is a reduced P q´1 Ă P n´q . In both cases, since the coordinate rings of the subschemes Υ pkq are graded Cohen-Macaulay rings, we have gpdimpO Υ pkq q " codimpΥ pkq q. Therefore gpdimpT σ q `2 is the maximum of the gpdimpO Υ pkq q for different values of k. Since n ´q ´1 ě q, in both cases we obtain the equality: gpdimpT σ q " n ´q ´1. Let us show that, unless Σ " r1 p , 1 q s or Σ " rp2 q , 1 p qs, we have H 2 pT σ p´1qq ‰ 0, which implies gpdimpT σ q ě n ´2. It suffices to show H 1 pM σ p´1qq ‰ 0, which in turn holds true if h 0 pQ σ p´1qq ą 2. But by (49), we have h 0 pQ σ p´1qq ą 2 unless ℓ " 2 and s 1 " s 2 " a 1,1 " a 2,1 " 1, or ℓ " 1, s 1 P t1, 2u, a 1,1 " 2. Since these two cases correspond to the Segre symbols Σ " r1 p , 1 q s for some p, q ě 1 or Σ " rp2 q , 1 p qs for some q ě 1, p ě 0, we get gpdimpT σ q ě n ´2 except in these cases. Now we prove that, for regular pencils, we have gpdimpT σ q ď n ´2. It suffices to show that the module Q σ contains no copy of the residual field κ. Indeed, otherwise there would be a non-zero element of Q σ , represented by ph, kq P Rp1q 2 , whose annihilator contains the maximal ideal px 0 , . . . , x n q. Up to switching the factors, we may assume h ‰ 0. Also, since the pencil σ is regular, we may choose the generators pf, gq of the pencil to be both associated to smooth quadrics. Also, we may select coordinates x 0 , . . . , x n of P n so that 2f " x 2 0 `¨¨¨`x 2 n .

Then, for the pair ph, kq P Rp1q 2 with h ‰ 0, there must be a matrix pa i,j q 0ďi,jďn , with a i,j P R for all 0 ď i, j ď n such that:

(51) ˆh k ˙`x 0 ¨¨¨x n ˘" ˆx0 ¨¨¨x n g 0 ¨¨¨g n ˙¨a 0,0 ¨¨¨a 0,n . . . . . . a n,0 ¨¨¨a n,n ‹ ',
where σ " pf, gq and we wrote g i " Bg{Bx i , for all i P t0, . . . , nu. We used here 2f " x 2 0 `¨¨¨`x 2 n . Note that, by the symmetric role of f and g, we may assume h ‰ 0. Hence, (51) implies that the matrix A " pa i,j q 0ďi,jďn is hI n`1 and is thus invertible in κpx 0 , . . . , x n q. Hence, we may rewrite (51) in κpx 0 , . . . , x n q as:

ˆh k ˙`x 0 ¨¨¨x n ˘A´1 " ˆx0 ¨¨¨x n g 0 ¨¨¨g n ˙.
Therefore J σ should have generic rank 1, which is impossible by the Euler relation since f, g are not proportional.

Summing up, we have proved Hom R pκ, Q σ q " 0. Therefore, applying Ext Rpκ, ´q to the above sequence we get Ext q R pκ, T σ q " 0 for q ď 2. Hence depthpT σ q ě 3 and finally gpdimpT σ q ď n ´2. This concludes the proof for regular pencils.

It remains to carry out the proof if σ is irregular. In view of the filtration (50) and since the coordinate rings of the primary components of Ξ are graded Cohen-Macaulay rings, we get: gpdimpO Ξq " codimp Ξq ď codimpY q " n ´r1 . Now, if c i " 1 for all i P t1, . . . , r 1 u, the sheaf O Y pF q has a minimal Buchsbaum-Rim resolution of length equal to codimpY q. This is obtained by the Buchsbaum-Rim resolution of O Y pF q in the linear span L " P v´1 Ă P n of Y seen as cokernel of a matrix of linear forms of size 2 ˆv over L, combined with the Koszul complex of L in P n ; we refer to [5, Theorem A2.10, Exercice A2.19] for Buchsbaum-Rim complexes and matrices associated with scrolls.

We deduce gpdimpO Y pF qq " codimpY q, which in turn yields: gpdimpT σ q " maxpgpdimpO Ξq, gpdimpO Y pF qq ´2 " n ´r1 ´2.

This proves the last part of ii).

To conclude the proof, let us assume that there is i P t1, . . . , r 1 u such that c i ě 2 and show that gpdimpT σ q " n ´1. It is enough to show that H 1 pT σ q ‰ 0. Note that H 0 pT σ q " 0 by incompressibility of σ, hence: h 1 pT σ q ě 2pn `1q ´h0 pQ σ q ´pn `1q ě n `1 ´h0 pQ σ q, so it suffices to check h 0 pQ σ q ă n `1.

To show this inequality we recall the notation pu, vq for the splitting type of σ and take cohomology of (50) twisted by O P n p1q. Since the linear span of the residual subscheme Ξ of Y in Ξ σ is PpH 0 pC t qq and since h k pO Y pF `Hqq " 0 for k ą 0 and h 0 pO Y ptF `Hqq " v `tr 1 , for all t ě 0, we obtain: h 0 pQ σ q " h 0 pO Y pF `Hqq `h0 pO Ξp1qq " " r1 ÿ i"1 pc i `2q `h0 pC t q " pv `2r 1 q `pu ´vq.

We are reduced to show u `2r 1 ă n `1 and, since n `1 " u `v `r1 , this amounts to v ą r 1 . But the inequality v " ř r1 i"1 c i ą r 1 takes place precisely if there is i P t1, . . . , r 1 u such that c i ě 2, so the non-vanishing H 1 pT σ q ‰ 0 is established. The proof of the theorem is achieved.

Example 4.14. Let σ be a regular pencil of quadrics with r 0 " 1. Then the module of global sections T σ of T σ has the Buchsbaum-Rim type resolution:

0 Ð T σ Ð Rp´2q p n`1
3 q Ð ¨¨¨Ð Rp´1 ´jq jp n`1 j`2 q Ð ¨¨¨Ð Rp´nq n´1 Ð 0, with j P t1, . . . , n ´1u. This resolution is minimal and linear of length n ´2.

For a regular pencil of quadrics, the Buchsbaum-Rim complex in the above display is exact if and only if dimpQ σ q " 0. This happens if and only if r 0 " 1.

Pencils of quadrics in dimension 3

We can arrive at a full classification for pencils of quadrics in P 3 over an algebraically closed field κ of characteristic different from 2. The result is the following. Theorem 5.1. Let σ be a pencil of quadrics in P 3 . Then the following holds.

i) The pencil σ is free if and only if it is locally free. This happens: a) If σ has Segre symbol rp1, 1q, p1, 1qs, in which case T σ » O P 3 p´1q '2 ; b) If σ has Segre symbol rp2, 2qs, in which case T σ » O P 3 p´1q '2 ; c) If σ is irregular and incompressible, in which case T σ » O P 3 p´1q '2 ; d) If σ is compressible, in which case T σ » O P 3 ' O P 3 pe ´2q, where e P t0, 1, 2u is the number of double planes in the pencil. ii) In all other cases σ is regular, pdimpT σ q " 1 and the sheafified minimal graded free resolution of T σ reads: a) If r 0 " 1:

0 Ñ O P 3 p´3q '2 Ñ O P 3 p´2q '4 Ñ T σ Ñ 0.
b) If r 0 " 2 and the Segre symbol is not rp1, 1q, p1, 1qs or rp2, 2qs:

0 Ñ O P 3 p´3q '2 Ñ O P 3 p´2q '2 ' O P 3 p´1q Ñ T σ Ñ 0.
The proof of the theorem is by inspection of the different Segre symbols. It follows from the analysis appearing in the next subsubsections. In the column labelled stable, we wrote s or sss according to whether T σ is stable or strictly semi stable (in the sense of the slope), and free when T σ is split. In the description of Ξ σ we let the subschemes Υ pkq show up when a primary component of Ξ σ has a non trivial filtration as in the proof of Theorem 4.2. In the column labelled Chern we write the triple pc 1 pT σ q, c 2 pT σ q, c 3 pT σ qq. Some comments are in order.

i) When σ is free, we have T σ » O P 3 p´1q '2 . ii) When r 0 " 1, the sheaf Q σ has a Buchsbaum-Rim resolution that induces a sheafified minimal graded free resolution:

0 Ñ O P 3 p´3q '2 Ñ O P 3 p´2q '4 Ñ T σ Ñ 0.
This gives the Chern classes of T σ when r 0 " 1. iii) When r 0 " 2, there are lines M, L Ă P 3 , not necessarily distinct, with L Ă Ξ σ , and a finite length subscheme W Ă M , such that T σ fits into (18). Note that, since r 0 " 2, there is a quadric in the pencil, say f 2 , which is a rank-2 quadric in the coordinates x 0 , x 1 , up to homography. So, setting L " V px 0 , x 1 q and composing the Jacobian matrix with the projection O '2 P 3 p1q Ñ O P 3 p1q onto the second factor and with the obvious quotient O P 3 p1q Ñ O L p1q we obtain explicitly the morphism Q σ Ñ O L p1q required to get (18), so Lemma 3.4 holds also for pn, ̺q " p3, 2q.

We have R M » R L » O P 3 p´2q and the length of W is either 2 or 0, according to whether pdimpT σ q is 1 or 0. So T σ is polystable in the free case, otherwise it is strictly slope semistable and Gieseker-unstable.

In the latter case, we have I W {M » O M p´2q and the morphism R L p1q Ñ I W {M p1q of (18) is the natural surjection O P 3 p´1q Ñ O M p´1q. Therefore the term G 2 appearing in the sequence (20) fits into:

0 Ñ O P 3 p´3q Ñ O P 3 p´2q '2 Ñ G 2 Ñ 0.
Then we have a sheafified minimal graded free resolution of T σ of the form:

0 Ñ O P 3 p´3q Ñ O P 3 p´2q '2 ' O P 3 p´1q Ñ T σ Ñ 0.
This gives the Chern classes of T σ when r 0 " 2. iv) Stability of T σ for r 0 " 3 follows from Proposition 3.1. v) One can put two quadrics of σ in normal form. This is done in [6]. 5.2. Irregular pencils. We gave in Example 4.12 the list of numerical invariants of irregular pencils in P 3 . We observed that there is only one irregular incompressible pencil in P 3 . Note that in any case the number of points in the support of C t is at most 3, so irregular pencils have a normal form pf 1 , f 2 q which is completely determined up to SL 2 -action as SL 2 is 3-transitive on P 1 . Note that we can assume that this support is contained in tp1 : 0q, p0 : 1q, p1 : ´1qu. The column labelled m displays the compressibility of the pencil.

We are going to see that, for irregular pencils of quadrics σ in P 3 , the sheaf T σ is always free, with exponents as in the following table. The pair pa, bq in the column exponents indicates that T σ » O P 3 paq ' O P 3 pbq. The splitting type of σ is p2, 1q and the regular part of σ vanishes at a single point λ P P 1 which gives a single quadric of corank 2 in the pencil, so r 0 " 2. This gives a component Ξ p1q Ă Ξ σ which is a reduced line. The component Y of Ξ σ is a line which meets Ξ p1q at λ. In the normalized form appearing in the proof of Theorem 4.2, we have λ " p0 : 1q and the matrix ρ reads:

r1 pu, vq exponents m Segre f1 f2 1 p3, 0q p0, ´2q 1 r1, 1, 1s x 2 0 `x2 2 x 2 1 `x2 2 1 p3, 0q p0, ´2q 1 r2, 1s x0x1 x 2 0 `x2 2 1 p3, 0q p0, ´2q 1 r3s 2x0x2 `x2 1 x0x1 1 p3, 0q p0, ´1q 1
rp2, 1qs 2x0x1 `x2 2 x 2 0 1 p3, 0q p0, ´1q 1 r1 2 , 1s x 2 0 x 2 1 `x2 2 1 p2, 1q p´1, ´1q 0 r1s x0x2 2x0x1 `x2 3 2 p2, 0q p0
¨0 z 1 z 2 0 z 1 0 0 0 z 2 0 0 0 0 0 0 z 1 ‹ ‹ ' .
The associated pencil is p2x 0 x 2 , 2x 0 x 1 `x2 3 q and, up to dividing by 2, the Jacobian matrix reads: ˆx2 0 x 0 0 x 1 x 0 0 x 3 Ṫhe kernel of this matrix is T σ » O P 3 p´1q '2 , the syzygy map being:

¨x0 0 ´x1 ´x3 ´x2 0 0 x 0 ‹ ‹ ' .
5.2.2. Compressible pencils. Assume σ is compressible. Then the sheaf T σ contains a copy of the trivial sheaf O P 3 which is thus a direct summand of T σ . Therefore T σ {O P 3 is a reflexive sheaf of rank one and thus isomorphic to O P 3 p´c 1 pT σ qq. For compressible pencils, we get the following. i) If σ contains no double plane, then T σ » O P 3 ' O P 3 p´2q.

ii) If σ contains precisely one double plane, then T σ » O P 3 ' O P 3 p´1q.

iii) If σ contains two double planes, then T σ » O P 3 ' O P 3 .

Locally free pencils of quadrics

Let us conclude the analysis of freeness and local freeness of pencils of quadrics over an algebraically closed field κ of characteristic different from 2. In the next table, the column labelled Σ displays the Segre symbol of the regular part of σ. In the column labelled exponents we write the sequence of degrees of the line bundles which are direct summands of T σ , for instance the sequence p0 n´3 , ´12 q means that T σ » O 'pn´3q P n ' O P n p´1q '2 . Recall from Section 2.5 that n " n ´h0 pT σ q. When n " 2 one should not consider the first three lines. Theorem 6.1. Let n ě 2. A pencil of quadrics σ is free if and only if σ is locally free. This happens if and only if, up to homography, σ " pf 1 , f 2 q is:

(52) f 1 f 2 exponents n r 0 r 1 Σ x 0 x 1
x 2 x 3 p0 n´3 , ´12 q 3 n ´2 n ´3 rp1 2 q, p1 2 qs x 0 x 1 `x2 x 3

x 2 0 `x2 1 p0 n´3 , ´12 q 3 n ´2 n ´3 rp2 2 qs x 0 x 2

x 0 x 1 `x2 x 0 x 1 p0 n´1 q 1 n n ´1 r2s

Proof. Let σ be a locally free pencil of quadrics. Following the notation of Section 2.5, set m for the compressibility of σ and n " n ´m. By Lemma 2.14, the sheaf T σ decomposes as O 'm P n ' E, thus E must be locally free. In addition, the associated incompressible pencil σ is also locally free, since T σ coincides with the restriction of E to some n-dimensional linear space.

If σ is regular, then Corollary 4.9 says that n ď 2 or n " 3 and σ has Segre symbol r1 2 , 1 2 s or r2 2 s. In the latter case the normal form of the quadrics of σ obtained as in the proof of Theorem 4.2 is the one displayed in the first two lines of the table in display (52). Since σ depends only on x 0 , . . . , x n, this is actually the normal form of the quadrics of σ.

On the other hand, if σ is irregular, then, since σ is locally free, setting r1 for the generic rank of σ, we must have r1 ě n ´2 by Corollary 4.6. Combining this with Lemma 3.10 gives n `1 ě 3n ´6 which implies n ď 3. If n " 3, we are in the situation of Subsection 5.2.1 and we obtain the third line of the above table.

It remains to treat the cases n ď 2. Let us assume n " 2. Since σ is incompressible, T σ is a reflexive sheaf of rank 1 with determinant equal to e ´2 where e is the number of double lines in σ. Note that e P t0, 1u as σ is incompressible.

Also, e " 0 if and only if r 0 " n ´1 and this corresponds to the Segre symbols r1, 1, 1s, r2, 1s and r3s, while e " 1 takes place when r 0 " n and the Segre symbol is rp2, 1qs or rp1, 1q, 1s. These Segre symbols are associated to unique pencils up to homography since ρ σ defines at most 3 distinct points in support of C t and PGL 2 pκq acts transitively on triplets of points of P 1 .

Finally, assume n " 1, so that T σ » O P n . Then there are only two possible Segre symbols, whose normal forms give the pencils px 2 0 , x 2 1 q and px 0 x 1 , x 2 0 q.

Locally free pencils of higher degree

In this section, κ is any field of characteristic 0. In contrast with the case of pencils of quadrics seen in the previous section, we will now show that there are locally free pencils of higher degree that are not free.

Before stepping into the general case, let us take a look at the case of pencils of cubics in detail.

Over the complex projective space P 3 , there are, according to [2], two non-normal cubic surfaces up to homography. In the homogeneous variables px 0 , . . . , x 3 q, the equations of these surfaces are:

f " x 3 1 `x2 0 x 2 `x2 1 x 3 , , g " x 3 1 `x2 0 x 2 `x0 x 1 x 3 .
Both surfaces are singular along the line L " V px 0 , x 1 q. The Jacobian matrix of the pencil of cubics σ " pf, gq reads:

J σ " ˆ2x 0 x 2 3x 2 1 `2x 1 x 3 x 2 0 x 2 1 2x 0 x 2 `x1 x 3 3x 2 1 `x0 x 3 x 2 0 x 0 x 1 ˙.
The sheaf Q σ has rank two over L and admits no zero-dimensional subsheaf. The first part of Lemma 2.2 implies that T σ is locally free. However, the scheme-theoretic locus where Q σ has rank two has an embedded point at p " p0 : 0 : 1 : 0q. In fact, the Jacobian scheme Ξ σ has 4 primary components P 1 , . . . , P 4 described by the next table.

Dimension degree radical ideal 1 5 px 0 , x 1 q 1 1 px 1 , x 3 q 1 1 px 0 ´x1 , x 3 q 0 20 px 0 , x 1 , x 3 q.

Note that pΞ σ q red consists of the union of the 3 lines V px 0 , x 1 q, V px 1 , x 3 q and V px 0 ´x1 , x 3 q; the first one appears with multiple structure of degree 5.

In this example, T σ p2q has c 1 pT σ p2qq " 0 and c 2 pT σ p2qq " 1, c 3 pT σ p2qq " 0. Also, we have H 0 pT σ p2qq " 0. Therefore T σ p2q is a null correlation bundle.

This example is generalized to degree k `3, for any k ě 0, in our next result. Then we have T σ » N p´k ´2q, where N is a null correlation bundle.

Proof. Set f " x 0 x k`2 x 3 and g " x 2 x 3 px k 2 ´xk 1 q. Observe that, in the algebraic closure of κ, the divisor V pgq is an arrangement of planes consisting of k `2 planes H 1 , . . . , H k`2 passing through the line L " V px 1 , x 2 q together with an extra plane H not containing L, namely the plane V px 3 q. This arrangement is free, more precisely, we have :

T g » O P 3 ' O P 3 p´1q ' O P 3 p´k ´1q.
Factoring out the trivial summand O P 3 of T g , we write explicitly the syzygy φ : O P 3 p´1q ' O P 3 p´k ´1q Ñ O '3 P 3 , independently on whether κ is closed or not:

φ " ¨x1 x k`1 2 x 2 x k 1 x 2 ´pk `2q x 3 ´xk 1 x 3 '.
The the Jacobian matrix J σ : O '4 P 3 Ñ O '2 P 3 pk `2q reads:

J σ " ˆxk`2 1 pk `2qx 0 x k`1 1 pk `3qx k`2 2 `pk `2qx k`1 2 x 3 x k`2 2 0 pk `1qx k 1 x 2 x 3 x k`1 1 x 3 ´pk `2qx k`1 2 x 3 x k`1 1 x 2 ´xk`2 2 ˙.
Note that this matrix has a vanishing entry at the bottom left corner and that the vanishing of this entry corresponds to the trivial summand of T g . Therefore, projecting onto the last three factors of O '4 P 3 and onto the second factor of O '2 P 3 pk`2q we get a commutative diagram, which is essentially a particular case of the diagram in display (13):

O P 3
The top arrow defines a surface D Ă P 3 of degree k `2, whose equation must sit in the top left corner of J σ . In other words, D " V px k`2 1 q is the pk `2q-tuple structure over the plane V px 1 q.

Also, we observe that the image of J g is I C{P 3 pk `2q, where the curve C is the scheme-theoretic singular locus of V pgq and is defined by the 3 minors of order 2 of φ. Incidentally, over the algebraic closure of κ, the curve C consists of k `2 reduced lines L 1 , . . . , L k`2 , with L i " H X H i for all i P t1, . . . , k `2u, together with a pk `1q-tuple complete intersection structure over L of degree pk `1q 2 defined by V px k 0 x 1 , x k`1 0 ´pk `2qx k`1 1 q. The rightmost column of diagram (53) gives a surjection M σ Ñ I C{P 3 pk `2q, whose kernel is a torsion free sheaf of rank 1, isomorphic to I B{P 3 pk `2q, where the subscheme B Ă D Ă P 3 is defined in P 3 by the homogeneous polynomials h of the form h " a 0 f 0 `¨¨¨`a 3 f 3 with a i P R " κrx 0 , . . . , x 3 s and satisfying a 1 g 1 `a2 g 2 `a3 g 3 " 0, where we put f i " Bf {Bx i and g i " Bg{Bx i , for i P t0, 1, 2, 3u. Since φ accounts for all relations of the homogeneous ideal of C, the homogeneous ideal of B is thus generated by pf 0 , f 1 φ 1,1 `f2 φ 2,1 `f3 φ 3,1 , f 1 φ 1,2 `f2 φ 2,2 `f3 φ 3,2 q and the matrix of these generators is:

¨xk`2 1 pk `2qx 0 x k`2 1 `pk `3qx k`3 2 pk `2qx 0 x k`1 1 x k`1 2 `pk `3qx k 1 x k`3 2 `pk `1qx k 1 x k`2 2 x 3
'.

Theorem 7 . 1 . 1 `xk` 3 2`xk` 2 2x 3 , g " x 2 x 3 px k` 1 2

 7113231 For any k ě 0, define the pencil σ " pf, gq as:f " x 0 x k`2

1 `xk` 3 2`xk` 2 2

 132

  L p1q, with cokernel G. Therefore, either this morphism is surjective, or G is supported on a linear space strictly containing L. However, this second possibility is excluded because L is maximal.

	I L p1q, and we get a second diagram:
					we get
	a morphism O	'pn`1q P n	Ñ I 'pn`1q P n	։ I L p1q factors through O	'pn`1´̺q P n	Ñ

Summing up, we obtain an epimorphism M σ ։ I L p1q. Since L is cut by n`1 ´̺ equations, the induced epimorphism O

  This is a block-diagonal matrix pencil having a block O

	which is ρ and a block O 'v P 1 p´1q Ñ O	'pv`r1q P 1	'pu´vq P 1	p´1q Ñ O	'pu´vq P 1

  5.1. Regular pencils. Let us write the table of possible Segre symbols of regular pencils, together with the description of Ξ σ arising from the previous sections.

	Segre	ℓ	Ξσ	r0	Chern	stable pdim
	r1, 1, 1, 1s 4	4 simple points	1 p´2, 3, 4q	s	1
	r2, 1, 1s	3	double point & 2 simple points	1 p´2, 3, 4q	s	1
	r2, 2s	2	2 double points	1 p´2, 3, 4q	s	1
	r3, 1s	2	triple point & simple point	1 p´2, 3, 4q	s	1
	r4s	3	quadruple point	1 p´2, 3, 4q	s	1
	rp1 2 q, 1, 1s 2	line & 2 simple points	2 p´2, 2, 2q	sss	1
	rp1 2 q, 2s	2	line & double point	2 p´2, 2, 2q	sss	1
			line Υ p2q &
	rp2, 1q, 1s 2	simple point Υ p1q &	2 p´2, 2, 2q	sss	1
			simple point
	rp3, 1qs	1	line Υ p2q & double point Υ p1q	2 p´2, 2, 2q	sss	1
	rp1 2 q, p1 2 qs 2	2 disjoint lines	2 p´2, 1, 0q	free	0
	rp2 2 qs	1	double line Υ p1q	2 p´2, 1, 0q	free	0
	rp1 3 q, 1s	2	plane & simple point	3 p´1, 1, 1q	s	1
	rp2, 1 2 qs	1	plane Υ p2q & simple point Υ p1q	3 p´1, 1, 1q	s	1

Therefore, the homogeneous ideal

We have an exact sequence: 0 Ñ I B{P 3 pk `2q Ñ M σ Ñ I C{P 3 pk `2q Ñ 0, and thus, from the leftmost column of diagram (53): 0 Ñ T σ Ñ O P 3 p´1q ' O P 3 p´k ´1q Ñ I B{D pk `2q Ñ 0.

The morphism O P 3 p´1q Ñ I B{D pk `2q is given by the generator x k`3 2 of the ideal of B in D as in (54). It defines a curve section A of D of degree k `3 which contains B. We get an exact sequence:

where the curve E Ă P 3 is defined by the sequence and is cut in D as the residual scheme of B with respect to the complete intersection

From the diagram (53), using the snake lemma we also get:

We compute the equations of E from (54) as `IB{D : px k`3 2 q ˘and get:

Therefore, the curve E is a double structure of arithmetic genus ´1 over the line L. We conclude from (55) that T σ pk `2q is a null correlation bundle.

Remark 7.2. By the previous theorem, for any degree k `3 there is a pencil σ which is locally free but not free. Also, we have gpdimpT σ q " 2. More precisely, the sheafified minimal graded free resolution of T σ reads:

This is in contrast with the case of pencils of quadrics, where local freeness is equivalent to freeness and where gpdimpT σ q ď n ´2.

Regular sequences of length 2 and rational foliations

We complete this paper by looking at arbitrary regular sequences of length 2 and showing how these are related to rational 1-forms, which we now introduce.

Let ω P H 0 pΩ 1 P n pd `2qq be a rational 1-form of type pd 1 `1, d 2 `1q, where 0 ď d 1 ď d 2 , given by ω " af 1 ¨df 2 ´bf 2 ¨df 1 , where f 1 and f 2 are homogeneous polynomials with no common factors of degree d 1 `1 and d 2 `1, respectively, with d 1 `d2 " d, and a and b are relatively prime integers such that pd 1 `1qb " pd 2 `1qa. Remark that σ :" pf 1 , f 2 q is a regular sequence in R " κrx 0 , . . . , x n s.

Regarding ω as an element of Hom P n pTP n , O P n pd `2qq, we set K ω :" kerpωq. Since ω vanishes along the complete intersection scheme C :" V pσq, the image of the morphism ω : TP n Ñ O P n pd`2q is actually contained in the ideal sheaf I C pd`2q. Applying the functor Hom P n pO P n p1q 'n`1 , ´q to the resolution of I C pd `2q

where η " ppd 1 `1qf 1 pd 2 `1qf 2 q t , we check that the composed morphism

lifts to a unique morphism µ : O P n p1q 'n`1 Ñ O P n pd 1 `1q ' O P n pd 2 `1q, since Hom P n pO P n p1q 'n`1 , O P n q " Ext 1 P n pO P n p1q 'n`1 , O P n q " 0. Therefore we obtain the commutative diagram:

This proves that K ω » kerpµq. We argue that µ " J σ , thus in fact K ω » T σ p1q. Indeed, note that ω "

which means that the entries of the morphism α : O P n p1q 'n`1 Ñ O P n pd `2q given by the composition

Since, on the other hand, α " p´qf 2 pf 1 q˝µ, we conclude that µ " J σ , as desired. Conversely, given a regular sequence σ " pf 1 , f 2 q with degpf i q " d i `1, we follow the proof of Lemma 2.3 in Section 2.2 and consider the associated codimension 1 distribution D σ as presented in display (8); in the case at hand, this simplifies to (setting

where Γ σ Ă P n is a (possibly not pure) 2-codimensional subscheme of P n , and l " c 1 pQ σ q; this is precisely the codimension one distribution associated to the (possibly non saturated) twisted rational 1-form ω " pd 1 `1qf 1 ¨df 2 ´pd 2 `1qf 2 ¨df 1 P Hom P n pTP n , I Γ pd ´l `2qq Ă H 0 pΩ 1 P n pd `2qq. Moreover, the bottom line of the diagram in display (7) yields the following description for the singular scheme Γ σ of ω:

In particular, we have that degpΓ σ q " degpQ σ q `degpCq " degpQ σ q `pd 1 `1qpd 2 `1q.

Summarizing, we have established a natural 1-1 correspondence between regular sequences of length 2 and rational codimension one foliations as follows.

Lemma 8.1. There exists a 1-1 correspondence between regular sequences σ " pf 1 , f 2 q on R and rational codimension one foliations D on P n of type pdegpf 1 q, degpf 2 qq such that T σ p1q " T D and SingpDq " Γ σ .

The previous statement has the following two important applications when n " 3. First, as an immediate consequence of [4, Theorem 6.3], we obtain the following stability result for logarithmic sheaves associated to regular sequences of length 2 on P 3 . Corollary 8.2. Let σ " pf 1 , f 2 q be a regular sequence in κrx 0 , x 1 , x 2 , x 3 s and let d i :" degpf i q ´1; assume that d 1 `d2 ą 0 and c 1 pQ σ q " 0.

(1) If d 1 `d2 is even, then

, then T σ is slope-stable. In particular, if the Jacobian scheme is 0-dimensional, then T σ is slope-stable.

We remark that the previous result is not sharp, and it is not hard to find examples of regular sequences with slope-stable logarithmic sheaves whose degrees do not satisfy the numerical inequalities above. Indeed, if σ corresponds to a pencil of quadrics, so that d 1 " d 2 " 1, with dim G σ " 0, then Corollary 8.2 only implies that T σ is slope-semistable; however, as we have seen in Section 5.1, T σ is actually slope-stable in this case. Note that the case d 1 " d 2 " 1 is the only one for which the right hand sides of the inequalities is not positive.

In addition, the higher degree pencils provided in Theorem 7.1 yield yet another set of examples showing that the converse of Corollary 8.2 does not hold.

Finally, as a second application, we give a negative answer to a problem posed by Calvo-Andrade, Cerveau, Giraldo and Lins Neto, see [3, Problem 2]. To be preceise, these authors asked whether the tangent sheaf of a codimension one foliation on P 3 splits as a sum of line bundles whenever it is locally free. Indeed, in light of the proof of Lemma 8.1, the pencils presented in Theorem 7.1 provide examples, for each k ě 0, of rational foliations of type pk `3, k `3q on P 3 whose tangent sheaves are slope-stable locally free sheaves.