Exceptional bundles of homological dimension ${k}$

We characterize exceptional vector bundles on P of arbitrary homological dimension defined by a linear resolution. Moreover, we determine all Betti numbers of such resolution.


Introduction
Let E be a vector bundle on P n .A resolution of E is an exact sequence where F i splits as a direct sum of line bundles.The minimal length of such resolutions is called the homological dimension of E and it is denoted by hd(E).Besides, it is well-known that hd(E) ≤ n − 1.
We say that the resolution of E is linear if it is of the form for some d ∈ Z, i.e. the entries of the matrices associated to the morphisms α i are linear forms.The exponents a i are called the Betti numbers of E.
In the present work we will focus our attention on vector bundles E on the projective space P n with hd(E) ≤ k, for some k ∈ {1, . . ., n − 1} a positive integer, and defined by a linear resolution of type Our purpose is to characterize those E which are exceptional, i.e whose only endomorphisms are the homotheties and satisfying Ext q (E, E) = 0, for all q ≥ 1. Exceptional bundles are a powerful tool in the study of stable sheaves and the derived category of coherent sheaves.They first appeared in a paper by Drézet and Le Potier [DLP85], who used them to describe the possible ranks and Chern classes of semi-stable vector bundles on P 2 .Their study was then formulated within the setting of derived categories, due to mathematicians such as Gorodentsev and Kuleshov [GK04].This gave rise to a technique called Helix Theory whose main idea is to describe the set of exceptional bundles over a variety and to produce new ones by means of mutations.The first non-trivial example of vector bundles on P n of type (1) is the case of homological dimension 1, that is when E is defined by a resolution 0 → O P n (−1) a 1 → O a 0 P n → E → 0. These bundles were introduced by Dolgachev and Kapranov [DK93] and they are usually called Steiner bundles.Herein we will call them classical Steiner bundles and will consider them as a particular case of the more general notion of a Steiner bundle on algebraic varieties (see Definition 2.2).
The study of classical Steiner bundles is extensive and has been addressed from very different points of view.With respect to the scope of this paper we refer the work of Brambilla and Soares ([Bra04], [Bra08] and [Soa08]).The first proved that any exceptional general classical Steiner bundle E is characterised by χ(End E) = 1.In the same spirit, Soares [Soa08] introduced an appropriate generalization of the concept of a Steiner bundle on an algebraic variety X and showed that an exceptional Steiner bundle E on X is also characterized by χ(End E) = 1, provided E is general.In both cases, a complete description of the resolution of E is provided.
It is a natural step to work towards the characterization of exceptional bundles E of higher homological dimension (we refer to [MP14], where this problem is also discussed).So in section 2 we first recall the main concepts and results on vector bundles on algebraic varieties of homological dimension 1 with a linear resolution, the so-called Steiner bundles, needed later on.
We address the characterization of exceptional bundles of arbitrary homological dimension with a resolution of type (1) in section 3.Such resolution splits into k short exact sequences.Denoting S i = coker α i+1 , i = 1, . . ., k − 1, the farther vector bundle S k−1 is a Steiner bundle, so we know exactly when it is exceptional.This fact will clearly play a role in achieving our goal.
Since any exceptional bundle F satisfies χ(End F ) = 1 we compute the Euler characteristic of End E. We get an iterative formula that interrelates the Euler characteristics χ(End E) and χ(End S i ) (Proposition 3.3) and we compute χ(End E) in terms of the Betti numbers of E (Proposition 3.4).Our first important result (Proposition 3.5) will give sufficient conditions so that χ(End S i ) = 1, for all i = 1, . . ., k − 1, and it will be the main tool to prove our main theorem (Theorem 3.7).We prove that a vector bundle E defined by (1) is exceptional if and only if χ(End E) = 1 and all S i are exceptional, provided we impose a cohomological vanishing condition on E. Furthermore, if this is the this case, we are able to describe all Betti numbers in (1).They are completely determined by the exponents that define the resolution of the exceptional Steiner bundle S k−1 (see Corollary 3.8).
In the last section, section 4, we give some examples of exceptional bundles E of homological dimension k ≥ 2. For instance, a theorem by Ellia and Hirchowitz (see Theorem 4.1) will help us to explicitly construct a large family of exceptional vector bundles of homological dimension 2. In the case when hd(E) ≥ 3 we provide several examples with the help of Macaulay2 [GS] which give support to the conjecture stated in the end of the paper.
Notation 1.1.Given a smooth algebraic variety X of dimension n and a coherent sheaf E on X, we denote . Recall furthermore that when E is a locally free sheaf, we have an isomorphism Throughout this paper we will consider K a fixed algebraically closed field of characteristic 0.

Exceptional vector bundles of homological dimension 1
Let E be a vector bundle on P n , n ≥ 3, with a linear resolution P n → E → 0, and we will call E a classical Steiner bundle.
In this section we recall the main results which we will be using throughout the present paper on vector bundles of homological dimension 1.Although we will only deal with vector bundles on the projective space, we introduce all concepts and state their properties in the broadest setting.
Definition 2.1.Let X be a smooth algebraic variety.A coherent sheaf E on X is called simple if Hom(E, E) K. If, furthermore, it satisfies Ext q (E, E) = 0, for all q ≥ 1, then we say that E is exceptional.
An ordered pair (E, F ) of coherent sheaves on X is strongly exceptional if both E and F are exceptional and Ext q (F, E) = 0, ∀ q ≥ 0, Ext q (E, F ) = 0, ∀ q = 0.
In [Bra08], Brambilla characterized simple and exceptional classical Steiner bundles.Her work was generalised in [Soa08] to vector bundles on smooth irreducible algebraic varieties.So, a natural definition of Steiner bundles in this general context was introduced: Definition 2.2.A vector bundle E on a smooth irreducible algebraic variety X is called a Steiner bundle if it is defined by an exact sequence of the form where s, t ≥ 1 and (F 0 , F 1 ) is an ordered pair of vector bundles on X satisfying the following two conditions: It is immediate from Definition 2.1 that any exceptional vector bundle E satisfies χ(End E) = 1.If E is a Steiner bundle on X then the Euler characteristic of End E has a very simple formula and it is possible to describe all solutions of the equation χ(End E) = 1.
Lemma 2.3 ([Soa08], Lemma 2.2.3).Let E be a Steiner bundle on a smooth irreducible algebraic variety X defined by the exact sequence (2) and let λ = h 0 (F ∨ 0 ⊗ F 1 ).Then Moreover, where {u k } k≥0 is the sequence defined recursively by (3) Remark 2.4.We say that a vector bundle E on X fitting in a short exact sequence of the form As we mentioned above, the Euler characteristic of End E of any exceptional vector bundle E is always 1.The converse is not true in general.In spite of that, in the case of Steiner bundles the equation χ(End E) = 1 indeed characterizes exceptional bundles, provided E is general (Remark 2.4).
Theorem 2.5 ([Soa08], Theorem 2.2.7).Let E be a general Steiner bundle on a smooth irreducible algebraic variety X defined by an exact sequence of type (2).As- Equivalently, E is exceptional if and only if it is of the form for some k ∈ N, where {u k } k≥1 is the sequence in (3).

Exceptional vector bundles of homological dimension k
The main purpose in this section is to study exceptional bundles of arbitrary homological dimension and to generalize in some way the results of the previous section on vector bundles of homological dimension 1.
Let E be a vector bundle on P n , n ≥ 3, with linear resolution In particular, hd(E) ≤ k ≤ n − 1. Cut this long exact sequence by setting We thus get the following k short exact sequences: Set S 0 = E, so we can write more generally S i = coker α i+1 , i = 0, . . ., k − 1.We start with two lemmas regarding some cohomological properties of the vector bundles E and S i that will be useful in the sequel.
Lemma 3.1.Let E be a vector bundle on P n with linear resolution (4) and let we have for each i = 0, . . ., k − 1.
Amongst the set of vector bundles with resolution of type (4) we are interested in studying those which are exceptional.Computing χ(End E) is thus a natural step.We present two formulas for the Euler characteristic of End E. The first is an iterative formula and it will be especially useful in the proof of Proposition 3.5.Proposition 3.3.Let E be a vector bundle on P n with linear resolution (4) and let S i = coker α i+1 , i = 0, . . ., k − 1.Then, for all i = 0, . . ., k − 2, Proof.We first note that according to Definition 2.2 the vector bundle S k−1 is a Steiner bundle on P n , so we know by Lemma 2.3 that , for every i = 0, . . ., k − 2. This will immediately imply (10).We will prove the statement by induction on k.
If k = 1 then E = S 0 = S k−1 is a classical Steiner bundle on P n with resolution and we already saw that χ(End E) = a 2 0 + a 2 1 − (n + 1)a 0 a 1 .Now suppose that the statement holds for every vector bundle F on P n with a linear resolution of the form P n → F → 0, and let us prove it for any vector bundle E of homological dimension at most k and linear resolution of type (4).If E is such a vector bundle then S 1 (1) is defined by the exact sequence Hence hd(S 1 (1)) ≤ k − 1 and by the induction hypothesis we know that, for any i = 1, . . ., k − 2, So the only case left to prove is when i = 0. Dualising and twisting by E the last short exact sequence in (6) we get On the other hand, from the same sequence in (6) we deduce that and also (twisting it by S ∨ 1 ) ) .Now, applying Lemma 3.1 to S 1 , we know that χ(S 1 ) = h 0 (S 1 ) + (−1) n−k+1 h n−k+1 (S 1 ) + (−1) n h n (S 1 ).
An alternative formula for χ(End E) can be given depending only on the Betti numbers of E.
Proposition 3.4.Let E be a vector bundle on P n with linear resolution (4).Then Proof.We will prove the statement by induction on k.If k = 1 then E is a classical Steiner bundle on P n and according to Lemma 2.3 we have )a 0 a 1 .Suppose that every vector bundle F with a linear resolution of type Let E be a vector bundle on P n defined by (4) and let S i be the vector bundles obtained by cutting the resolution of E into short exact sequences as in (6).In particular, S 1 (1) is a vector bundle defined by The previous proposition gives us a formula tying χ(End E) in with χ(End S 1 ): So we need to compute χ (S ∨ 1 ) = (−1) n χ (S 1 (−n − 1)).We have h 0 (S 1 (−n − 1)) = 0 and hence, applying both Lemmas 3.1 and 3.2, we obtain Therefore, At last, we get We now state the main result that will allow us to achieve a characterization of exceptional bundles which have a linear resolution of length k ≥ 2. Note that the case of homological dimension 1 was already characterized, as recalled in Theorem 2.5.Proposition 3.5.Let E be a vector bundle on P n with linear resolution (4) and let Suppose that E is simple, χ(End E) = 1 and Proof.We claim that the hypothesis , where the second vanishing can be obtained from the Serre duality The other cohomology groups, H q (S ∨ k−1 (−k + 3 − q)), q > 0, all vanish by Lemma 3.1.Therefore, the vector bundle S ∨ k−1 (−k + 3) is m-regular for every m ≥ 0 and in particular we have By Proposition 3.3, we know that and, since we are supposing χ(End E) = χ(End S 0 ) = 1, we then may conclude that completing the proof of the proposition.

One gets from cohomology the exact sequence
It turns out that this condition is not too restrictive due to Ellia and Hirschowitz's theorem (see Theorem 4.1 in section 4): given a general morphism its kernel is a globally generated vector bundle with natural cohomology.Especially, the corresponding morphism H(φ) in cohomology is surjective.
We are now able to state our main theorem which under a certain cohomological assumption characterizes exceptional vector bundles of any homological dimension.
Theorem 3.7.Let E be a vector bundle on P n with linear resolution (4) and let Proof.We first prove (a), so suppose that H n−k (E(k−3−n)) = 0 and E is exceptional.Set S 0 = E and consider the sequence with i = 1, . . ., k − 1. Dualise and twist this sequence by S i , so we get Lemma 3.2 gives us Applying cohomology to (14) we thus get Now, consider the sequence obtained after twisting (13) by We claim that H q S ∨ i−1 (−i + 1) = 0, for all q ≥ 0. In fact, we saw in the proof of Proposition 3.5 that h 0 S ∨ i−1 (−i + 1) = 0. Also, using (11) with m = k − 2 − i, we obtain , for all q ≥ 0, as claimed, and together with (15), we get Since E is exceptional by hypothesis, we conclude that S i is also exceptional, for each i = 1, . . ., k − 1.
We next prove (b).Suppose that χ(End E) = 1 and S i is exceptional, for all i = 1, . . ., k − 1.We have that is, a 0 = χ (S ∨ 1 ).This implies that H 0 (E ∨ ) = 0.But H k (E ∨ ) = H n (E ∨ ) = 0 and hence H q (E ∨ ) = 0, for all q.From the sequence , for q ≥ 0. Note that the proof of (15) holds in general and does not depend of the hypothesis in (a), so . Thus E is exceptional, for we are supposing S 1 is exceptional.
It now results that under the hypothesis of Theorem 3.7 the Betti numbers of an exceptional vector bundle E with a linear resolution of type (4) are completely determined.The pair (a k−1 , a k ) is a pair of consecutive terms of the sequence (3) (recall Lemma 2.3) and determines all the other exponents in the resolution of E.
Corollary 3.8.Let E be an exceptional vector bundle on P n with linear resolution (4) and Then the Betti numbers of E satisfy the following relations: (a) a k and a k−1 are two consecutive terms, u s and u s+1 respectively, of the sequence {u s } s≥0 defined by is a Steiner bundle and by Theorem 3.7 it is exceptional so χ(End S k−1 ) = 1.Therefore, it follows directly from Theorem 2.5 that the pair (a k , a k−1 ) is of the form (u s , u s+1 ), for some s ≥ 2, where {u s } s≥0 is the sequence In the course of the proof of Proposition 3.5 we saw that allow us to write a k−2 in terms of a k−1 and a k : For 1 ≤ i ≤ k − 3, we apply Lemma 3.2 (ii) to obtain a general formula for a i : To compute h n (S k−1 (i − n − 1)) we note that h n−1 (S k−1 (i − n − 1)) = 0 (recall (11)) and use Lemma 3.2 (i) again.We thus obtain We would like to point out that the previous corollary would still hold under the weaker assumptions of Proposition 3.5.However, we think that the statement becomes more interesting with the hypotheses of Theorem 3.7.
Remark 3.9.Let E be a vector bundle on P n with linear resolution (4) and such that If the Betti numbers of E satisfy (a) and (b) in Corollary 3.8 then E may not be exceptional.Nevertheless, a converse statement of the referred corollary could be as follows: Let E be a vector bundle on P n with linear resolution (4) and then E is exceptional (note that the condition that (a k , a k−1 ) = (u s , u s+1 ), where {u s } s≥0 is the sequence (16), is automatically satisfied if S k−1 is exceptional).
The next example shows that the cohomological vanishing H n−k (E(k − 3 − n)) = 0 is indeed necessary.We construct a vector bundle of homological dimension 2 whose Betti numbers do not satisfy conditions (a) and (b) in Corollary 3.8, although it is exceptional.
Example 3.10.Set R = K[x 0 , x 1 , x 2 , x 3 , x 4 ].Let A be a homogeneous 3 × 7 matrix with general linear entries and I 3 (A) the ideal generated by the 3 × 3 minors of A. Hence, by [MR08], Proposition 1.2.16,R/I 3 (A) has a minimal free R-resolution of the following type: Since R/I 3 (A) is an artinian ring then the sheafification of R/I 3 (A) is trivial and the corresponding complex in P 4 is as follows: Twist it by O P 4 (5) and cut it into short exact sequences: In particular, the vector bundle E is defined by the linear resolution 0 → O P 4 (−2) 15 → O P 4 (−1) 70 → O 126 P 4 → E → 0, and hd E ≤ 2. Observing that F ∨ is a general Steiner bundle which is exceptional by Theorem 2.5, we infer that F is exceptional.
More generally, we are able to construct several examples in the software system Macaulay2 ( [GS]).