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«GIANFRANCO CASNATI, DANIELE FAENZI, FRANCESCO MALASPINA Abstract. A del Pezzo threefold F with maximal Picard number is isomor- phic to P1 × P1 × ...»

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Abstract. A del Pezzo threefold F with maximal Picard number is isomor-

phic to P1 × P1 × P1. In the present paper we completely classify locally free

sheaves E of rank 2 such that hi F, E(t) = 0 for i = 1, 2 and t ∈ Z. Such a classification extends similar results proved by E. Arrondo and L. Costa regarding del Pezzo threefolds with Picard number 1.

1. Introduction and Notation Let k be an algebraically closed field of characteristic 0 and Pn the n-dimensional projective space over k. A well–known theorem of Horrocks (see [32] and the refer- ences therein) states that a locally free sheaf E on Pn splits as direct sum of invertible sheaves if and only if it has no intermediate cohomology, i.e. hi Pn, E(t) = 0 for 0 i n and t ∈ Z.

It is thus natural to ask for the meaning of such a vanishing on other kind of algebraic varieties F. Obviously such a vanishing makes sense only if a natural polarization is defined on F. For instance, if there is a natural embedding F ⊆ Pn, then one can consider OF (h) := OPn (1)⊗OF and we can ask for locally free sheaves E on F such that H∗ F, E := t∈Z H i F, E(th) = 0 for 0 i dim(F ) which i are called arithmetically Cohen-Macaulay (aCM for short) bundles. Obviously we are particularly interested in characterizing indecomposable aCM bundles, i.e.

bundles of rank r ≥ 2 which do not split as sum of invertible sheaves. Among aCM 0 bundles, there are bundles E such that H∗ F, E has the highest possible number of generators in degree 0. After [34] such bundles are simply called Ulrich bundles.

Ulrich bundles have many good properties, thus their description is of particular interest.

There are a lot of classical and recent papers devoted to the aforementioned topics. For example, in the case of smooth quadrics and Grassmannians, there are some classical result generalizing the above Horrocks’ criterion (see [1], [6], and [33]). More recently, in the same setup, several authors dealt with the case of hypersurfaces (see e.g. [26], [27], [28], [13], [29], [14], [12], [25]) and of Segre products (see [8]). Nevertheless the Ulrich property has been recently object of deep inspection (see, e.g. [10], [11], [16], [17]).

Another case which could be worth of a particular attention is the case of Fano and del Pezzo n–folds. We recall that a smooth n–fold F is Fano if its anticanonical −1 sheaf ωF is ample (see [24] for results about Fano and del Pezzo varieties). The 2000 Mathematics Subject Classification. Primary 14J60; Secondary 14J45.

All the authors are members of GRIFGA–GDRE project, supported by CNRS and INdAM. The first and third authors are supported by

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greatest positive integer r such that ωF ∼ OF (−rh) for some ample class h ∈ Pic(F ) = is called the index of F and one has 1 ≤ r ≤ n + 1. If r = n − 1, then F is called del Pezzo. For such an F, the group Pic(F ) is torsion–free so that OF (h) is uniquely determined. By definition, the degree of F is the integer d := hn.

Let us restrict to the case n = 3. It is well–known that 1 ≤ d ≤ 8. When d ≥ 3 the sheaf OF (h) is actually very ample so h is the hyperplane class of a natural embedding F ⊆ Pd+1. If d = 8, then F is P3, and OF (h) gives the second Veronese embedding.

If 1 ≤ d ≤ 5, the complete classification of aCM bundles of rank 2 on F can be found in [3], [5], [9], [26], [19], [4].

A fundamental hypothesis in all the aforementioned papers, both on hypersurfaces and on Fano threefolds, which considerably simplifies the proofs, is that the varieties always have Picard number (F ) := rk(Pic(F )) = 1. Indeed in this case both c1 and c2 can be handled as integral numbers. In the cases we are interested in this is no longer possible.

When d = 6, then F ⊆ P7 is either the Segre product P1 ×P1 ×P1, thus (F ) = 3, or a hyperplane section of the Segre product P2 × P2 ⊆ P8, thus (F ) = 2. If d = 7, then F is the blow up of P3 at a point p embedded in P8 via the linear system of quadrics through p, thus again (F ) = 2.

In the present paper we focus our attention on the del Pezzo threefold with the highest Picard number, namely the Segre embedding F := P1 × P1 × P1 ⊆ P7.

In [17] it was proved that F (and any other Segre product except P1 × P1 ) is of wild representation type i.e. there are -dimensional families of non-isomorphic indecomposable aCM sheaves, for arbitrarily large ∈ Z.

Let A(F ) be the Chow ring of F, so that Ar (F ) denotes the set of cycles of codimension r. We have three different projections πi : F → P1 and we denote by hi the pull–back in A1 (F ) of the class of a point in the i–th copy of P1. The exterior product morphism A(P1 ) ⊗ A(P1 ) ⊗ A(P1 ) → A(F ) is an isomorphism (see [21], Example 8.3.7), thus we finally obtain A(F ) ∼ Z[h1, h2, h3 ]/(h2, h2, h2 ).

= 1 2 3 In particular Pic(F ) ∼ Z⊕3, whence (F ) = 3. Such an equality characterizes F = among del Pezzo threefolds.

The aim of this paper is to classify the rank two aCM bundles on F. In Section 2 we first recall some general definitions and facts on locally free sheaves. The next three Sections, 3, 4 and 5, are devoted to the proof of our first main result.

Theorem A. Let E be an indecomposable aCM bundle of rank 2 on F and let c1 (E) = α1 h1 + α2 h2 + α3 h3. Assume that h0 F, E = 0 and h0 F, E(−h) = 0 (we

briefly say that E is initialized). Then:

(1) the zero locus E := (s)0 of a general section s ∈ H 0 F, E has codimension 2 inside F ;

(2) either 0 ≤ αi ≤ 2, i = 1, 2, 3, or c1 = h1 + 2h2 + 3h3 up to permutations of the hi ’s;

(3) if c1 = 2h, or c1 = h1 + 2h2 + 3h3 up to permutations of the hi ’s, then E is Ulrich.

Going back to the classification given in [3] of indecomposable initialized aCM bundles of rank 2 on del Pezzo threefolds of degree d = 3, 4, 5, let us note two things. First, such a characterization can be easily generalized also to the cases


d = 1, 2, and the bound 0 ≤ c1 ≤ 2 remains true in these cases. Second, from such a viewpoint the statement of Theorem A can be read as follows: either c1 satisfies the aforementioned “standard ” bound, or E is a “sporadic”bundle.

In Section 6 we provide a complete classification of bundles satisfying the standard bound. Then we examine the sporadic case in Section 7. Finally, we will answer to the natural question of determining which intermediate cases are actually admissible in Section 8. We summarize this second main result in the following simplified statement (see Theorems 6.1, 6.6, 7.1, 8.1 for an expanded and detailed statement).

Theorem B. There exists an indecomposable and initalized aCM bundle E of rank 2 on F with c1 (E) = α1 h1 + α2 h2 + α3 h3 if and only if (α1, α2, α3 ) is one of the

following, up to permutations:

(0, 0, 0), (2, 2, 2), (1, 2, 3), (0, 0, 1), (1, 1, 1), (1, 2, 2).

Moreover, denote by E the zero–locus of a general section of such an E. Then (1) if α1 = α2 = α3 = 0, then E is a line and any line on F can be obtained in such a way;

(2) if α1 = α2 = α3 = 2, then E is an elliptic normal curve of degree 8 in P7 ;

(3) if αi = i up to permutation, then E is a rational normal curve of degree 7 in P7 ;

(4) if α1 = α2 = 0, α3 = 1, then E is a line and any line on F can be obtained in such a way;

(5) if α1 = α2 = α3 = 1, then E is a, possibly reducible, reduced conic;

(6) if α1 = 1, α2 = α3 = 2, then E is a, possibly reducible, quintic with arithmetic genus 0.

In particular, for each line E ⊆ F we show the existence of exactly four indecomposable non–isomorphic aCM bundles of rank 2 whose general section vanishes exactly along E.

The case of del Pezzo threefolds with Picard number 2 and the study of moduli spaces of aCM bundles of rank 2 on del Pezzo threefolds will be the object of forthcoming papers.

1.1. Notation. Let R be a Noetherian local ring. We denote by d(R) its depth and by dim(R) its Krull dimension. R is Cohen–Macaulay (reps. Gorenstein) if and only if d(R) = dim(R) (reps. its injective dimension as R–module is finite).

Let R be a Noetherian ring. We say that R is Cohen–Macaulay (resp Gorenstein) if its localization RM is Cohen–Macaulay (resp. Gorenstein) for all the maximal ideals M ⊆ R.

If X and Y are schemes over k and X is a closed subscheme of Y we will denote 2 by IX|Y the sheaf of ideals of X inside Y. The sheaves CX|Y := IX|Y /IX|Y and its ˇ OX dual NX|Y := CX|Y are respectively called the conormal and the normal sheaf of X inside Y.

Let X ⊆ PN be a subvariety, i.e. an integral closed subscheme. We set OX (h) :=

OPN (1) ⊗ OX. X is arithmetically Cohen–Macaulay (resp. arithmetically Gorenstein) if its homogeneous coordinate ring is Cohen–Macaulay (resp. Gorenstein).

In this case we will say that X is aCM (resp. aG).

The variety X is aCM if and only if the natural restriction maps H 0 PN, OPN (t) → H X, OX (th) are surjective and hi X, OX (th) = 0, 1 ≤ i ≤ dim(X) − 1. X is 0 4 G. CASNATI, D. FAENZI, F. MALASPINA aG if and only if it is aCM and α–subcanonical, i.e. its dualizing sheaf satisfies ωX ∼ OX (αh) for some α ∈ Z.

= A vector bundle on X is locally free sheaf on X of finite rank. If X is a scheme, then Ar (X) denotes the Chow group of codimension r cycles on X up to rational +∞ equivalence and A(X) := r=0 Ar (X). By a vector bundle, we mean a coherent locally free sheaf. For all the other unmentioned definitions, notations and results see [24] and [23].

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We conclude that the assumption E = ∅ yields a contradiction. This completes the proof of the statement.

We are now ready to prove the claimed effectiveness of c1 − D.

Proposition 3.4. If E is an indecomposable initialized aCM bundle of rank 2 on F, s ∈ H 0 F, E and D denotes the component of codimension 1 of (s)0, if any, then c1 ≥ c1 − D ≥ 0.

In particular if c1 = 0, then the zero–locus of each section s ∈ H 0 F, E has codimension 2.

Proof. The equality E = ∅ implies IE|F ∼ OF. Sequence (2) thus becomes = 0 −→ OF (D) −→ E −→ OF (c1 − D) −→ 0.

(7) If D ∈ |2h2 + 2h3 |, then the cohomology of Sequence (7) twisted by OF (−2h) would give 0 = h0 F, OF (c1 − D) ≥ h0 F, OF (c1 − D − 2h) = h1 F, OF (D − 2h) = 1, a contradiction. If D ∈ |2h3 |, then h1 F, OF (c1 −D−2h) = h2 F, OF (D−2h) = 1 contradicting the condition 1 ≤ −1.

In the remaining cases we know that hi F, OF (D − h) = hi F, OF (D − 2h) = 0, i = 1, 2. Taking the cohomology of Sequence (7) suitably twisted, we obtain hi F, OF (c1 − D − h) = hi F, OF (c1 − D − 2h) = 0 in the same range, because both E and OF (D) are aCM.

If Sequence (7) does not split, then H 1 F, OF (2D − c1 ) = Ext1 OF (c1 − D), OF (D) = 0.

F Thus δj − j ≤ −2 for exactly one j = 1, 2, 3. Since δ1 ≥ 0 and 1 ≤ −1 (as we assumed above) such a j is not 1. We can assume δ3 − 3 ≤ −2, whence we obtain 3 ≥ 2. If 2 ≥ 1, then h1 F, OF (c1 − D − h) = 0. If 2 ≤ 0, then h2 F, OF (c1 − D − 2h) = 0. In both cases we have a contradiction.

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5. The proof of the theorem A In this section we will complete the proof of Theorem A stated in the introduction. It only remains to show that if E is an indecomposable, initialized, aCM bundle, then its general section vanishes exactly along a curve (see Proposition 3.4).

We already checked in the previous sections (see Propositions 3.4 and 4.2) that either c1 ≥ 0 and 2h − c1 ≥ 0 or c1 = h1 + 2h2 + 3h3 up to permutations of the hi ’s for such kind of bundles and that the general section vanishes exactly along a curve when either c1 = 0, or 2h − c1 = 0, or c1 = h1 + 2h2 + 3h3 up to permutations of the hi ’s.

It follows that we can restrict our attention to the remaining cases. If we assume α1 ≤ α2 ≤ α3, such cases satisfy α1 ≤ 1 ≤ α3.

Lemma 5.1.

Let E be an indecomposable initialized aCM bundle of rank 2 on F whose general section s ∈ H 0 F, E satisfies (s)0 = E ∪ D where E has codimension 2 (or it is empty) and D ∈ |δ1 h1 + δ2 h2 + δ3 h3 | is non–zero.


RANK TWO ACM BUNDLES 13 (1) up to permutations (δ1, δ2, δ3 ) ∈ { (0, 0, 1), (0, 1, 1), (0, 0, 2), (0, 1, 2) } ;

(2) E is not globally generated;

(3) E = ∅.

Proof. We already know that (δ1, δ2, δ3 ) ∈ { (0, 0, 0), (0, 0, 1), (0, 1, 1), (0, 0, 2), (0, 1, 2), (0, 2, 2) } (see Relation (4)). Since D = 0, it follows that (δ1, δ2, δ3 ) = (0, 0, 0). Let D ∈ |2h2 + 2h3 | and look at the cohomologies of Sequences (2) and (3) respectively twisted by OF (−2h) and OF (c1 − D − 2h). On the one hand, taking into account that h0 F, E(−2h) = h1 F, E(−2h) = 0 by hypothesis we obtain 1 = h1 F, OF (D − 2h) = h0 F, IE|F (c1 − D − 2h) ≤ h0 F, OF (c1 − D − 2h).

On the other hand we also know that the last dimension is zero, due to the restrictions D ≥ 0, 2h − c1 ≥ 0 and c1 = 2h. It follows that equality (δ1, δ2, δ3 ) = (0, 2, 2) cannot occur.

If E would be globally generated, then the general section s ∈ H 0 F, E should vanish on a curve, contradicting the hypothesis.

Finally, assume that E = ∅. Thus IE|F (c1 − D) ∼ OF (c1 − D) in Sequence = (2). We obviously have that OF (D) is globally generated and h1 F, OF (D) = 0.

Moreover OF (c1 − D) is globally generated too by Proposition 3.4. We conclude that E is globally generated too thanks to Sequence (2) contradicting what we just proved above.

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When the class of E is h1 h3 + h1 h2, then we cannot repeat the above argument.

In this case we have that h2 F, OF (D − h) = h3 F, OF (D − 2h) = 0. Moreover E is aCM, hence we obtain h1 F, IE|F (c1 − D − h) = h2 F, IE|F (c1 − D − 2h) = 0.

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