Classification of rank two weak Fano bundles on del Pezzo threefolds of degree four

We classify rank two vector bundles on a given del Pezzo threefold of degree four whose projectivizations are weak Fano into seven cases. We also give an example for each of these seven cases.

1. Introduction 1.1. Motivation. The motivation of this study comes from the classification of Fano manifolds. Since smooth Fano 3-folds were classified (see [18] and references therein), many researchers have treated the classification of Fano 4-fold having projective bundle structures. In 1990, Szurek and Wiśniewski called a vector bundle whose projectivization is Fano a Fano bundle and gave a classification of rank 2 Fano bundles on P 3 or a smooth hyperquadric Q 3 [37]. After that, the classification of rank 2 Fano bundles over smooth Fano 3-folds has been addressed by many researchers (e.g. [37,22,28,27]). In particular, Muñoz, Occhetta, and Solá Conde classified rank 2 Fano bundles on Fano 3-folds of Picard rank 1 [27].
On the other hand, after the establishment of the classification of Fano 3-folds, several papers also started classifying weak Fano 3-folds. In 1989, K. Takeuchi developed his 2-ray game method by considering not only Fano 3-folds but also weak Fano 3-folds of Picard rank 2 [38]. By this 2-ray game method, he successfully gave a concise proof of the existence of a line on a Mukai 3-fold, which is a new perspective on the classification of Fano 3-folds. Since the establishment of the 2-ray game method, classifying weak Fano 3-folds of Picard rank 2 has been considered to be significant and treated by many researchers (e.g. [38,22,19,8]).
In view of these previous researches, classifying weak Fano 4-folds with Picard rank 2 would be important to investigate Fano 4-folds. Our approach to this problem is to consider weak Fano 4-folds with P 1 -bundle structures, as Szurek-Wiśniewski did [37].
1.2. Known classification of weak Fano bundles. In this paper, we say that a vector bundle E on a smooth projective variety X is weak Fano if its projectivization P X (E) is a weak Fano manifold. Weak Fano bundles are firstly introduced by Langer [22] as generalizations of Fano bundles. Until now, rank 2 weak Fano bundles are classified when the base space is the projective space [22,39,30] or a hypercubic in P 4 [16].
We quickly review these known results. On P 2 , Langer [22] and Ohno [30] classified weak Fano bundles whose 1st Chern class is odd. Yasutake [39] classified rank 2 weak Fano bundles over P 2 whose 1st Chern class is even except for the existence of a weak Fano rank 2 bundle E with c 1 (E) = 0 and c 2 (E) = 6. Meanwhile, Cutrone-Marshburn [8, P.2, L.14] pointed out that such a weak Fano bundle exists as the Bordiga scroll [32]. Yasutake also classified rank 2 weak Fano bundles on P n with n ≥ 3. Thus the rank 2 weak Fano bundles over projective spaces are classified. Rank 2 weak Fano bundles on smooth cubic hypersurfaces in P 4 are also classified by the 3rd author of this article [16]. 1.3. Instanton bundles and Ulrich bundles. Besides, weak Fano bundles have emerged in the study of different contexts. For example, minimal instanton bundles and special Ulrich bundles are most studied weak Fano (but not Fano) bundles on del Pezzo 3-folds.
Let X be P 3 or a del Pezzo 3-fold of Picard rank 1, i.e., a smooth Fano 3-fold of Picard rank 1 whose canonical divisor is divisible by 2. Then an instanton bundle on X is defined to be a rank 2 slope stable vector bundle E with c 1 (E) = 0 and H 1 (E( KX 2 )) = 0 [3,21,9]. The moduli space of instanton bundles on P 3 is a significant object for mathematical physics. Indeed, a certain subset of this moduli space corresponds to self-dual solutions of the SU(2) Yang-Mills equations on the 4-sphere S 4 up to gauge equivalence [2,3]. After this, Faenzi [9] and Kuznetsov [21] defined a generalized notion for instanton bundles E on a del Pezzo 3-fold X as above. By the above reason, moduli spaces of instanton bundles on P 3 and del Pezzo 3-folds have been studied deeply (e.g. [9,21]).
For an instanton bundle E on X, it is known that −K X .c 2 (E) ≥ 4 (cf. [21,Corollary 3.2], [9,Lemma 1.2]). Thus an instanton bundle E is said to be minimal if −K X .c 2 (E) = 4. When X is a del Pezzo 3-fold, a vector bundle E is a minimal instanton bundle if and only if E(1) is a special Ulrich bundle of rank 2 [21,Lemma 3.1], which is defined to be a rank 2 vector bundle F such that det F ≃ O X (−K X ) ≃ O X (2) and H • (F (−j)) = 0 for every j ∈ {1, 2, 3}. When d := O X (1) 3 ≥ 3, the above definition is equivalent to say that F has the following linear resolution on P d+1 : [4]. Moreover, Beauville showed that every special Ulrich bundle F of rank 2 is isomorphic to a unique non-trivial extension of I C (2) by O X , where C is a normal elliptic curve in X [4, Remark 6.3]. Since every normal elliptic curve C is defined by quadratic equations [14], F is globally generated. Since det F ≃ O(−K X ), we conclude that F is a weak Fano bundle.
In summary, on a del Pezzo 3-fold of degree d ≥ 3, minimal instanton bundles are the most well-studied examples of weak Fano bundles. In this article, we supplementally show that every slope stable weak Fano bundle E with rk E = 2 and c 1 (E) = 0 is an instanton bundle (see Corollary 4.7).

Main Result.
In this article, we classify rank 2 weak Fano bundles over a del Pezzo 3-fold of degree 4, which is a smooth complete intersections of two hyperquadrics in P 5 [10]. Our classification is given as follows.
Theorem 1.1. Let X be a smooth complete intersection of two hyperquadrics in P 5 . For a normalized bundle E of rank 2, E is a weak Fano bundle if and only if E is one of the following.
, where C is a nondegenerate smooth elliptic curve in X of degree 6. (vii) A unique non-trivial extension of I C (1) by O X (−1), where C is a nondegenerate smooth elliptic curve in X of degree 7 which is defined by quadratic equations.
In the above statement, a unique non-trivial extension E of F by G means that dim Ext 1 (F , G) = 1 and E fits into an exact sequence 0 → G → E → F → 0 such that the extension class is non-zero in Ext 1 (F , G). Furthermore, on an arbitrary smooth complete intersection of two hyperquadrics in P 5 , there exist examples for each case of (i) -(vii). Remark 1.2. We give some remarks about the above result.
(1) The vector bundles of type (i), (ii) and (v) are Fano bundles [27]. Others are not Fano bundles but weak Fano bundles. Our result is essentially different from the classification of rank 2 weak Fano bundles on a hypercubic in P 4 [16], which was done by the 3rd author of this article. Briefly speaking, he proved that slope stable rank 2 weak Fano bundles are always minimal instanton [16,Theorem 1.1]. In contrast, on a complete intersection of two hyperquadrics, other stable bundles appear. Indeed, every weak Fano bundle E of type (vii) in Theorem 1.1 is a non-minimal instanton bundle. Our result also shows that the zero scheme of a general section of E(1) is a non-projectively normal elliptic curve C defined by quadratic equations. Hence E(1) is not 0-regular but globally generated. Note that we can not remove the condition that C is defined by quadratic equations (cf. Theorem 1.5 and Remark 5.8).
1.5. Key results for proving Theorem 1.1. Here we collect the key ingredients for the proof of Theorem 1.1.
Let X be a smooth complete intersection of two hyperquadrics in P 5 . Let H X be a hyperplane section of X and L X a line on X. Then Pic(X) ≃ H 2 (X, Z) ≃ Z[H X ] and H 4 (X, Z) ≃ Z[L X ]. Using the above isomorphisms, we identify the cohomology classes with integers. Note that the property to be weak Fano is slope stable under taking tensor products with line bundles. To classify rank 2 weak Fano bundles E, we may assume that E is normalized, i.e., c 1 (E) ∈ {0, −1}.
Theorem 1.4. Let X be a del Pezzo 3-fold of degree 4. Let E be a rank 2 vector bundle on X. Then the following conditions are equivalent.
(2) E is a Fano bundle with (c 1 (E), c 2 (E)) = (−1, 2). In particular, by the classification of Fano bundles on a del Pezzo 3-fold of degree 4 [27], E is the restriction of the Spinor bundle [31] on a smooth hyperquadric in P 5 containing X, or the pull-back of the Spinor bundle on the 3-dimensional smooth hyperquadric Q 3 under a double covering X → Q 3 . Thirdly, we characterize weak Fano bundles E with (c 1 (E), c 2 (E)) = (0, 2) or (0, 3) as follows.
Theorem 1.5. Let X be a del Pezzo 3-fold of degree 4. Let E be a rank 2 vector bundle on X. Then the following assertions are equivalent.
Then by Theorems 1.3, 1.4, and 1.5, we obtain the characterization part of Theorem 1.1.
Finally, we show the existence of E for each condition in Theorem 1.1, which is namely the following theorem. To show Theorem 1.4, we will use the fact that h 0 (E(1)) > 0 for every normalized weak Fano bundle E of rank 2 (=Proposition 2.2 (1)). When (c 1 , c 2 ) = (−1, 2), the zero scheme of a global section of E(1) is a conic and hence we can show that E(1) is globally generated. Moreover, this fact implies that E is a Fano bundle, which is already classified in [27].
To show Theorem 1.5, we will prove the following theorem. If F is not globally generated, then X is a del Pezzo 3-fold of degree 1 and Applying this result, we see that E(1) is globally generated for every weak Fano bundle E with c 1 (E) = 0 on a del Pezzo 3-fold of degree 4, which will imply Theorem 1.5. As another corollary of Theorem 1.7, we will also see that every weak Fano bundle E with det E = O X on a del Pezzo 3-fold X of Picard rank 1 is an instanton bundle (=Corollary 4.7).
Our proof of Theorem 1.7 goes as follows. Let X be a Fano 3-fold of Picard rank 1 and F a vector bundle with c 1 (F ) = c 1 (X). Let π : M := P X (F ) → X be the projectivization and ξ a tautological divisor. Assume that Bs |ξ| = ∅ and ξ is nef and big. Since −K M = (rk F )ξ, M is a weak Mukai manifold. Then by using some results about ladders of Mukai varieties [36,17,33,35,24,25], we can obtain a smooth K3 surface S ⊂ M by taking general (rk F ) elements of |ξ| (=Theorem 4.1). Since the complete linear system |ξ| S | still has a non-empty base locus, there is an elliptic curve B and a smooth rational curve Γ on S such that ξ| S ∼ gB + Γ, where g = (1/2)ξ rk F +2 + 1 > 0 [34,36]. Note that Bs |ξ| = Γ.
Let us suppose that Γ is contracted by π : M → X. Then it is easy to see that Γ is a line in a fiber of π. As in [17, § 6], we consider the blowing-up of M along Γ. Since Γ is a line of a π-fiber, we can compute the conormal bundle of Γ and hence conclude that the anti-canonical image of Bl Γ M is the join of P 1 × P 2 and some linear space. This structure enables us to show X is a del Pezzo 3-fold of degree 1 and . The most technical part of this proof is to show that Γ is contracted by π : M → X. For more precise, see § 4.3.4. 1.6.3. Outline for Theorem 1.6. Fix an arbitrary smooth complete intersection X of two hyperquadrics in P 5 . The existence of (i), (ii) and (iii) in Theorem 1.1 is obvious. For the existence of (iv), (v), (vi), and (vii) in Theorem 1.1, it suffices to find a line, a conic, a non-degenerate sextic elliptic curve, and a non-degenerate septic elliptic curve defined by quadratic equations respectively. Indeed, if such a curve is found, the Hartshorne-Serre correspondence gives a vector bundle which we want to find. The existence of lines and conics are obvious since a general hyperplane section, which is a del Pezzo surface, contains such curves. The existence of a nondegenerate sextic elliptic curve was already proved by [4,Lemma 6.2]. Note that such a curve is always defined by quadratic equations (cf. [14]).
The remaining case is the existence of a septic elliptic curve defined by quadratic equations. Note that there is a non-degenerate septic elliptic curve in P 5 defined by quadratic equations (cf. [7,Theorem 3.3], [15, § 2]). Hence an example of a del Pezzo 3-fold of degree 4 containing such a curve was already known. In this article, we will slightly improve this known result by showing that every del Pezzo 3-fold of degree 4 contains such a curve.
To obtain this result, we study a non-degenerate septic elliptic curve C in P 5 and show that C is defined by quadratic equations if and only if C has no trisecants (=Proposition 5.2). For this characterization, we will use Mukai's technique [29], which interprets the property of being defined by quadratic equations into the vanishing of the 1st cohomologies of a certain family of vector bundles on C. Then as in [4, Lemma 6.2] and [12], we will produce such a curve C by smoothing the union of a conic and an elliptic curve of degree 5, which has no trisecants. For more precise, see § 5.3.

1.7.
Organization of this article. In § 2, we give certain numerical conditions for rank 2 weak Fano bundles over del Pezzo 3-folds of degree 4 and prove Theorem 1.3. In § 3, we prove Theorem 1.4. In § 4, we prove Theorem 1.7 and Theorem 1.5 as its corollary. In § 5, we find a septic elliptic curve C on an arbitrary del Pezzo 3-fold of degree 4 and prove Theorem 1.6.
Notation and Convention. Throughout this article, we will work over the complex number field. We regard vector bundles as locally free sheaves. For a vector bundle E on a smooth projective variety X, we define P X (E) := Proj Sym E. In this article, we only deal with slope stability among the stabilities. For a given ample When X is of Picard rank 1, we abbreviate H-stable (resp. H-semi-stable) to simply stable (resp. semistable).
We say that E is weak Fano if P X (E) is a weak Fano manifold, that is, −K PX (E) is nef and big. When X is a smooth Fano 3-fold of Picard rank 1, we often identify H 2i (X, Z) ≃ Z by taking an effective generator. We also say a rank 2 vector bundle E on X is normalized when c 1 (E) ∈ {0, −1}.

Numerical bounds: Proof of Theorem 1.3
In this section, X denotes a smooth del Pezzo 3-fold of degree 4 and H X a class of a hyperplane section. Let us recall that H 2 (X, Z) is generated by H X and H 4 (X, Z) is generated by H 2 X /4. If we naturally identify H 2 (X, Z) as Z and H 4 (X, Z) as Z by these generators, we can take integers c 1 , c 2 as c 1 (E) = c 1 ·H X and c 2 (E) = c 2 ·H 2 X /4 for a given vector bundle E. We also call this integer c i the i-th Chern class of E for each i ∈ {1, 2}. Let π : P X (E) → X be the projectivization, ξ the tautological line bundle, and H := π * H X . In this setting, The goal of this section is to prove Theorem 1.3. First of all, it is easy to check that the vector bundle E = O X ⊕ O X (a) with a ∈ Z ≥0 is weak Fano if and only if a ∈ {0, 1, 2} since ξ and H generates the nef cone of P X (E) and −K PX (E) = 2ξ + (2 − a)H. This means that, for a rank 2 normalized bundle E on X, the conditions (i-iii) in Theorem 1.1 are equivalent to E being a decomposable weak Fano bundle. Thus, to prove Theorem 1.3 for a given rank 2 weak Fano normalized bundle E on X, we will characterize E with global sections as the condition (iv) in Theorem 1.1, and compute the Chern classes of E with no global sections.
Lemma 2.1. Let X be a del Pezzo 3-fold of degree 4 and E a rank 2 weak Fano normalized bundle with Chern classes c 1 , c 2 ∈ Z.
Then we obtain the following assertions.
Following [28, Section 1.1], we introduce the invariant β := min{n ∈ Z | h 0 (E(n)) > 0} for a given vector bundle E. Then E has O(−β) as a subsheaf, which is saturated by the definition of β. Hence we have the following exact sequence where Z is a closed subscheme of X of purely codimension 2 or the empty set. This exact sequence (2.2) means that c 2 (E(β)) = [Z] belongs an effective class. Now we bound c 2 and β for a normalized weak Fano bundle of rank 2 as follows.
Proposition 2.2. Let X be a del Pezzo 3-fold of degree 4. Let E be a rank 2 normalized weak Fano bundle on X with Chern classes c 1 and c 2 . Set β := min{n ∈ Z | H 0 (X, E(n)) = 0}. Then we obtain the following assertions.
Proof. We use the same notation as in Lemma 2.1. Note that On the other hand, by (2.1), we have (1). Hence c 2 is an even integer, which implies c 2 ≤ 2.
Proof of Theorem 1.3. Let X be a del Pezzo 3-fold of degree 4. Let E be a normalized rank 2 weak Fano bundle on X.
(1) It is easy to see that if E is one of (i-iv) in Theorem 1.1, then h 0 (E) > 0, which implies β ≤ 0. Hence it suffices to show the converse direction.
Suppose that β ≤ 0. If β < 0, then by Proposition 2.2 (1), we have c 1 = 0 Hence we may assume β = 0. Then we have Hence we may assume that c 2 > 0. Since β = 0, E has a global section, which implies that ξ is linearly equivalent to an effective divisor. Then by Lemma 2.1 (2), we have Since c 2 > 0, we have c 1 = 0 and c 2 = 1. Hence Z is a line on X. Then the exact sequence (2.2) gives the description (iv) in Theorem 1.1 (cf. [16,Lemma 3.2]). Hence E is one of (i-iv) in Theorem 1.1. We complete the proof of (1). Here we note that the extension 0 → O X → E → I L → 0 for an arbitrary line L on X is uniquely determined by L since Ext By the Riemann-Roch theorem and the additivity of the Euler character, we have that the Hilbert polynomial of C is 2t + 1, where t is a variable. Hence C is a conic curve that may be singular.
By the above argument, in order to show that E is (v) in Theorem 1.1, it is enough to show that E(1) is globally generated. Let P be the linear span of C in P 5 . Since C is a conic, P is a plane (i.e. two-dimensional linear subspace). Choose two smooth quadrics Q 1 and Q 2 such that X = Q 1 ∩ Q 2 . Then at least one of Q 1 and Q 2 does not contain P . Indeed, if both of Q 1 and Q 2 contain P , then P is a divisor of X, which contradicts that Pic(X) ≃ Z generated by a hyperplane section H X . Therefore we may assume that the plane P is not a subscheme of Q 1 . In this case, the intersection Q 1 ∩ P is a conic in P . Since C ⊂ X ∩ P ⊂ Q 1 ∩ P , we notice that C = X ∩ P . This means that C is defined by linear equations in X, and hence I C/X (1) is globally generated. Then the exact sequence (3.1) implies that E(1) is globally generated. We complete the proof of this implication (1) ⇒ (3).
Finally, we show the implication (3) ⇒ (2). Let E be (v) in Theorem 1.1, that is, E is the non-trivial extension of I C by O X (−1), where C is a smooth conic. As we saw in the proof of the implication (1) ⇒ (3), I C (1) is globally generated and hence so is E(1). Since E(1) is globally generated, it is nef and hence ξ + H is a nef divisor on P X (E). Thus −K PX (E) = 2ξ + 3H is ample, which means that E is a Fano bundle.
Here we note that the extension (3.1) is uniquely determined for a smooth conic C since  Thus, by repeating this argument and using Reid's result [33,Theorem], we have a ladder M n−2 ⊂ M n−3 ⊂ · · · ⊂ M 1 ⊂ M 0 = M such that each M i+1 ∈ |A| Mi | has only Gorenstein canonical singularities. We may assume that M i is smooth on the outside of Bs |A| by Bertini's theorem. We set S = M n−2 and X = M n−3 . Note that X is a weak Fano 3-fold such that A| X = −K X and Bs |A| = Bs | − K X |. By Shin's result [35,Theorem 0.5], it is known that dim Bs | − K X | ∈ {0, 1}. Now we show dim Bs |A| = 0. If dim Bs |A| = 0, then it also follows from [ibid.,Theorem 0.5] that Bs |A| = Bs | − K X | consist of a single point p ∈ M and p is an ordinary double point of S. Moreover, p must be a singular point of X. Then by the same argument as [25, Proof of Theorem 3.1], we can show that ψ| S : S → ψ(S) is isomorphic at p. Since S = ψ −1 (ψ(S)), ψ is isomorphic at p. In particular, M is smooth at p := ψ(p). By the same argument as [24, Proof of Theorem 2.5], there is a member D ∈ |A| which is smooth at p. Since ψ is isomorphic at p, D := ψ −1 (D) ∈ |A| is smooth at p = Bs |A|. Hence general members of |A| are smooth, which contradicts that X = M n−3 is singular at p.
Therefore, dim Bs | − K X | = dim Bs |A| = 1. Again by [35, Theorem 0.5], Bs | − K X | is a smooth rational curve and | − K X | has a member which is smooth along Bs | − K X |. Hence a general member of |A| is smooth. We complete the proof.

4.2.
Lower bound of the degree of elliptic curves on Fano 3-folds. As another preliminary for proving Theorem 1.7, we give a (non-optimal) lower bound of the anti-canonical degree of an elliptic curve on a Fano 3-fold as follows.
(3): When r(X) ≥ 3, it is known by Kobayashi and Ochiai that X is isomorphic to a hyperquadric Q 3 in P 4 or the projective 3-space P 3 . It is easy to see that the degree of every elliptic curve B on Q 3 (resp. P 3 ) is greater than or equal to 4 (resp. 3). Thus we obtain (3).

4.3.
Proof of Theorem 1.7. Let X be a smooth Fano 3-fold of Picard rank 1 and F a nef bundle with det F = O(−K X ) and rk F ≥ 2. Let π : M := P X (F ) → X be the natural projection and ξ a tautological divisor. Set n = dim M = 2 + rk F . Recall the assumption ξ dim M = c 1 (F ) 3 − 2c 1 (F )c 2 (F ) + c 3 (F ) > 0. Since −K M ∼ (rk F )ξ, M is a weak Mukai manifold. Then Theorem 4.1 gives the following smooth ladder of |ξ|: where M i is a smooth prime member of |ξ| Mi−1 |.
Until the end of this proof, we suppose that F is not globally generated. We proceed in 5 steps.

4.3.1.
Step 1. In Step 1 and Step 2, we prepare some facts on the weak Fano 3-fold M n−3 and the K3 surface M n−2 . Claim 4.3. Set X := M n−3 , which is a weak Fano 3-fold. Then π X := π| X : X → X is the blowing-up along a (possibly disconnected) smooth curve C on X. Moreover, the following assertions hold.
(2) Every connected component of C is of genus g ≥ 1.
Proof. Let s 1 , . . . , s rk F −1 ∈ H 0 (X, F ) ≃ H 0 (M, O(ξ)) be the sections defining X. Set s = (s 1 , . . . , s rk F −1 ) : O ⊕ rk F −1 → F and let C := {x ∈ X | rk s(x) < rk F − 1} be the degeneracy locus of s, which is possibly empty. Since X is a smooth variety, X is the blowing-up of X along C with dim C ≤ 1 (see [11,Lemma 6.9]). Moreover, we obtain the following exact sequence: By the exact sequence (4.2), we have , which is globally generated since X is of Picard rank 1 (cf. [18,Corollary 2.4.6]). Hence we may assume that C is not empty. Moreover, it follows from [26,Theorem (3.3)] that C is smooth.
(1) The equality C ≡ c 2 (F ) follows from the exact sequence (4.2). Moreover, since c 1 (F ) 3 (2) We assume that a connected component C i of C is a smooth rational curve. Taking the restriction of the exact sequence (4.2) to C i , we obtain an exact se- by the adjunction formula, the degree of Im(s| Ci ) is −2, which contradicts that Im(s| Ci ) is a quotient of O ⊕ rk F −1 P 1 . We complete the proof.
(1) There is an elliptic fibration f S : S → P 1 and a section Γ of f S such that ξ| S ∼ gB + Γ for a general fiber B of f S . Moreover, it holds that S is normal and contains the curve C defined in Claim 4.3. (4) If Γ is not contracted by π S , then π S is isomorphic along a general f S -fiber B. In particular, a smooth elliptic curve B := π S (B) is an effective Cartier divisor on S. Moreover, it holds that Proof. Since we assume that Bs |ξ| is not empty, so is Bs |ξ| S |, where ξ| S is a nef and big divisor on S. Then (1) is known by [36,Lemma 2.3] and [34, (2.7 (3) Set E S := Exc(π X ) ∩ S where π X is the blowing-up defined in Claim 4.3. Then E S is a member of some tautological bundle of the P 1 -bundle Exc(π X ) → C.
Hence S contains C and every fiber of S → S is connected, which implies S is normal.

4.3.3.
Step 3. In this step, we show the following claim.
Claim 4.5. If Γ is not contracted by π S , then, for a general f S -fiber B, we have Proof. Since Γ is not contracted by π S , B := π S (B) is a smooth elliptic curve by Claim 4.4 (4). The inequality (−K X ).B ≥ (1/4)(−K X ) 3 follows from Lemma 4.2 when r(X) ≤ 2. Assume that r(X) ≥ 3, that is, X = P 3 or Q 3 . Let H X be an ample generator of Pic(X) and r := r(X). Set H = (π * X H X )| S and A := rH ∼ (π * X (−K X ))| S . To obtain a contradiction, we assume (−K X ).B < (1/4)(−K X ) 3 . Then Lemma 4.2 gives (−K X ).B = 12. When X = P 3 (resp. Q 3 ), B is a plane cubic curve (resp. a complete intersection of two quadrics in P 3 ) and hence H − B is effective. Let C ⊂ S be the proper transform of C ⊂ S. Then C is a nef divisor on S by

4.3.5.
Step 5. Finally, we prove Theorem 1.7. Let τ : M := Bl Γ M → M = P X (F ) be the blowing-up. Set D := Exc(τ ) and ξ := τ * ξ − D. Since ξ| S ∼ gB + Γ, | ξ| is base point free. Let Ψ : M → P g+rk F be the morphism given by | ξ| and W its image. Then, for the proper transform S ⊂ M of S, the morphism S → Ψ( S) is given by |gB|. Since Ψ( S) is a linear section of W , W is of degree g. In particular, the ∆-genus of W ⊂ P g+rk F is 0. Moreover, D → W is birational. Let us compute g = deg W and the normal bundle N Γ/M . By Claim 4.6, Γ ⊂ X is contracted by π X . Then Claim 4.3 implies that Γ is a line in a fiber of M = P X (F ) → X. Therefore, it holds that → W is the morphism given by the complete linear system of the tautological bundle and W is a join of P rk F −3 and P 1 × P 2 in P rk F +3 .
Then we let V = M n−4 ⊂ M = P X (F ) be the intersection of general (r − 2) members of |ξ|. V contains Γ since Γ is the base locus of |ξ|. Let τ V : V := Bl Γ V → V be the blowing-up and Since D V ≃ P 1 × P 2 and D V → Γ coincides with the first projection, it holds that F 1 ∩ F 2 = ∅. Hence |F i | is base point free and induces a morphism h : V → P 1 . Thus we obtain the following commutative diagram: Recall that the h-section Γ is contracted by π| V : V → X, which is an adjunction theoretic scroll. We set x := π| V (Γ) and J := π| −1 V (x). Then J is a projective space containing Γ. If dim J ≥ 2, then h| J : J → P 1 must contract the whole J, which contradicts that Γ is a section of h. Hence J = Γ. Therefore, the induced morphism (π| V , h) : V → X × P 1 is birational. Since every fiber of π| V : V → X is a projective space, V is isomorphic to X × P 1 .
Let α : O ⊕r−2 X → F be the morphism corresponding to (r − 2) members of |ξ| which define V ⊂ P X (F ). Since V ≃ X × P 1 , α is injective and Cok α ≃ L ⊕2 for some line bundle L. Since det F = O(−K X ), −K X is divided by 2 in Pic(X) and . By the classification of del Pezzo 3-folds, X is a del Pezzo 3-fold of degree 1 (cf. [10]). We complete the proof of Theorem 1.7.

4.4.
Proof of Theorem 1.5. In this section, we give a corollary as consequences of the above results and show Theorem 1.5.
Corollary 4.7. Let X be a del Pezzo 3-fold of Picard rank 1. Let E be a rank 2 weak Fano bundle on X with c 1 (E) = 0. Then the zero scheme of a general global section of E(1) is a connected smooth elliptic curve and h 1 (E(−1)) = 0. Especially, if E is slope stable, then E is an instanton bundle [21,9].
Proof. First, we treat the case X is of degree 1 and E = O ⊕2 X . In this case, it is well-known that X is a smooth sextic hypersurface of P (1, 1, 1, 2, 3) [10] and hence the zero scheme of a general section s ∈ H 0 (E(1)) = H 0 (O X (1)) ⊕2 , which is a general complete intersection of two members of |O X (1)|, is a connected smooth elliptic curve. Hence, by Theorem 1.7, we may assume that F := E(1) is globally generated. Let C be the zero scheme of a general s ∈ H 0 (F ). Since C is a smooth curve with ω C ≃ O C , C is a disjoint union of smooth elliptic curves. We show that C is connected. Let d be the degree of X and ξ F be the tautological bundle of P X (F ). Since ξ 4 we have deg C < 2d. By Lemma 4.2 (2), the degree of every connected component of C is greater than or equal to d. Therefore, C must be connected. Using an exact sequence 0 → O → F → I C (2) → 0, we have H 1 (E(−1)) = H 1 (F (−2)) = H 1 (I C ) = 0 since C is a smooth connected curve and H 1 (O X ) = 0. Therefore, if E is slope stable, then E is an instanton bundle by definition.

Existence: Proof of Theorem 1.6
In this section, we show the existence of vector bundles for each case in Theorem 1.1 on an arbitrary del Pezzo 3-fold of degree 4. The existence of the cases (i-iii) is clear, and that of the case (iv) and (v) was proved in Proof of Theorem 1.3 in Section 2 and Proof of Theorem 1.4 in Section 3 respectively. The existence of the case (vi) is equivalent to the existence of a special Ulrich bundle of rank 2 (cf. Remark 1.2 (3)), which is proved by [4, Proposition 6.1]. Thus, the remaining case is only (vii). The main purpose of this section is to prove the existence of a bundle belonging to (vii), which is equivalent to the following theorem by the Hartshorne-Serre correspondence.
Theorem 5.1 (=Theorem 1.6 for (vii)). Let X ⊂ P 5 be an arbitrary smooth complete intersection of two hyperquadrics. Then X contains an elliptic curve C of degree 7 defined by quadratic equations.
The key proposition is the following.
Proposition 5.2. Let C ⊂ P 5 be an elliptic curve of degree 7. Then the following assertions are equivalent.
(1) C is defined by quadratic equations.
By Proposition 5.2, it suffices to show that every smooth complete intersection X of two hyperquadrics in P 5 contains an elliptic curve C of degree 7 having no trisecants. We will obtain such a curve by smoothing the union of a conic and an elliptic curve of degree 5. [29]. We quickly review his technique.

Mukai's technique. Our proof of the implication (3) ⇒ (1) in Proposition 5.2 is based on Mukai's technique
Let C be an elliptic curve of degree 7 in P 5 . Let σ : P := Bl C P 5 → P 5 be the blowing-up. Set H := σ * O P 5 (1), E = Exc(σ), and e := σ| E . Let i : C ֒→ P 5 and j : E ֒→ P be the inclusions. Note that |2H − E| is base point free if and only if C is defined by quadratic equations. Consider the morphism i × σ : C × P → P 5 × P 5 . We denote the diagonal in P 5 × P 5 by ∆ and its pull-back via i × σ by E. Note that E is isomorphic to E. On P 5 × P 5 , there is the following natural exact sequence Ω P 5 (1) ⊠ O P 5 (−1) → I ∆/P 5 ×P 5 → 0. Taking the pull-back under i × σ, we have Since E is of codimension 2 in C × P, the kernel of the above surjection, say E, is locally free. Thus, we obtain an exact For each closed point x ∈ Bl C P 5 , we set and regard it as a locally free sheaf on C. In this setting, Mukai [29] proved the following theorem: We show the implication (2) ⇒ (3). Let Φ : C ֒→ P 6 be the embedding given by |O C (1)|. Then there is a point p ∈ P 6 \ Φ(C) such that the image of Φ(C) by the projection from p is C ⊂ P 5 . Hence we obtain the following exact sequence: Assume that Ω P 5 (1)| C has a rank 3 subsheaf F such that µ(F ) ≥ −7/5. Since F is also a subsheaf of a slope stable bundle Ω P 6 (1)| C [6, Theorem 1.3], we have −7/5 ≤ µ(F ) < −7/6, which implies that deg F = −4. Let F ⊂ Ω P 5 (1)| C be the saturation of F in Ω P 5 (1)| C . Since F is also a rank 3 subsheaf of Ω P 6 (1)| C , we also have deg( F ) = −4 by the same argument as above. Hence it follows that F = F , i.e., F is saturated. Let G := Ω P 5 (1)| C /F , which is locally free. Considering the dual, we obtain an exact sequence 0 → G ∨ → T P 5 (−1)| C → F ∨ → 0. We set V = C 6 and P 5 := P(V ). Note that there is a surjection V ∨ ⊗O C ։ T P 5 (−1)| C . Let Taking the restriction on C, we obtain surjections Finally, we show the implication (3) ⇒ (1). Assume that Ω P 5 (1)| C is slope stable for rank 3 subsheaves. Let x ∈ Bl C P 5 be a point and set E x as in (5.1). Then by Theorem 5.3 and the exact sequence (5.2), we obtain inclusions E x ֒→ Ω P 5 (1)| C ֒→ Ω P 6 (1)| C . Note that the determinant bundles of the above three vector bundles are isomorphic.
(i) If (r i ) = (1, 1, 1, 1), then we have d i < −7/6 and hence d i ≤ −2, which contradicts that Since µ(E ∨ 2 ) = 5/3, we have Hom(ξ, E ∨ 2 ) = 0. (iv) If (r i ) = (2, 2), then d i ≤ −3 for each i. Hence deg E 1 = −3 and deg E 2 = −4. Since E 1 is slope stable and E 2 is slope semi-stable, there are exact where L i is a line bundle of degree i and M −2 is a line bundle of degree −2. Let ξ be a line bundle of degree 2 such that ξ −1 is not equal to L −2 or M −2 . Then It is easy to see We complete the proof of Proposition 5.2. 5.3. Proof of Theorem 5.1. Let X ⊂ P 5 be an arbitrary complete intersection of two hyperquadrics. We will construct an elliptic curve C of degree 7 on X such that C is defined by quadratic equations. We proceed with 4 steps.
Step 1. First of all, we confirm the following lemma.
Lemma 5.4. For an arbitrary closed point x ∈ X, lines on X passing through x are finitely many.
Proof. Let ψ : Bl x X → P 4 be the restriction of Bl x P 5 → P 4 . Then the Stein factorization of ψ is a crepant birational contraction. Since the proper transforms of the lines passing through x are contracted by ψ, if there are infinitely many lines on X passing through x, then ψ contracts a divisor D, which contradicts the classification of weak Fano 3-folds having crepant divisorial contractions [19].
Step 2. In this step, we show the following lemma.
Lemma 5.5. Let Γ be a smooth conic on X. Then there exists a smooth elliptic curve C of degree 5 satisfying the following.
(i) The scheme-theoretic intersection C ∩ Γ is reduced one point.
(ii) C ∪ Γ has no trisecants on X.
Proof. Let Γ be a smooth conic on X. Since the linear span Γ ⊂ P 5 is not contained in X, we have Γ ∩ X = Γ. In particular, for any two points on Γ, the line passing through them is not contained in X. Let x 1 ∈ Γ be a point. Let S 0 ⊂ X be a general hyperplane section such that S 0 has no lines on X passing through x 1 . Set {x 1 , x 2 } = S 0 ∩Γ. Let l := x 1 , x 2 . Then l ∩ X = {x 1 , x 2 }. Note that Λ := |O X (1) ⊗ m x1,x2 | is a 3-dimensional linear system whose general members are smooth. By Lemma 5.4, the lines passing through x 1 or x 2 on X are finitely many. Hence a general member S ∈ Λ does not contain any lines passing through x 1 or x 2 .
Let ε : S → P 2 be the blowing-up of P 2 at 5 points. Let h be the pull-back of a line and e 1 , . . . , e 5 the exceptional curves. Let σ : S → S be the blowing-up at x 1 and e 0 the exceptional curve. Note that S is a del Pezzo surface by (5.3). We take a general member C ∈ |3h − (e 0 + e 1 + · · · + e 4 )|. Set C = σ( C), which is smooth and passing through x 1 . Note that C is an elliptic curve of degree 5 on S. By taking general C, we may assume that C does not pass through x 2 . Since S ∩ Γ = {x 1 , x 2 }, we have C ∩ Γ = x 1 , which implies (i).
We show (ii). First of all, there are no bisecants of Γ on X since Γ ∩ X = Γ. Moreover, C has no trisecants on X. Indeed, since S is a complete intersection of two hyperquadrics in P 4 , if C has a trisecant l, then l is contained in S. Since every line l in S is linear equivalent to e i for 1 ≤ i ≤ 5, h − (e i + e j ) for 1 ≤ i < j ≤ 5, or 2h − (e 1 + · · · + e 5 ), we have C.l ≤ 2.
Hence, if there is a trisecant l of C ∪Γ, l must be a bisecant of C. Note that every bisecant l of C is contained in S. In fact, since l = C ∩ l ⊂ C ⊂ S , we have l = l ∩X ⊂ S ∩X = S. Then by (5.3), we have ∅ = l ∩{x 1 , x 2 } = l ∩Γ∩S = l ∩Γ.
Claim 5.7. The following assertions hold.
Let C 12 ∈ |I Z/Q1 (2)| be a general smooth member. Recall the birational map P 3 X in the diagram (5.4). We show that the proper transform C 12 ⊂ X of C 12 is a non-degenerate septic smooth elliptic curve on X which meets the line l at 3 points transversally.
Next, we show C 12 is a smooth elliptic curve meeting l transversally at 3 points. By the condition Claim 5.7 (2) (b), the proper transform E ⊂ Bl B P 3 of Q 0 meets C 12 transversally at 3 points. As we saw in the diagram (5.4), E ≃ Q 0 is contracted by σ : Bl B P 3 → X and σ| E : E → l corresponds to the 1st projection pr 1 : (P 1 ) 2 → P 1 under the isomorphism (5.5). Then by the condition Claim 5.7 (2) (c), σ| C12 : C 12 → C 12 is isomorphic. Moreover, C 12 meets l at the image of the 3 points q 1 , q 2 , q 3 . Therefore, C 12 is a smooth elliptic curve of degree 7 which meets l at 3 points.
Finally, we confirm that the embedding C 12 ⊂ X ⊂ P 5 is non-degenerated. Assume that there is a hyperplane section H of X containing C 12 . Since l is a trisecant of C 12 , H must contain l. Thus the proper transform H of H on P 3 is a hyperplane of P 3 . Then H contains C 12 , which contradicts that C 12 is of degree 4 in P 3 . We complete the proof of Proposition 5.6.
Remark 5.8. Let X ⊂ P 5 be an arbitrary smooth complete intersection of two hyperquadrics. By Proposition 5.6, there is a smooth elliptic curve C of degree 7 having a trisecant. By the Hartshorne-Serre construction, there is a rank 2 vector bundle E fitting into 0 → O X (−1) → E → I C/X (1) → 0. This vector bundle E is an instanton since H 1 (E(−1)) = H 1 (I C/X ) = 0 and c 2 (E) = 3. However, E is not a weak Fano bundle since there are surjections E| l ։ I C/X (1) ։ O l (−2).