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VECTOR BUNDLES ON FANO 3-FOLDS

WITHOUT INTERMEDIATE COHOMOLOGY

Enrique ARRONDO ∗

AND

Laura COSTA †

e-mail: enrique@sunal1.mat.ucm.es
e-mail: costa@cerber.mat.ub.es

DRAFT:5/4/98

1 Introduction

A well-known result of Horrocks (see [Ho]) states that a vector bundle on a projective
space has not intermediate cohomology if and only if it decomposes as a direct sum of
line bundles. There are two possible generalizations of this result to arbitrary varieties.
The first one consists of giving a cohomological characterization of direct sums of line
bundles. This has been done for quadrics and Grassmannians by Ottaviani (see [O1],
[O2]) and for rank-two vector bundles on smooth hypersurfaces in P

4 by Madonna
(see [Ma]).

The second generalization, to which we make some contribution in this paper, is
to characterize vector bundles without intermediate cohomology. Besides the result
of Horrocks for projective spaces, there is such a characterization for smooth quadrics
due to Knörrer (see [Kn]). In fact, it was shown by Buchweitz, Greuel and Schreyer
that only in the above two kind of varieties (projective spaces and quadrics) there is,

∗Partially supported by DGICYT PB96-0659.
†Partially supported by DGICYT PB94-0850 and DGICYT PB96-0166.

1

http://arXiv.org/abs/math/9804033v1


up to a twist by a line bundle, a finite number of indecomposable vector bundles with-
out intermediate cohomology (see [BGS]). As a consequence, one should not expect
to find a precise characterization of vector bundles without intermediate cohomology
on arbitrary varieties. As far as we know, there is only a result in that direction, for
the Grassmann variety G(1, 4) of lines in P

4, due to Graña and the first author (see
[AG]).

The present paper deals with vector bundles without intermediate cohomology on
some Fano threefolds with the Betti number b2 = 1. Specifically, we will work over
a smooth cubic in P

4, a smooth complete intersection of type (2, 2) in P
5 and over a

smooth intersection of the Grassmannian G(1, 4) ⊂ P
9 with three hyperplanes. For

rank-two vector bundles, we give a complete classification (see Theorem 3.4). For
arbitrary rank, we characterize which are the Chern classes of vector bundles without
intermediate cohomology and verifying some general conditions (see Theorem 4.9).

Next we outline the structure of the paper. In section 2, we recall some general
background. In section 3, we introduce some standard rank-two vector bundles on
our Fano threefolds without intermediate cohomology, and we prove that any inde-
composable rank-two vector bundle is, up to a twist, one of these. In section 4, we
give the characterization of the Chern classes of “sufficiently general” vector bundles
without intermediate cohomology and having arbitrary rank.

Acknowledgements: The authors want to express their gratitude to the Diparti-
mento di Matematica of the Università degli Studi di Milano, where most of the work
has been developed. They also want to stress the important help that the Maple
package Schubert, created by S. Katz and S.A. Strømme, has provided for some of
the computations needed in this work.

2 Generalities.

For d = 3, 4, 5, let Vd be a Fano 3-fold of degree d, index 2, with the Betti number
b2 = 1 and such that one of the two generators of its Picard group is spanned. Then,
it has to be

V3 = a smooth cubic in P
4;

V4 = a smooth complete intersection of type (2, 2) in P
5;

2



V5 = a smooth intersection of the Grassmannian G(1, 4) ⊂ P
9 with three hyper-

planes.

Indeed, it is known (see e.g. [Is]) that there exist, up to a deformation type, only four
Fano threefolds of index two, the above ones and a double covering V2 of P

3 ramified
along a quartic surface.

Notation 2.1 Let OVd
(1) be the ample generator of Pic(Vd). In our situation, OVd

(1)
is in fact very ample, and corresponds to the hyperplane section H of Vd in P

d−1. Also,
the canonical divisor of Vd is given by K = −2H . By a line on Vd we mean a rational
curve C such that C · OVd

(1) = 1 and we shall denote it by L, whereas a conic is a
rational curve C with C · OVd

(1) = 2 and we shall denote it by C.

Since H2(Vd, Z) is generated by the class of H , H4(Vd, Z) by a line L, and H6(Vd, Z)
by a point P , we may identify the Chern classes ci ∈ H2i(Vd, Z) of vector bundles
with integers, namely the coefficients of H , L, and P . With this notation, HL = 1,
H2 = dL and H3 = d.

Given F a coherent sheaf on Vd we will write F (t) instead of F ⊗ OVd
(tH).

Definition 2.2 A vector bundle F on Vd has not intermediate cohomology if and
only if H i(Vd, F (t)) = 0, for all t ∈ Z and i = 1, 2.

We will end this section by recalling the formulas for Chern classes of a twist of
a rank-r vector bundle, the Riemann-Roch formula for rank-r vector bundles on Vd

and the well-known regularity criterion of Castelnuovo-Mumford.

2.3 Given F a rank-r vector bundle on Vd with Chern classes c1(F ) = c1, c2(F ) = c2

and c3(F ) = c3 it holds:
c1(F (t)) = c1 + rt;

c2(F (t)) = c2 + (r − 1)tc1d +

(

r

2

)

t2d;

c3(F (t)) = c3 + (r − 2)c2t +

(

r − 1

2

)

c1t
2d +

(

r

3

)

t3d;

χ(F ) =
dc3

1 − 3c1c2

6
+

dc2
1 − 2c2

2
+

(d + 3)c1

3
+

c3

2
+ r.

3



In particular,

χ(OVd
(t)) =

dt3

6
+

dt2

2
+

(d + 3)t

3
+ 1.

2.4 Let O(1) be an ample invertible sheaf on a variety X which is generated by
global sections. Let F be a vector bundle with

H i(X, F (−i)) = 0, i = 1, 2, · · · , dimX.

Then,

i) H i(X, F (k)) = 0, for any i = 1, 2, · · · , dimX and k ≥ −i;

ii) F is generated by global sections.

3 Rank-two vector bundles without intermediate

cohomology

The aim of this section is to characterize, up to twist by line bundles, rank-two vector
bundles on Vd without intermediate cohomology.

In the next examples we will summarize several facts about vector bundles on Vd

that will be used in the sequel. Some of these results are due to M. Szurek and J.A.
Wísniewski ([SW]).

Example 3.1 Let L be a line in Vd. Serre’s correspondence provides a vector bundle
SL fitting in an exact sequence:

0 → OVd
→ SL → IL → 0 (1).

The rank-two vector bundle SL has Chern classes (c1, c2) = (0, 1), is semistable and
has not intermediate cohomology. It holds that SL has only one section, whereas
SL(1) is generated by global sections. The latter comes from the exact sequence (1),
the fact that OVd

(1) and IL(1) are generated by its sections and the vanishing of
H1(OVd

(1)).

4



Example 3.2 If C is a conic in Vd, let SC be the rank-two vector bundle obtained
from Serre’s construction and fitting in a non-trivial extension:

0 → OVd
(−1) → SC → IC → 0 (2).

It holds that SC is stable and generated by its global sections. It has not intermediate
cohomology and has Chern classes (c1, c2) = (−1, 2).

When d = 4, SC is related to the spinor bundles of the quadrics containing V4.
More precisely, let Π be the unique plane containing C. Then, only one quadric Q4

containing V4 contains Π. If Q4 is smooth, then SC(1) is the restriction to Vd of one
of the spinor bundles of Q4. In other words, if we identify Q4 with the Grassmann
variety G(1, 3) of lines in P

3, then SC(1) is the restriction to V4 of one of the universal
bundles of G(1, 3). On the other hand, when d = 5, SC(1) is the restriction to V5 of
the rank-two universal quotient bundle on G(1, 4).

Example 3.3 Using again Serre’s correspondence, an elliptic curve E of degree d+2
gives rise to a stable, rank-two vector bundle SE, with Chern classes (c1, c2) = (0, 2),
without intermediate cohomology and appearing in an exact sequence:

0 → OVd
(−1) → SE → IE(1) → 0 (3).

It also holds that SE(1) is generated by its global sections.

Next we will prove that the above examples are essentially the only rank-two
vector bundles on Vd without intermediate cohomology.

Theorem 3.4 An indecomposable rank-two vector bundle F on Vd with d = 3, 4, 5
has not intermediate cohomology if and only if it is a twist of either SL, or SC, or
SE.

Proof: Let F be an indecomposable rank-two vector bundle on Vd with Chern classes
c1(F ) = c1 and c2(F ) = c2, which has not intermediate cohomology. Without loss of
generality we can assume that F (−1) has no sections while F has. Hence, any section
of F gives rise to an exact sequence:

0 → OVd
→ F → ID(c1) → 0 (4)

where D is a scheme of pure dimension one. Observe that it cannot be D = ∅, since
Ext1(OVd

(c1), OVd
) ∼= H1(OVd

(−c1)) = 0, and this would imply that F decomposes.

Claim:

5



F (2 − c1) is generated by its global sections and therefore c1 ≤ 2.

Proof of the Claim: Using Serre’s duality and the natural identification F ∗ ∼= F (−c1)
we have:

h3(F (−1 − c1)) = h0(F (−1)) = 0.

Hence, since F has not intermediate cohomology, using Castelnuovo-Mumford crite-
rion 2.4, we obtain the claim.

We will prove the theorem distinguing two cases.

Case A: Assume that c1 ≤ 0. From Serre’s duality and (4) twisted with OVd
(−c1 −

1), we get that the only cohomology of F (−1) is h3(F (−1)) = h0(F (−c1 − 1)) =
h0(OVd

(−c1 − 1)). Computing χ(F (−1)) and χ(OVd
(−c1 − 1)) from 2.3 we get:

χ(F (−1)) =
c3
1d

6
−

c1c2

2
−

dc1

6
+ c1;

χ(OVd
(−c1 − 1)) = −

c3
1d

6
+

dc1

6
− c1.

Hence, from χ(OVd
(−c1 − 1)) = −χ(F (−1)) we obtain the identity c1c2 = 0. Since

D 6= ∅, we have c2 6= 0, and hence c1 = 0. Again from 2.3 and the fact that F has
not intermediate cohomology we get 2 − c2 = χ(F ) = h0(F ) ≥ 1, from which c2 = 1.
This means that D is a line L, and therefore F ∼= SL.

Case B: If c1 > 0, we know from the claim that the only possibilities are c1 = 1, 2.
In fact, this can be obtained, together with a much more precise information, with
the same techniques as in case A. Indeed, we have now that h0(F (−c1)) = 0, so that
h3(F (−1)) = h3(F (−2)) = 0. Therefore, χ(F (−1)) = χ(F (−2)) = 0. Again, using
2.3, we have

χ(F (−2)) =
c3
1d

6
−

c1c2

2
+ c1 −

dc2
1

2
+ c2 − 2 +

dc1

3
.

Comparing with the above expression for χ(F (−1)), we obtain that c2−
dc2

1

2
−2+ dc1

2
=

0, and therefore the only possibilities are (c1, c2) = (1, 2), (2, d + 2).

In the first case, D has degree c2 = 2, and by Serre’s correspondence ωD
∼=

OD(c1 − 2) = OD(−1), which means that D is a plane conic C (maybe singular or
even non-reduced). Therefore, F ∼= SC(1).

6



In the second case, by the claim, F is then generated by its global sections. This
implies that a general section of F vanishes on a smooth curve D. Since ωD =
OD(c1 − 2) = OD, this means that D is an elliptic curve. Its degree is c2 = d + 2.
Therefore, F ∼= SE(1) where SE is the vector bundle on Vd associated to an elliptic
curve E of degree d + 2.

4 Vector bundles of higher rank without interme-

diate cohomology

In this section we will study vector bundles of rank r ≥ 3 without intermediate
cohomology. While in the previous section we gave a complete classification, in this
section we will just characterize the possible Chern classes. We will also need to make
some general assumptions.

Convention: For simplicity, we will frequently use (⋆) to denote the following con-
ditions on a rank-r vector bundle F on Vd:

(⋆) The vector bundle F has not trivial summands, h0(F ) ≥ r, h0(F (−1)) = 0 and
F has r − 1 sections whose dependency locus has codimension two.

Lemma 4.1 Let F ′ and F ′′ be two vector bundles on Vd with respective ranks r′ and
r′′. If F ′ and F ′′ verify (⋆), then also F ′ ⊕ F ′′ verifies (⋆).

Proof: It is clear from the definition that, since F ′ and F ′′ verify (⋆), then their
global sections generate them up to probably a subvariety of codimension at least
two. Therefore, the same holds for F ′ ⊕ F ′′. This implies that the dependency locus
of r′ + r′′ − 1 general sections of F ′ ⊕ F ′′ has codimension two. From this, it follows
immediately that F ′ ⊕ F ′′ verifies (⋆).

Before stating and proving our main theorem, we will give a series of examples.
Eventually this list will become the complete list for small rank.

Example 4.2 Let D be a rational curve of degree r ≤ d+1 in Vd. Serre’s construction
yields a rank-r vector bundle F fitting in an exact sequence

0 → Or−1
Vd

→ F → ID(1) → 0.

7



Since D is projectively normal in P
d+1 if and only if r ≤ d + 1, then F has not

intermediate cohomology. From the above exact sequence it holds that c1(F ) = 1,
c2(F ) = r and c3(F ) = r − 2, and also h0(F ) = d. Notice that, when d = 5 and
r = 3, F is the restriction to V5 of the dual of the rank-three universal subbundle on
G(1, 4), and in particular it is generated by its global sections.

Example 4.3 Take F to be the rank-three vector bundle given by the above example
when r = 3. Then

∧2 F ∼= F ∗(1) has not intermediate cohomology, has general
sections and Chern classes c1 = 2, c2 = d + 3 and c3 = 2. Since, for d = 5, F is
generated by its global sections, then the same holds for

∧2 F .

Example 4.4 Consider the elliptic curve E given in Example 3.3. Exact sequence
(3) implies that there is a section s ∈ H0(SL(2)) vanishing on E and not vanishing
on a divisor of Vd. If D′ is the residual curve of E inside the zero locus of s, it turns
out that D′ has degree 3d− 1 and (arithmetic) genus 2d− 2. The standard mapping
cone construction (see for instance Lemma 3.2 in [AS]) provides an exact sequence

0 → SE(−3) → OVd
(−2) ⊕ SL(−2) → ID′ → 0.

We use now this exact sequence to see that there is a section s′ ∈ H0(SC(3)) vanishing
on D′ and not vanishing on a divisor of Vd. We have now the followig exact sequences
for D, the residual curve of D′ inside the zero locus of s′:

0 → SL(−3) ⊕ OVd
(−3) → SE(−2) ⊕ SC(−2) → ID → 0 (5)

and its dual

0 → OVd
→ SE(−2) ⊕ SC(3) → SL(3) ⊕ OVd

(3) → ωD → 0. (6)

The curve D has degree 3d+3 and arithmetic genus 2d+4. From the exact sequence
(6) tensored with OVd

(−3) it follows that h0(ωD(−1)) = 2, so that Serre’s construction
yields a rank-three vector bundle F with Chern classes c1 = 3, c2 = 3d+3, c3 = d+3
and fitting in an exact sequence

0 → O2
Vd

→ F → ID(3) → 0. (7)

Comparing with (5) it easily follows that H0(F (−1)) = 0, H1(F (l)) = 0 for any
l ∈ Z, and H2(F (l)) = 0 for any l ≥ −1. It also follows from Serre’s construction

8



that H2(F (−2)) = 0. Hence, Mumford-Castelnuovo criterion applied to F ∗(2) will
imply that F has not intermediate cohomology as soon as we prove that h1(F ∗(1)) =
h2(F (−3)) = 0. We observe that, from the exact sequences (6) and the dual of (7)
and our construction, there is a commutative diagram

0 → OVd
(−2) → F ∗(1) → OVd

(1)2 → ωD → 0
|| ↓ ↓ ||

0 → OVd
(−2) → SE ⊕ SC(1) → SL(1) ⊕ OVd

(1) → ωD → 0

The vanishing we want to prove reduces to the surjectivity of the map H0(OVd
(1)2) →

H0(ωD), but this is proved by just taking cohomology in the above diagram.
Notice that the above construction does not imply a priori that the curve D

is smooth. However, once the vector bundle F is constructed, it follows by the
Castelnuovo-Mumford criterion that F is generated by its global sections. Hence, the
dependency locus of two general sections of F is a smooth curve D.

When d = 5, such a vector bundle F can be constructed in a more direct way.
Indeed, in this case SC(1) is the restriction to V5 of the universal rank-two quotient
bundle Q on G(1, 4). Since S2Q has not intermediate cohomology, neither F =
S2(SC(1)) has. It follows at once that F is a rank-three vector bundle, generated by
its global sections and its Chern classes are c1 = 3, c2 = 18 and c3 = 8.

Example 4.5 Let G be the rank-three vector bundle obtained in Example 4.3 when
d = 5. Since G was generated by its global sections and h0(G) = 10, then there is an
exact sequence

0 → K → O10
V5

→ G → 0

where K is defined as a kernel. It follows that F = K∗ is a rank-seven vector bundle
without intermediate cohomology and generated by its global sections. It also holds
that c1(F ) = 2, c2(F ) = 12 and c3(F ) = 10.

9



Table 4.6 From Lemma 4.1 and the above examples, we can give the following list
of rank-r vector bundles Fr,c without intermediate cohomology, verifying (⋆) and with
c1(Fr,c) = c

d r (c, c2, c3) Fr,c

3,4,5 3 (1, 3, 1) Example 4.2
3,4,5 3 (2, d + 3, 2) Example 4.3
3,4,5 3 (3, 3d + 3, d + 3) Example 4.4
4,5 4 (1, 4, 2) Example 4.2

3,4,5 4 (2, d + 4, 4) SC(1) ⊕ SC(1)
3,4,5 4 (3, 3d + 4, d + 6) SC(1) ⊕ SE(1)
3,4,5 4 (4, 6d + 4, 4d + 8) SE(1) ⊕ SE(1)

5 5 (1, 5, 3) Example 4.2
3,4,5 5 (2, d + 5, 6) SC(1) ⊕ F3,1

3,4,5 5 (3, 3d + 5, d + 9) SC(1) ⊕ F3,2

3,4,5 5 (4, 6d + 5, 4d + 2) SC(1) ⊕ F3,3

3,4,5 5 (5, 10d + 5, 10d + 15) SE(1) ⊕ F3,3

3,4,5 6 (2, d + 6, 8) F3,1 ⊕ F3,1

3,4,5 6 (3, 3d + 6, d + 12) SC(1) ⊕ SC(1) ⊕ SC(1)
3,4,5 6 (4, 6d + 6, 4d + 16) SC(1) ⊕ SC(1) ⊕ SE(1)
3,4,5 6 (5, 10d + 6, 10d + 20) SC(1) ⊕ SE(1) ⊕ SE(1)
3,4,5 6 (6, 15d + 6, 20d + 24) SE(1) ⊕ SE(1) ⊕ SE(1)
4,5 7 (2, d + 7, 10) F4,1 ⊕ F3,1

3,4,5 7 (3, 3d + 7, d + 15) SC(1) ⊕ SC(1) ⊕ F3,1

3,4,5 7 (4, 6d + 7, 4d + 20) SC(1) ⊕ SE(1) ⊕ F3,1

3,4,5 7 (5, 10d + 7, 10d + 25) SE(1) ⊕ SE(1) ⊕ F3,1

3,4,5 7 (6, 15d + 7, 20d + 30) SE(1) ⊕ SE(1) ⊕ F3,2

3,4,5 7 (7, 21d + 7, 35d + 35) SE(1) ⊕ SE(1) ⊕ F3,3

Remark 4.7 Notice that in the above table, vector bundles with the same invariants
can be constructed in different ways. Moreover, in some cases it is possible to consider
vector bundles given by non-trivial extensions instead of direct sums. For instance,

10



it is immediate to see that Ext1(SE(1), SE(1)) 6= 0. In fact, the dimension of this
space of extensions is five, the same as the dimension of the space of elliptic curves
of degree d + 2 contained in Vd. Clearly, a vector bundle obtained from a non-trivial
extension also verifies condition (⋆) and has not intermediate cohomology.

Remark 4.8 In case of rank r = 3, it is clear that the vector bundles in the above
list are indecomposable, since otherwise any direct summand of rank one would be
necessarely trivial. In case r = 4 and c1 = 1 (even when d = 3, as constructed in
Example 4.2) the corresponding vector bundle in the above list is also indecompos-
able. Indeed, we can rule out (using the classification given in the previous section)
all its possible rank-two direct summands. What seems to be difficult is to decide in
general when it is possible to find an indecomposable vector bundle without interme-
diate cohomology and having some of the above invariants. This is so even when we
know that we can take non-trivial extensions as in the above remark.

We can now state and prove the main result of this section.

Theorem 4.9 Let r ≥ 3 be an integer. There exists a rank-r vector bundle F on Vd

without intermediate cohomology, verifying (⋆) and with Chern classes c1(F ) = c1,
c2(F ) = c2 and c3(F ) = c3 if and only if

(i) c2 =
dc2

1

2
+ r −

dc1

2
;

(ii) c3 = −2c1 + c1r −
dc2

1

2
+

dc3
1

6
+

dc1

3
;

(iii)
r

d
≤ c1 ≤ r.

Moreover, in this situation, the dependency locus of r − 1 general sections of such F

is a curve D of degree c2 and arithmetic genus

pa(D) = (c1 − 1)(r − 1) − dc2
1 +

dc3
1

3
+

2dc1

3
.

11



Proof: Let F be a rank-r vector bundle on Vd verifying (⋆), with Chern classes
c1(F ) = c1, c2(F ) = c2 and c3(F ) = c3, which has not intermediate cohomology.
By assumption, F has r − 1 sections whose dependency locus has codimension two.
These sections of F give rise to an exact sequence:

0 → OVd

r−1 → F → ID(c1) → 0 (8)

where D is a scheme of pure dimension one. Observe that it cannot be D = ∅,
since Ext1(OVd

(c1), OVd

r−1) ∼= H1(OVd
(−c1))

⊕(r−1) = 0, and this would imply that (8)
splits, and hence F would have a trivial summand.

Let us see that c1 > 0. Indeed, assume that c1 ≤ 0. From Serre’s duality and (8)
twisted by OVd

(−1), the only cohomology of F (−1) is h3(F (−1)) = h3(OVd
(c1 − 1)).

Since c1(F (−1)) = c1 − r, c2(F (−1)) = c2 − (r − 1)c1d +
(

r

2

)

d and c3(F (−1)) =

c3 − (r − 2)c2 +
(

r−1
2

)

c1d −
(

r

3

)

d, computing χ(F (−1)) and χ(OVd
(c1 − 1)) from 2.3

we get:

χ(F (−1)) =
c3
1d

6
−

c1c2

2
−

dc1

6
+ c1 +

c3

2
;

χ(OVd
(c1 − 1)) =

c3
1d

6
−

dc1

6
+ c1.

Hence, from χ(OVd
(c1 − 1)) = χ(F (−1)) we obtain the identity c3 − c1c2 = 0.

Similarly, we have χ(F (−c1)) = (r − 1)χ(OVd
(−c1)), which is equivalent to:

(−
(r − 1)c3

1

6
−

(r − 1)c1

3
+

(r − 1)c2
1

2
)d +

c1c2

2
− c2 +

c3

2
− rc1 + c1 + r =

(r − 1)((−
c3
1

6
+

c2
1

2
−

c1

3
)d − c1 + 1)

and χ(F (−c1 − 1)) = (r − 1)χ(OVd
(−c1 − 1), which is equivalent to:

(−
(r − 1)c3

1

6
+

(r − 1)c1

6
)d +

c1c2

2
+

c3

2
− rc1 + c1 =

(r − 1)((−
c3
1

6
+

c1

6
)d − c1)

from which we respectively obtain the identities c1c2 + c3 = c2 − 1 and c1c2 + c3 = 0.
Putting together the three identities, we get c1 = c3 = 0 and c2 = 1, so that in
particular D is a line L.

12



The dual of (8) is

0 → OVd
(−c1) → F ∗ → OVd

⊕(r−1) → ωD(2 − c1) → 0. (9)

Since ωD(2 − c1) is trivial in our case, it follows that F ∗ has r − 2 independent
sections such that the composed map OVd

⊕(r−2) → F ∗ → OVd

⊕(r−1) defines a direct
summand. But this implies that F ∼= OVd

⊕(r−2) ⊕ SL, which is a contradiction as
r ≥ 3. Hence, c1 > 0.

Since c1 > 0, we have that h3(F (−1)) = h3(F (−2)) = 0 (the latter equality is
equivalent to h0(F ∗) = 0, which is obtained from (9) as above, by using that F has
no trivial summands). Therefore, χ(F (−1)) = χ(F (−2)) = 0. Again, using 2.3, we
have:

χ(F (−1)) =
c3
1d

6
−

c1c2

2
−

dc1

6
+ c1 +

c3

2
;

χ(F (−2)) =
c3
1d

6
−

c1c2

2
+ c1 −

dc2
1

2
+ c2 − r +

dc1

3
+

c3

2
.

The vanishing of the above two terms immediately implies that c2(F ) and c3(F ) verify
the identities (i) and (ii).

Using once more 2.3, we obtain that −h3(F (−3)) = χ(F (−3)) = d(c1 − r). This
implies that c1 ≤ r. The other inequality in (iii) follows from the fact that h0(F ) ≥
r − 1, Riemann-Roch equality 2.3 and the identities (i) and (ii).

Finally, the genus of the curve D is χ(ID), which can be computed from (8), and
it is

(c1 − 1)(r − 1) − dc2
1 +

dc3
1

3
+

2dc1

3
.

Conversely, let r, c1, c2 and c3 be integers verifying (i), (ii) and (iii). We will
prove by induction on r that there exists a rank-r vector bundle F on Vd without
intermediate cohomology, verifying (⋆) and with Chern classes c1(F ) = c1, c2(F ) = c2

and c3(F ) = c3. For r ≤ 7 this is showed in Table 4.6, so let us assume r ≥ 8. In
this case, the proof goes as for r ≤ 7, using Lemma 4.1. We will distinguish three
cases:

Case A) If c1 = r, we can take F = SE(1) ⊕ F ′, where F ′ is a rank-(r − 2) vector
bundle on Vd without intermediate cohomology, verifying (⋆) and with first Chern
class c1(F

′) = r − 2. Indeed, such F ′ can be constructed by induction hypothesis.

13



Case B) If r−2
d

+ 1 ≤ c1 ≤ r − 1, we take F = SC(1) ⊕ F ′, where F ′ is a rank-(r − 2)
vector bundle on Vd without intermediate cohomology, verifying (⋆) and with first
Chern class c1(F

′) = c1 − 1. This vector bundle F ′ exists by induction hypothesis,
since F ′ verifies (iii).

Case C) If r
d
≤ c1 ≤

r−1
d

+ 1, we take now F = F ′ ⊕ F ′′, where F ′ and F ′′ are vector
bundles on Vd of respective ranks r − d and d without intermediate cohomology,
verifying (⋆) and with Chern classes c1(F

′) = c1 − 1, c1(F
′′) = 1. Once again, F ′

verifies the inductive assumption since r ≥ 8, and F ′′ was constructed in Example 4.2.

Remark 4.10 If we do not make the assumption that our vector bundles have at
least r independent sections, the same proof is clearly valid to see that in that case (i)
and (ii) holds, and instead of (iii) we would have that r−1

d
≤ c1 ≤ r. What presents

some difficulty is to prove the actual existence of vector bundles with those invariants.
This is due to the fact that we cannot make use of Lemma 4.1. In fact, we are still
able to prove the existence of most of them, but some cases still remain unknown to
us.

Remark 4.11 Our results are in fact valid over any Fano threefold of degree d and
index two and arbitrary Betti number. In that case, the only assumption we need
on the vector bundles is their determinant to be a multiple of OVd

(1), the hyperplane
section of Vd in P

d−1.

14



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