HILBERT SCHEMES OF SOME THREEFOLD SCROLLS OVER Fe

MARIA LUCIA FANIA AND FLAMINIO FLAMINI

Abstract. Hilbert schemes of suitable smooth, projective threefold scrolls over the Hirze- bruch surface F

, e ≥ 2, are studied. An irreducible component of the Hilbert scheme parametrizing such varieties is shown to be generically smooth of the expected dimension and the general point of such a component is described.

1. Introduction

Projective varieties are distributed in families, obtained by suitably varying the coefficients of their defining equations. The description of such families and, in particular, of the properties of their parameter spaces is a central theme in algebraic geometry.

Milestones to approach such problems have been both the introduction of technical tools, like flatness, base change, Hilbert polynomial, etc., and the proof (due to Grothendieck with refinements by Mumford) of the existence of the so called Hilbert scheme, a closed, pro- jective scheme, parametrizing families of projective varieties with suitable constant numeri- cal/projective invariants, together with some other fundamental universal properties.

Since then, Hilbert schemes of projective varieties with given Hilbert polynomial have intere- sted several authors over the years, especially because of the deep connections of the subject with several other important theories in algebraic geometry: zero-dimensional schemes on smooth projective varieties, Brill-Noether theory of line bundles on curves, moduli spaces of genus g curves and their stratifications in terms of suitable subvarieties, vector bundles on smooth projective varieties, just to mention a few (for an overview the reader is referred, for instance, to the bibliography in [38]).

For particular cases of projective varieties, one can find in the literature sufficiently detailed descriptions of their Hilbert schemes. For example special classes of threefolds in P ⁵ were studied in [20]; results for codimension–two projective varieties are due to [17, 14, 15]; in codimension three, [32] considered the case of arithmetically Gorenstein closed subschemes in a projective space, whereas [31] dealt with determinantal schemes. For codimension greater than or equal to two, Hilbert schemes of Palatini scrolls in P ⁿ , with n odd, have been treated in [18] while in [19] Hilbert schemes of varieties defined by maximal minors were considered.

We also mention results in [33] concerning Hilbert schemes of determinantal schemes.

An important class of projective varieties is that of r-scrolls in P ⁿ , namely ruled varieties over a smooth base which are embedded in P ⁿ in such a way that the rulings are r-dimensional linear subspaces of P ⁿ . This class is important not only because it usually comes out as a fundamental special case from problems in classical adjunction theory (cf. e.g. [5, 36]), but mainly because it is strictly related to the study of vector bundles of rank (r + 1) over smooth projective varieties.

2000 Mathematics Subject Classification. Primary 14J30, 14J27, 14J60, 14C05; Secondary 14M07, 14N25, 14N30.

Key words and phrases. Ruled varieties, Vector bundles, Rational surfaces, Hilbert scheme.

The authors thank C. Ciliberto and E. Sernesi for having pointed out questions on Hilbert schemes of three- fold scrolls over F

, with e ≥ 2, during the talk of the first author at the Workshop ”Algebraic geometry: two days in Rome two”, held in Rome in February 2012. The authors are also greateful to the referee for helpful comments and for having posed a question which allowed us to realize that there was a mistake in the first version of the paper. Both authors are members of GNSAGA-INdAM. We acknowledge partial support from MIUR funds, PRIN 2010-2011 project “ Geometria delle Variet` a Algebriche”.

1

For rank-two, degree d vector bundles over genus g curves (equivalently, surface scrolls of degree d and sectional genus g), apart from the classical approach of C. Segre ([37]) and of some other more recent partial results as, for instance, in [27, 3, 25, 26], a systematic study of Hilbert schemes of such surface scrolls has been developed in the series of papers [10, 11, 12, 13], where the authors bridged the Hilbert scheme approach with the vector-bundle one, showing in particular how projective geometry and degeneration techniques can be used in order to improve some known results about rank-two vector bundles on curves and also to obtain some new ones.

A similar approach has been used to study Hilbert schemes of r-scrolls, r ≥ 1, over smooth projective surfaces S, with S either a K3 ([21]) or the Hirzebruch surfaces F 0 and F 1 ([6, 7]).

In the authors’ opinion, it would be interesting to develop the use of projective geometry and of degeneration techniques in order to study possible limits of vector-bundles, of any rank, on classes of smooth, projective varieties.

In this paper we focus on some classes of 1–scrolls over Hirzebruch surfaces F e , with e ≥ 2.

Rank–two vector bundles on Hirzebruch surfaces are classified in [9]; some of their cohomo- logical and ampleness properties are studied in [1]; moduli spaces of rank-two vector bundles on Hirzebruch surfaces are considered, for example, in [2]. On the other hand, very little is known about Hilbert schemes of 1–scrolls over F e .

We consider vector bundles E e arising as extensions of suitable line bundles over F e and with Chern classes c ₁ ( E e ) = 3C _e + b _e f , c ₂ ( E e ) = k _e , where C _e and f are respectively the section of minimal self-intersection and a fiber of F e , whereas b e and k e are integers suitably chosen (cf. Assumptions 3.1, 4.3). Such a choice of c ₁ ( E e ) = 3C _e + b _e f and of the integers b _e , k _e gives the first case for which the bundle E e is both uniform and very-ample (cf.§ 4 and Remark 4.2).

Let therefore X _e be a threefold in P ⁿ

which is a scroll over F e , n _e ≥ 6, e ≥ 2, that is X _e ∼ = P(E e ) is the projectivization of a rank–two vector bundle E e over F e as above. We assume n e ≥ 6 because it is known that there are no such scrolls when n _e ≤ 5, see [36].

If one wants to parametrize varieties X _e of this type, the first tasks to be tackled are:

(i) looking at [X _e ] as a point of a component of H ^d ₃

^,n

, the Hilbert scheme parametrizing 3-dimensional subvarieties of P ⁿ

of degree d e having same Hilbert polynomial P X

(T ) as that of X _e , and

(ii) understanding the general point of such a component in H ^d ₃

^,n

.

For e = 0, 1, the above problems have been considered in [6, 7], where the Hilbert schemes of threefold scrolls X 0 and X 1 were studied. Namely, it was proved that the irreducible component containing such scrolls is generically smooth, of the expected dimension, and its general point is actually a threefold scroll, that is the component is filled up by scrolls. The aim of this paper is to see what happens if the base of the scroll is F e , with e ≥ 2.

Our main results, Theorems 5.1, and 5.7, in particular answer a question on Hilbert schemes of threefold scrolls over F e , e ≥ 2, pointed out to us by C. Ciliberto and E. Sernesi and for which we thank them.

In this paper, we prove that there exists an irreducible component X e of H ^d 3

^,n

, containing such scrolls, which is generically smooth, of the expected dimension and such that [X _e ] belongs to the smooth locus of X e (cf. Theorem 4.5). In contrast with the e = 0, 1 cases, we show that the family of constructed scrolls X _e ’s surprisingly does not fill up the component X e (cf.

Theorem 5.1).

We thus exhibit a smooth variety X  ⊂ P ⁿ

, which is a candidate to represent the general point of X e . More precisely, we show that X _ corresponds to the general point of an irreducible component, of the same Hilbert scheme H ^d ₃

^,n

, which is generically smooth and of the expected dimension. We then show that X  flatly degenerates in P ⁿ

to a general threefold scroll X e

as above, in such a way that the base–scheme of the flat, embedded degeneration is entirely

contained in X e . By the generic smoothness of X e , we can conclude that X  is actually the

general point of X e (cf. §’s 5.1, 5.2).

The paper is structured in the following way. In Section 2 notation is fixed. In Section 3, following [6, 7], we consider suitable rank-two vector bundles over F e , with e ≥ 2. In Section 4 we consider Hilbert schemes parametrizing families of 3-dimensional scrolls over F e , e ≥ 2. In Section 5 a description of the general point of the component X e determined in Theorem 4.5 is presented. More precisely, in § 5.1 we first construct the candidate X  and analyze some of its properties, similar to those investigated for X _e in Sections 3, 4; then, in § 5.2, we show that X  actually corresponds to the general point of X e . Finally, Section 6 contains some concrete examples of Hilbert scheme of scrolls over some F e , with e ≥ 2 and e both even and odd.

2. Notation and Preliminaries The following notation will be used throughout this work.

X is a smooth, irreducible, projective variety of dimension 3 (or simply a threefold);

χ(F) = P(−1) ⁱ h ⁱ (F), the Euler characteristic of F, where F is any vector bundle of rank r ≥ 1 on X;

c _i ( F), the i-th Chern class of F;

F |

the restriction of F to a subvariety Y ;

K _X the canonical bundle of X. When the context is clear, X may be dropped, so K X = K;

c _i = c _i (X), the i-th Chern class of X;

d = deg X = L ³ , the degree of X in the embedding given by a very-ample line bundle L;

g = g(X), the sectional genus of (X, L) defined by 2g − 2 = (K + 2L)L ² ;

if S is a smooth surface, ≡ will denote the numerical equivalence of divisors on S.

For non-reminded terminology and notation, we basically follow [29].

Definition 2.1. A pair (X, L), where L is an ample line bundle on a threefold X, is a scroll over a normal variety Y if there exist an ample line bundle M on Y and a surjective morphism ϕ : X → Y with connected fibers such that K _X + (4 − dim Y )L = ϕ ^∗ (M ).

In particular, if Y is smooth and (X, L) is a scroll over Y , then (see [5, Prop. 14.1.3]) X ∼ = P(E), where E = ϕ ∗ (L) and L is the tautological line bundle on P(E). Moreover, if S ∈ |L| is a smooth divisor, then (see e.g. [5, Thm. 11.1.2]) S is the blow up of Y at c 2 (E) points; therefore χ( O Y ) = χ( O S ) and

(2.1) d := L ³ = c ² ₁ ( E) − c 2 ( E).

Throughout this work, the scroll’s base Y will be the Hirzebruch surface F e = P(O _P

⊕ O _P

(−e)), with e ≥ 0 an integer.

Let π _e : F e → P ¹ be the natural projection onto the base. Then Num(F e ) = Z[C e ] ⊕ Z[f ], where:

• C _e denotes the unique section corresponding to the morphism O _P

⊕ O _P

(−e) → → O _P

(−e) on P ¹ , and

• f = π ^∗ (p), for any p ∈ P ¹ . In particular

C _e ² = −e, f ² = 0, C _e f = 1.

Let E e be a rank-two vector bundle over F e and let c _i ( E e ) be its i ^th -Chern class. Then c 1 ( E e ) ≡ aC e + bf , for some a, b ∈ Z, and c 2 ( E e ) ∈ Z.

3. Some rank-two vector bundles over F e , for e ≥ 2

In [6, 7] the authors considered suitable rank-two vector bundles over F e , for e = 0, 1. In

this and the following section, we will focus on the case e ≥ 2. Therefore, unless otherwise

stated, from now on we will use the following:

Assumptions 3.1. Let e ≥ 2, b _e , k _e be integers. Let E e be a rank-two vector bundle over F e , with

c 1 ( E e ) ≡ 3C e + b e f and c 2 ( E e ) = k e , such that

(i) h ⁰ ( E e ) ≥ 7 (ii) b e ≥ 3e + 1 (iii) k _e + e > b _e

(cf. § 4 below and [1, Prop.7.2], for motivation). Moreover, there exists an exact sequence

(3.1) 0 → A _e → E e → B _e → 0,

where A _e and B _e are line bundles on F e such that

(3.2) A _e ≡ 2C _e + (2b _e − k _e − 2e)f and B _e ≡ C _e + (k _e − b _e + 2e)f (cf. [1, Prop.7.2] and [9]).

From (3.1), in particular, one has c ₁ ( E e ) = A _e + B _e and c ₂ ( E e ) = A _e B _e .

Exact sequence (3.1) gives important preliminary information on the cohomology of E e , A e

and B _e . Indeed, one has

Lemma 3.2. With Assumptions 3.1, one has

h ^j ( E e ) = h ^j (A _e ) = 0, for j ≥ 2, h ⁱ (B _e ) = 0, for i ≥ 1, h ⁰ (A e ) = 6b e − 3k _e − 9e + 3 + h ¹ (A e ), h ⁰ (B e ) = 2k e − 2b _e + 3e + 2 and

(3.3) h ⁰ (E e ) = 4b e − k _e − 6e + 5 + h ¹ (E e ).

Proof. For dimension reasons, it is clear that h ^j ( E e ) = h ^j (F e , A e ) = h ^j (F e , B e ) = 0, j ≥ 3.

By Serre duality on F e ,

h ² (A _e ) = h ⁰ (−4C _e − (2b _e − k _e − e + 2)f ) = 0 and h ² (B _e ) = h ⁰ (−3C _e − (k _e − b _e + 3e + 2)f ) = 0, since K _F

≡ −2C _e − (e + 2)f . In particular, this implies that also h ² ( E e ) = 0.

We claim that, under Assumptions 3.1, we also have h ¹ (B e ) = 0. Indeed, since B e ≡ C _e + (k _e − b _e + 2e)f , it follows that R ¹ π ∗ (B _e ) = 0 and thus by Leray’s isomorphism,

h ¹ (B e ) = h ¹ (P ¹ , ( O _P

⊕ O _P

(−e)) ⊗ O _P

(k e − b _e + 2e))

= h ¹ (P ¹ , O _P

(k e − b _e + 2e)) + h ¹ (P ¹ , O _P

(k e − b _e + e)) = 0, by Assumptions 3.1-(iii).

Thus we have

(3.4) χ(A _e ) = h ⁰ (A _e ) − h ¹ (A _e ), χ(B _e ) = h ⁰ (B _e ), χ( E e ) = h ⁰ ( E e ) − h ¹ ( E e ).

From the Riemann-Roch formula, we have χ(A _e ) = 1

2 A _e (A _e − K _F

) + 1 = 1

2 (2C _e + (2b _e − k _e − 2e)f ) (4C _e + (2b _e − k _e − e + 2)f ) + 1 = 6b _e − 3k _e − 9e + 3, whereas

χ(B e ) = h ⁰ (B e ) = 1

2 B e (B e − K _F

) + 1 = 1

2 (C _e + (k _e − b _e + 2e)f ) (3C _e + (k _e − b _e + 3e + 2)f ) + 1 = 2k _e − 2b _e + 3e + 2.

Since χ( E e ) = χ(A _e ) + χ(B _e ), the remaining statements follow from the cohomology sequence

associated with (3.1) and from (3.4). 

From Lemma 3.2 we have:

(3.5) 0 → H ⁰ (A _e ) → H ⁰ ( E e ) → H ⁰ (B _e ) −→ H ^{∂ 1} (A _e ) → H ¹ ( E e ) → 0, where ∂ is the coboundary map determined by the extension (3.1). Thus

(3.6) h ¹ ( E e ) ≤ h ¹ (A _e ).

Remark 3.3. From (3.3), Assumption 3.1(i) is equivalent to 4b e − k _e − 6e + 5 + h ¹ (E e ) ≥ 7, that is k _e ≤ 4b _e − 6e − 2 + h ¹ ( E e ).

3.1. Vector bundles in Ext ¹ (B _e , A _e ). This subsection is devoted to an analysis of vector bundles fitting in the exact sequence (3.1). We need the following:

Lemma 3.4. With Assumptions 3.1, one has

(3.7) dim(Ext ¹ (B e , A e )) =



 

 



 

 



0 for b _e − e < k _e < ^3b

^+2−5e ₂ 5e + 2k e − 3b e − 1 for ^3b

^+2−5e ₂ ≤ k e < ^3b

^+2−4e ₂ 9e + 4k e − 6b e − 2 for ^3b

^+2−4e ₂ ≤ k e ≤ 4b e − 6e − 2 + h ¹ ( E e ).

Proof. By standard facts, Ext ¹ (B e , A e ) ∼ = H ¹ (A e − B _e ). From (3.2), (3.8) A _e − B _e ≡ C _e + (3b _e − 2k _e − 4e)f.

Now R ⁱ π _e∗ (C _e + (3b _e − 2k _e − 4e)f ) = 0, for i > 0, and π _e∗ (C _e + (3b _e − 2k _e − 4e)f ) ∼ = (O _P

⊕ O _P

(−e)) ⊗ O _P

(3b e − 2k _e − 4e), hence, from Leray’s isomorphism we have

h ¹ (A e − B _e ) = h ¹ (P ¹ , ( O _P

⊕ O _P

(−e)) ⊗ O _P

(3b e − 2k _e − 4e))

= h ¹ ( O _P

(3b e − 2k _e − 4e)) + h ¹ ( O _P

(3b e − 2k _e − 5e)) By Serre’s duality on P ¹ , the previous sum coincides with

h ⁰ ( O _P

(2k _e + 4e − 3b _e − 2)) + h ⁰ ( O _P

(2k _e + 5e − 3b _e − 2)).

Put α := 2k e + 4e − 3b e − 2 and β := 2k _e + 5e − 3b e − 2; note that β = α + e.

• If β < 0 then also α < 0 and thus h ¹ (A _e − B _e ) = 0.

• If β ≥ 0 and α < 0 then h ¹ (A _e − B _e ) = β + 1.

• Finally, if α ≥ 0 then β > 0 and thus h ¹ (A _e − B _e ) = α + β + 2.

Now observe that

β < 0 ⇔ k e < 3b e + 2 − 5e

2 and α < 0 ⇔ k e < 3b e + 2 − 4e

2 .

Moreover, since e ≥ 2, by Assumptions 3.1-(ii) one easily verifies that all such numerical conditions are compatible with Assumptions 3.1-(i) and (iii) (cf. also Rem. 3.3), in other words one has

b _e − e < 3b _e + 2 − 5e

2 < 3b _e + 2 − 4e

2 < 4b _e − 6e − 2 ≤ 4b _e − 6e − 2 + h ¹ ( E e ).

Hence (3.7) follows. 

Corollary 3.5. With Assumptions 3.1, for b e − e < k _e < ^3b

^+2−5e ₂ , one has E e = A e ⊕ B _e . In § 5 (cf. the proof of Theorem 5.1), we shall also need to know dim(Aut( E e )) = h ⁰ ( E e ⊗ E ^∨ e ).

Lemma 3.6. With Assumptions 3.1, take any E e ∈ Ext ¹ (A _e , B _e ). Then:

(3.9) h

( E

⊗ E

) =



 

 



 

 



6b

− 4k

− 9e + 4 for b

− e < k

<

3b

− 2k

− 4e + 2 for

≤ k

≤

and E

general

1 for

< k

≤ 4b

− 6e − 2 + h

(E

) and E

general.

Proof. (i) According to Corollary 3.5, for b _e − e < k _e < ^3b

^+2−5e ₂ , E e = A _e ⊕ B _e . Therefore E e ⊗ E ^∨ e ∼ = O ^⊕2 _F

⊕ (A _e − B _e ) ⊕ (B _e − A _e ).

From (3.2),

(3.10) B _e − A _e ≡ −C _e + (2k _e − 3b _e + 4e)f,

so it is not effective, since it negatively intersects the irreducible, moving curve f . From (3.8) and from the proof of Lemma 3.4, one has

h ⁰ (A e − B _e ) = h ⁰ (C e + (3b e − 2k _e − 4e)f ) = h ⁰ ( O _P

(3b e − 2k _e − 4e)) + h ⁰ ( O _P

(3b e − 2k _e − 5e)).

Put α ⁰ := 3b _e − 2k _e − 4e and β ⁰ := 3b _e − 2k _e − 5e; note that β ⁰ = α ⁰ − e

Since k _e < ^3b

^−5e+2 ₂ , O _P

(3b _e − 2k _e − 4e) is always effective whereas O _P

(3b _e − 2k _e − 5e) is effective unless 3b e − 2k _e − 5e = −1. So h ⁰ ( O _P

(3b e − 2k _e − 4e)) + h ⁰ ( O _P

(3b e − 2k _e − 5e)) = 6b _e − 4k _e − 9e + 2; taking into account also h ⁰ ( O ^⊕2 _F

), we conclude in this case.

(ii)-(iii) We treat here the remaining cases in (3.9). Recall that the upper-bound k e ≤ 4b _e − 6e − 2 + h ¹ ( E e ) comes from Assumptions 3.1-(i) (cf. Remark 3.3).

According to Lemma 3.4, when k e ≥ ^3b

^+2−5e ₂ , one has dim(Ext ¹ (B e , A e )) > 0. Therefore, let E e ∈ Ext ¹ (B _e , A _e ) be general. Using the fact that E e is of rank two and fits in the exact sequence (3.1), we have

E ^∨ e ∼ = E e ⊗ O(−A e − B _e ),

since c 1 (E e ) = A e + B e . Tensoring (3.1) respectively by E ^∨ _e , −B e , −A e , we get the following exact diagram

(3.11)

0 0 0

↓ ↓ ↓

0 → A _e − B _e → E e (−B _e ) → O _F

→ 0

↓ ↓ ↓

0 → E e (−B _e ) → E e ⊗ E ^∨ e → E e (−A _e ) → 0

↓ ↓ ↓

0 → O _F

→ E e (−A _e ) −→ B _e − A _e → 0

↓ ↓ ↓

0 0 0

We want to compute both h ⁰ ( E e (−B _e )) and h ⁰ ( E e (−A _e )).

From the cohomology sequence associated to the first row of diagram (3.11) we get 0 → H ⁰ (A _e − B _e ) → H ⁰ ( E e (−B _e )) → H ⁰ ( O _F

) −→ H ^{∂ b 1} (A _e − B _e ).

Observe that the coboundary map

H ⁰ ( O _F

) −→ H ^{∂ b 1} (A _e − B _e ),

has to be injective since it corresponds to the choice of the non-trivial extension class η _E

∈ Ext ¹ (B e , A e ) associated to E e general. Thus

h ⁰ ( E e (−B _e )) = h ⁰ (A _e − B _e )) = h ⁰ ( O _P

(α ⁰ )) + h ⁰ ( O _P

(β ⁰ )), with α ⁰ and β ⁰ as in Case (i) above.

Since k _e ≥ ^3b

^+2−5e ₂ , then β ⁰ ≤ −2 hence h ⁰ ( O _P

(β ⁰ )) = 0. Thus, h ⁰ ( E e (−B _e )) = h ⁰ ( O _P

(α ⁰ )).

Morover, h ⁰ ( O _P

(α ⁰ )) = 0 if and only if k _e > ^3b

₂ ^−4e ; thus

(3.12) h ⁰ ( E e (−B e )) =

 3b e − 2k e − 4e + 1 for ^3b

^+2−5e ₂ ≤ k e ≤ ^3b

₂ ^−4e 0 for k _e > ^3b

₂ ^−4e

From the third row of diagram (3.11), since B _e − A _e is not effective (cf. (3.10)), it follows

that h ⁰ ( E e (−A e )) = h ⁰ ( O F

) = 1, thus H ⁰ ( E e (−A e )) ∼ = C.

From the second column of diagram (3.11), we have

0 → H ⁰ ( E e (−B _e )) → H ⁰ ( E e ⊗ E ^∨ e ) −→ H ^{ψ 0} ( E e (−A _e )) ∼ = C → H ¹ ( E e (−B _e )) → · · · . Claim 3.7. The map ψ is surjective.

Proof of Claim 3.7. From the first two columns of diagram (3.11) and the fact that the coboundary map b ∂ is injective, as remarked above, we have

0 H ⁰ ( E e ⊗ E ^∨ _e )

↓ ↓ ^ψ

0 → H ⁰ ( O F

) −→ ^{∼ =} H ⁰ ( E e (−A e )) → 0

↓ ^{∂ b} ↓ ^{∂ ˜}

H ¹ (A e − B _e ) −→ H ¹ ( E e (−B e ))

Since H ⁰ (E e (−A e ))) ∼ = C, ψ is not surjective iff ψ ≡ 0, which is equivalent to ˜ ∂ injective and this is impossible since, from the first column of diagram (3.11), we have

H ⁰ (O _F

) −→ H ^{∂ b 1} (A e − B _e ) → H ¹ (E e (−B e ))

and the composition of the above two maps is ˜ ∂. This proves the claim.  From Claim 3.7, we conclude that

(3.13) h ⁰ ( E e ⊗ E ^∨ e ) = h ⁰ ( E e (−B _e )) + 1.

Combining (3.12) and (3.13) we determine h ⁰ ( E e ⊗ E ^∨ e ) in the case E e ∈ Ext ¹ (B _e , A _e ) is

general. 

Remark 3.8. (1) Note that when ^3b

^+2−5e ₂ ≤ k _e ≤ ^3b

₂ ^−4e (which makes sense only for e ≥ 2), any E e ∈ Ext ¹ (A e , B e ) is such that h ⁰ (E e ⊗ E ^∨ _e ) > 1, that is E e is not simple. This gives a different situation with respect to cases e = 0, 1. Indeed, for e = 1, b ₁ ≥ 4, when dim(Ext ¹ (B ₁ , A ₁ )) > 0, E 1 ∈ Ext ¹ (B ₁ , A ₁ ) general is always simple (cf. [7, Lemmas 3.4, 3.6]).

Similar computations hold for the case e = 0 (cf. (5.16) below).

(2) When h ⁰ ( E e ⊗ E ^∨ e ) = 1 (from (3.9) this, for instance, happens when E e ∈ Ext ¹ (B _e , A _e ) is general with ^3b

₂ ^−4e < k e ≤ 4b _e − 6e + 2 + h ¹ (E e )), E e has to be necessarily indecomposable.

3.2. Non-special bundles E e . For our analysis in § 4, it is fundamental to deal with vector bundles E e with no higher cohomology, in particular non-special that is with h ¹ (E e ) = 0.

Indeed, if E e turns out to be very-ample, the fact that E e has no higher cohomology not only implies that the ruled threefold P(E e ) isomorphically embeds via the tautological linear system as a smooth, linearly normal scroll X e in the projective space P ⁿ

of (the expected) dimension n _e := h ⁰ ( E e ) − 1, but mainly its non-speciality ensures good behavior of the Hilbert point [X _e ] in its Hilbert scheme (cf. proof of Claim 4.6).

From Lemma 3.2, having E e with no higher cohomology is equivalent to having E e non- special. In this subsection, we therefore find sufficient conditions for the non-speciality of E e , coming from (3.6) and the cohomology of A _e .

Lemma 3.9. With Assumptions 3.1, one has

(3.14) h ¹ (A e ) =



 

 

 

 



 

 

 

 



0 for b e − e < k e < 2b e + 2 − 4e

4e + k e − 2b e − 1 for 2b e + 2 − 4e ≤ k e < 2b e + 2 − 3e

7e + 2k _e − 4b e − 2 for 2b _e + 2 − 3e ≤ k _e < 2b _e + 2 − 2e

9e + 3k _e − 6b _e − 3 for 2b _e + 2 − 2e ≤ k _e ≤ 4b _e − 6e − 2 + h ¹ ( E e ).

Proof. Fom (3.2) π _e∗ (A _e ) ∼ = Sym ² ( O _P

⊕ O _P

(−e)) ⊗ O _P

(2b _e − k _e − 2e) and R ⁱ π _e∗ (A _e ) = 0 for i > 0. Hence by Leray’s isomorphism,

h ¹ (A e ) = h ¹ (Sym ² ( O _P

⊕ O _P

(−e)) ⊗ O _P

(2b e − k e − 2e))

= h ¹ (( O P

⊕ O P

(−e) ⊕ O P

(−2e)) ⊗ O P

(2b _e − k e − 2e))

= h ¹ ( O _P

(2b e − k e − 2e)) + h ¹ ( O _P

(2b e − k e − 3e)) + h ¹ ( O _P

(2b e − k e − 4e)) Let α ⁰ := 2e + k _e − 2b _e − 2. By Serre Duality theorem on P ¹ , from above we have

h ¹ (A e ) = h ⁰ ( O _P

(α ⁰ )) + h ⁰ ( O _P

(α ⁰ + e)) + h ⁰ ( O _P

(α ⁰ + 2e)).

• If α ⁰ +2e < 0, that is k e < 2b e +2−4e, then h ¹ (A e ) = 0 (observe that condition k e < 2b e +2−4e is compatible with k _e > b _e − e, because of Assumptions 3.1-(ii)).

• If α ⁰ + e < 0 ≤ α ⁰ + 2e, i.e. 2b e + 2 − 4e ≤ k e < 2b e + 2 − 3e, then

h ¹ (A e ) = h ⁰ (O _P

(α ⁰ + 2e))) = h ⁰ (O _P

(4e + k e − 2b _e − 2)) = 4e + k _e − 2b _e − 1.

• If α ⁰ < 0 ≤ α ⁰ + e, equivalently 2b e + 2 − 3e ≤ k e < 2b e + 2 − 2e, then

h ¹ (A _e ) = h ⁰ ( O _P

(α ⁰ + 2e)) + h ⁰ ( O _P

(α ⁰ + e)) = 2α ⁰ + 3e + 2 = 7e + 2k _e − 4b _e − 2.

• Finally, if α ⁰ ≥ 0, which is k _e ≥ 2b _e + 2 − 2e then

h ¹ (A _e ) = 3α ⁰ + 3e + 3 = 9e + 3k _e − 6b _e − 3

(notice that condition k e ≥ 2b _e + 2 − 2e is compatible with what computed in Remark 3.3; in other words one has 2b _e + 2 − 2e < 4b _e − 6e − 2 ≤ 4b _e − 6e − 2 + h ¹ ( E e ) because of Assumptions

3.1-(ii)). Hence h ¹ (A e ) is as in (3.14). 

Corollary 3.10. Assumptions 3.1 and k e < 2b e + 2 − 4e imply that any E e ∈ Ext ¹ (B e , A e ) is such that h ¹ ( E e ) = 0.

Remark 3.11. (1) Computations as in Remark 3.3 show that k _e < 2b _e + 2 − 4e implies h ⁰ ( E e ) = 4b e − k _e − 6e + 5 ≥ 2b _e − 2e + 3 which, from Assumption 3.1(iii) and e ≥ 2, turns out to be greater than or equal to 4e + 5 ≥ 13. Therefore, conditions b _e ≥ 3e + 1 and b e − e < k _e < 2b e + 2 − 4e are sufficient for Assumptions 3.1 to hold.

(2) When moreover b _e > 4e − 4, then ^3b

₂ ^−4e < 2b _e + 2 − 4e holds. In this case, as observed in Remark 3.8-(2), Lemmas 3.4 and 3.6 ensure that a general E e ∈ Ext ¹ (B _e , A _e ) is indecompo- sable.

Remark 3.12. As costumary, 0 ∈ Ext ¹ (B _e , A _e ) corresponds to the trivial bundle A _e ⊕ B _e . When k e ≥ 2b _e + 2 − 4e (i.e. when h ¹ (A e ) > 0), a given E e ∈ Ext ¹ (B e , A e ) \ {0} is non-special if and only if the coboundary map ∂ : H ⁰ (B _e ) → H ¹ (A _e ) (corresponding to the choice of E e ) is surjective. From (3.5), Im(∂) ∼ = Coker

n

H ⁰ ( E e ) → H ^{ρ 0} (B e ) o

; thus the surjectivity of ∂ can be geometrically interpreted with the fact that the linear system induced by the tautological line bundle O _P(E

) (1) onto the section Σ e ⊂ P(E e ), corresponding to the quotient line bundle E e → → B _e , is not complete with codim _H

_{( O}

₍₁₎₎ (Im(ρ)) = h ¹ (A _e ). When k _e ≥ 2b _e + 2 − 4e, it is a very tricky problem to find conditions granting the existence of a sublocus U ⊂ Ext ¹ (B _e , A _e ) s.t. h ¹ ( E e ) = 0 for any E e ∈ U.

4. 3-dimensional scrolls over F e and their Hilbert schemes

In this section, results from § 3 are used for the study of suitable 3-dimensional scrolls over F e in projective spaces and of some components of their Hilbert schemes.

The choice of c ₁ ( E e ) = 3C _e + b _e f and of the integers b _e , k _e (cf. Assumptions 3.1, 4.3), give the first case for which the bundle E e is both uniform and very-ample. Indeed, if E e is assumed to be ample with c ₁ ( E e ) = 3C _e + b _e f then the restriction of E e|f to any π _e -fiber f has to be ample; hence

E e|f = O f (a) ⊕ O f (b), with a, b > 0

and a + b = 3 because c ₁ ( E e )f = 3. Therefore, up to reordering, the only possibility is a = 2, b = 1 for any π e -fiber f , i.e. E e is uniform (cf. e.g. [35] and [2, Def. 3]). Moreover, c ₁ ( E e ) = 3C _e + b _e f , together with very-ampleness hypothesis, naturally lead to Assumptions 3.1.

Indeed, one has the following necessary condition for very-ampleness:

Proposition 4.1. (see [1, Prop. 7.2]) Let E e be a very-ample, rank-two vector bundle over F e

such that

c ₁ ( E e ) ≡ 3C _e + b _e f and c ₂ ( E e ) = k _e . Then E e satisfies all the hypotheses in Assumptions 3.1.

Remark 4.2. (1) By Lemma 3.4, when k e is such that b e − e < k _e < ^3b

^+2−5e ₂ the only bundle in Ext ¹ (B e , A e ) is E e := A e ⊕ B _e . The very-ampleness of B e and A e implies that of E e := A _e ⊕ B _e , [5, Lemma 3.2.3]. On the other hand the very-ampleness of E e := A _e ⊕ B _e implies the ampleness of B _e and A _e , but on F e ampleness of a line bundle is equivalent to very-ampleness, [29, V, Cor. 2.18], and thus E e := A e ⊕ B _e very-ample implies that both B _e and A _e are very-ample. Assumption 3.1(iii) (resp., k _e < 2b _e − 4e) is a necessary and sufficient condition for B e (resp., for A e ) to be very-ample. Since very-ampleness is an open condition, when dim(Ext ¹ (B _e , A _e )) > 0 and k _e < 2b _e − 4e holds, then the general bundle E e

in Ext ¹ (B _e , A _e ) is very-ample too.

(2) From the previous sections, condition b e − e < k _e < 2b e − 4e is compatible because of Assumption 3.1(ii) and gives also that any E e ∈ Ext ¹ (B e , A e ) is non-special.

(3) Comparing Lemmas 3.4 and 3.6 with this new bound on k _e , we notice that ^3b

^+2−5e ₂ <

2b _e − 4e holds if and only if b _e ≥ 3e + 3; similarly ^3b

^+2−4e ₂ < 2b _e − 4e holds if and only if b _e ≥ 4e + 3 and, finally, ^3b

₂ ^−4e < 2b _e − 4e holds if and only if b ≥ 4e + 1. In particular, when b _e ≥ 4e+1 and ^3b

₂ ^−4e < k _e < 2b _e −4e, Lemma 3.6 also ensures the existence of indecomposable bundles in Ext ¹ (B _e , A _e ) (cf. Remark 3.11(2)).

From Remark (4.2), it is clear that from now on we will focus on b e − e < k _e < 2b e − 4e. In other words, Assumptions 3.1 will be replaced by:

Assumptions 4.3. Let e ≥ 2, k _e , b _e be integers. Let E e be a rank-two vector bundle over F e

such that

c ₁ ( E e ) ≡ 3C _e + b _e f, c ₂ ( E e ) = k _e , with

(4.1) b _e ≥ 3e + 1 and b _e − e < k _e < 2b _e − 4e.

Let

(P(E e ), O _P(E

) (1))

be the 3-dimensional scroll over F e , and let π e : F e → P ¹ and ϕ : P(E e ) → F e be the usual projections.

Proposition 4.4. Let E e be as in Assumptions 4.3. Moreover, when dim(Ext ¹ (B _e , A _e )) > 0, we further assume that E e ∈ Ext ¹ (B e , A e ) is general. Then O _P(E

) (1) defines an embedding

(4.2) Φ _e := Φ _{| O}

P(Ee)

(1)| : P(E e ) ,→ X _e ⊂ P ⁿ

, where X e = Φ e (P(E e )) is smooth, non-degenerate, of degree d e , with

(4.3) n _e = 4b _e − k _e − 6e + 4 ≥ 4e + 4 ≥ 12 and d _e = 6b _e − 9e − k _e . Denoting by (X e , L e ) := (X e , O X

(H)) ∼ = (P(E e ), O _P(E

) (1)), one also has

(4.4) h ⁱ (X e , L e ) = 0, i ≥ 1.

Proof. The very-ampleness of L _e is equivalent to that of E e , and the latter follows from Remark 4.2(1) and Assumptions 4.3. The formula on the degree d e of X e in (4.3) follows from (2.1).

From Leray’s isomorphisms, Lemma 3.2 and Corollary 3.10 we get (4.4). Finally, since n _e +1 :=

h ⁰ (X _e , L _e ) = h ⁰ (F e , E e ), then n _e + 1 ≥ 4e + 5 ≥ 13 follows from Remark 3.11(2) and the fact

that e ≥ 2. 

4.1. The component X e of the Hilbert scheme containing [X _e ]. In what follows, we will be interested in studying the Hilbert scheme parametrizing subvarieties of P ⁿ

having the same Hilbert polynomial P (T ) := P X

(T ) ∈ Q[T ] of X e , which is the numerical polynomial defined by

(4.5) P (m) = χ(X _e , mL _e ) = 1

6 m ³ L ³ _e − 1

4 m ² L ² _e ·K + 1

12 mL _e ·(K ² +c ₂ )+χ( O X

), for all m ∈ Z, as it follows from [24, Example 15.2.5, pg 291].

For basic terminology and facts on Hilbert schemes we follow, for instance, [28, 38, 39].

The scroll X _e ⊂ P ⁿ

corresponds to a point [X _e ] ∈ H ^d ₃

^,n

, where H ^d ₃

^,n

denotes the Hilbert scheme parametrizing 3-dimensional subvarieties of P ⁿ

with Hilbert polynomial P (T ) as above (in particular of degree d e ), where n e and d e are as in (4.3). When [X e ] ∈ H ^d 3

^,n

is a smooth point, X _e is said to be unobstructed in P ⁿ

. Let

(4.6) N _e := N _X

_/P

be the normal bundle of X _e in P ⁿ

. From standard facts on Hilbert schemes (cf. e.g. [38, Corollary 3.2.7]), one has

(4.7) T _[X

_] ( H ^d ₃

^,n

) ∼ = H ⁰ (N _e ) and

(4.8) h ⁰ (N e ) − h ¹ (N e ) ≤ dim _[X

_] ( H ^d 3

^,n

) ≤ h ⁰ (N e ),

where the left-most integer in (4.8) is the expected dimension of H ₃ ^d

^,n

at [X e ] and where equality holds on the right in (4.8) iff X _e is unobstructed in P ⁿ

.

The next result shows that X _e is unobstructed and such that [X _e ] sits in an irreducible component of H ^d ₃

^,n

with “nice” behaviour.

THEOREM 4.5. There exists an irreducible component X e ⊆ H ^d ₃

^,n

, which is generically smooth and of (the expected) dimension

(4.9) dim( X e ) = n _e (n _e + 1) + 3k _e − 2b _e + 3e − 5, such that [X _e ] belongs to the smooth locus of X e .

Proof. By (4.7) and (4.8), the statement will follow by showing that H ⁱ (X e , N e ) = 0, for i ≥ 1, and conducting an explicit computation of h ⁰ (X _e , N _e ) = χ(X _e , N _e ).

To do this, let

0 −→ O X

−→ O X

(1) ^⊕(n

⁺¹⁾ −→ T _P

|X

−→ 0 (4.10)

be the Euler sequence on P ⁿ

restricted to X e . Since (X e , L e ) is a scroll over F e , H ⁱ (X e , O X

) = H ⁱ (F e , O _F

) = 0, for i ≥ 1.

(4.11)

From (4.4), (4.11), the cohomology sequence associated to (4.10) and from the fact that X e

is non–degenerate, one has:

(4.12) h ⁰ (X e , T _P

|X

) = (n e + 1) ² − 1 and h ⁱ (X e , T _P

|X

) = 0, for i ≥ 1.

The normal sequence

0 −→ T _X

−→ T

P

|X

−→ N _e −→ 0

(4.13)

gives therefore

H ⁱ (X _e , N _e ) ∼ = H ⁱ⁺¹ (X _e , T _X

) for i ≥ 1.

(4.14)

Claim 4.6. H ⁱ (X _e , N _e ) = 0, for i ≥ 1.

Proof of Claim 4.6. From (4.12), (4.13) and dimension reasons, one has h ^j (X _e , N _e ) = 0, for j ≥ 3. For the other cohomology spaces, we can use (4.14).

In order to compute H ^j (X _e , T _X

), j = 2, 3, we use the scroll map ϕ : P(E e ) −→ F e and we consider the relative cotangent bundle sequence:

0 → ϕ ^∗ (Ω ¹ _F

) → Ω ¹ _X

→ Ω ¹ _X

e

|F

−→ 0.

(4.15)

From (4.15) and the Whitney sum, one obtains

c ₁ (Ω ¹ _X

) = c ₁ (ϕ ^∗ (Ω ¹ _F

)) + c ₁ (Ω ¹ _X

_|F

) thus

Ω ¹ _X

_|F

= K _X

+ ϕ ^∗ (−c 1 (Ω ¹ _F

)) = K _X

+ ϕ ^∗ (−K _F

).

The adjunction theoretic characterization of the scroll gives

K _X

= −2L _e + ϕ ^∗ (K _F

+ c ₁ ( E e )) = −2L _e + ϕ ^∗ (K _F

+ 3C _e + b _e f ) thus

Ω ¹ _X|F

= K X

+ ϕ ^∗ (−K _F

) = −2L e + ϕ ^∗ (3C e + b e f ) which, combined with the dual of (4.15), gives

0 → 2L _e − ϕ ^∗ (3C _e + b _e f ) → T _X

→ ϕ ^∗ (T _F

) → 0.

(4.16)

In what follows, we compute the cohomology of the left and right-most bundles in (4.16).

(i) First we concentrate on ϕ ^∗ (T _F

). By Leray’s isomorphism, one has H ⁱ (ϕ ^∗ (T _F

)) ∼ = H ⁱ (T _F

), for any i ≥ 0.

Consider therefore the relative cotangent bundle sequence of π e : F e → P ¹ 0 → π ^∗ _e Ω ¹

P

→ Ω ¹

F

→ Ω ¹

F

|P

→ 0.

(4.17) Since Ω ¹

F

|P

= K _F

+ π _e ^∗ O _P

(2) = −2C _e − ef , dualizing (4.17) we get 0 → 2C _e + ef → T _F

→ π ^∗ _e T _P

→ 0.

(4.18)

Since π _e ^∗ T _P

∼ = π _e ^∗ O _P

(2), by Leray’s isomorphism

h ⁰ (π _e ^∗ T _P

) = 3, h ⁱ (π _e ^∗ T _P

) = 0, for i ≥ 1.

As in the proof of Lemma 3.9, Leray’s isomorphism gives

h ⁱ (2C _e + ef ) = h ⁱ (P ¹ , [ O _P

⊕ O _P

(−e) ⊕ O _P

(−2e)] ⊗ O _P

(e)), for any i ≥ 1.

Thus,

h ⁰ (2C e + ef ) = e + 2, h ¹ (2C e + ef ) = e − 1, h ^j (2C e + ef ) = 0, for j ≥ 2.

From [34, Lemma 10], one has

h ⁰ (F e , T _F

) = e + 5.

Therefore, putting all together in the cohomology sequence associated to (4.18), we get h ⁰ (X e , ϕ ^∗ (T _F

)) = h ⁰ (F e , T _F

) = e + 5,

h ¹ (X e , ϕ ^∗ (T _F

)) = h ¹ (F e , T _F

) = e − 1, (4.19)

h ^j (X e , ϕ ^∗ (T _F

)) = h ^j (F e , T _F

) = 0, for j ≥ 2.

(ii) We now devote our attention to the cohomology of 2L e − ϕ ^∗ (3C e + b e f ) in (4.16). Noticing that R ⁱ ϕ ∗ (2L _e ) = 0 for i ≥ 1 (see [29, Ex. 8.4, p. 253]), projection formula and Leray’s isomorphism give

(4.20) H ⁱ (X e , 2L e − ϕ ^∗ (3C e + b e f )) ∼ = H ⁱ (F e , Sym ² E e ⊗ (−3C _e − b _e f )), ∀ i ≥ 0.

Therefore

(4.21) h ^j (X e , 2L e − ϕ ^∗ (3C e + b e f )) = 0, j ≥ 3, for dimension reasons.

We now want to show that H ² (F e , Sym ² E e ⊗ (−3C _e − b _e f )) = 0. To do this, recall that E e

fits in the exact sequence (3.1), with A _e and B _e as in (3.2). By [29, 5.16.(c), p. 127], there is a finite filtration of Sym ² ( E e ),

Sym ² ( E e ) = F ⁰ ⊇ F ¹ ⊇ F ² ⊇ F ³ = 0 with quotients

F ^p /F ^p+1 ∼ = Sym ^p (A e ) ⊗ Sym ^2−p (B e ), for each 0 ≤ p ≤ 2. Hence

F ⁰ /F ¹ ∼ = Sym ⁰ (A e ) ⊗ Sym ² (B e ) = 2B e

F ¹ /F ² ∼ = Sym ¹ (A e ) ⊗ Sym ¹ (B e ) = A e + B e

F ² /F ³ ∼ = Sym ² (A _e ) ⊗ Sym ⁰ (B _e ) = 2A _e , that is F ² = 2A _e , since F ³ = 0. Thus, we get the following exact sequences

(4.22) 0 → F ¹ → Sym ² ( E e ) → 2B _e → 0

(4.23) 0 → F ² → F ¹ → A _e + B e → 0

(4.24) F ² = 2A _e

Twisting (4.22), (4.23) with −c ₁ ( E e ) = −3C _e − b _e f = −A _e − B _e and using (4.24) we get (4.25) 0 → F ¹ (−3C _e − b _e f ) → Sym ² ( E e ) ⊗ (−3C _e − b _e f ) → B _e − A _e → 0

(4.26) 0 → A e − B _e → F ¹ (−3C e − b _e f ) → O F

→ 0

First we focus on (4.26); from (3.8) and from the same arguments used in Lemma 3.4, one gets

h ⁱ (A e − B _e ) = h ⁱ (P ¹ , O _P

(3b e − 2k _e − 4e) ⊕ O _P

(3b e − 2k _e − 5e));

so, for dimension reasons, h ⁱ (A _e − B _e ) = 0, for any i ≥ 2. Since moreover h ⁱ ( O _F

) = 0 for i ≥ 1, then (4.26) gives

(4.27) h ² (F ¹ (−3C _e − b _e f )) = 0.

Passing to (4.25) observe that, from (3.10) and from the fact that K _F

≡ −2C _e − (e + 2)f , one gets

h ² (B _e − A _e ) = h ⁰ (−C _e + (3b _e − 2k _e − 5e − 2)f ) = 0.

Thus, from (4.27), (4.25) and (4.20), one has

(4.28) h ² (F e , Sym ² E e ⊗ (−3C _e − b _e f )) = h ² (X e , 2L e − ϕ ^∗ (3C e + b e f )) = 0.

Using (4.19), (4.21) and (4.28) in the cohomology sequence associated to (4.16), we get (4.29) h ^j (X _e , T _X

) = 0, for j ≥ 2.

Isomorphism (4.14) concludes the proof of Claim 4.6. 

Using (4.7) and (4.8), Claim 4.6 implies that there exists an irreducible component X e of H ^d ₃

^,n

containing [X _e ] as a smooth point.

Since smoothness is an open condition, X e is generically smooth. Moreover, always from

(4.8) and Claim 4.6, it follows that dim( X e ) = h ⁰ (X _e , N _e ) = χ(N _e ) i.e. X e has the expected

dimension.

The Hirzebruch-Riemann-Roch theorem gives χ(N _e ) = 1

6 (n ³ ₁ − 3n ₁ n ₂ + 3n ₃ ) + 1

4 c ₁ (n ² ₁ − 2n ₂ ) (4.30)

+ 1

12 (c ² ₁ + c 2 )n 1 + (n e − 3)χ( O X

) where n _i := c _i (N _e ) and c _i := c _i (X _e ).

If K := K X

, the Chern classes of N e can be obtained from (4.13):

n ₁ = K + (n _e + 1)L _e ; n ₂ = 1

2 n _e (n _e + 1)L ² _e + (n _e + 1)L _e K + K ² − c ₂ ; (4.31)

n ₃ = 1

6 (n _e − 1)n _e (n _e + 1)L ³ _e + 1

2 n _e (n _e + 1)KL ² _e + (n _e + 1)K ² L _e

−(n _e + 1)c ₂ L _e − 2c ₂ K + K ³ − c ₃ . The numerical invariants of X e can be easily computed by:

KL ² _e = −2d _e + 4b _e − 6e − 6; K ² L _e = 4d _e − 14b _e + 21e + 20;

c ₂ L _e = 2b _e − 3e + 10; K ³ = −8d _e + 36b _e − 54e − 48;

−Kc ₂ = 24; c 3 = 8.

Plugging these in (4.31) and then in (4.30), one gets

χ(N _e ) = (d _e + 3e − 2b _e + 5)n _e − 5 − 24e + 16b _e − 3d _e . From (4.3), one has d e = 6b e − 9e − k _e ; in particular

d e + 3e − 2b e + 5 = 4b e − 6e − k _e + 5 = n e + 1, as it follows from (4.3). Thus

χ(N e ) = (n e + 1)n e − 5 − 3(6b _e − 9e − k _e ) − 24e + 16b e = n e (n e + 1) + 3k e − 2b _e + 3e − 5,

as in (4.9). 

Remark 4.7. The proof of Theorem 4.5 gives

(4.32) h ⁰ (N _e ) = n _e (n _e + 1) + 3k _e − 2b _e + 3e − 5, h ⁱ (N _e ) = 0, i ≥ 1.

Using (4.12) and (4.32) in the exact sequence (4.13), one gets

(4.33) χ(T _X

) = h ⁰ (T _P

|

) − h ⁰ (N _e ) = 6b _e − 4k _e + 9 − 9e.

Moreover, from (4.13) and (4.12), one has:

(4.34) 0 → H ⁰ (T X

) → H ⁰ (T _P

|

) → H ^{α 0} (N e ) → H ^{β 1} (T X

) → 0,

In the sequel (cf. the proof of Theorem 5.1 below) we will make use of the following consequences of Theorem 4.5, interpreted via (4.34).

Corollary 4.8. When dim(Ext ¹ (B e , A e )) = 0, one has

h ⁰ (T X

) = 6b e − 4k _e − 8e + 8, h ¹ (T X

) = e − 1, h ^j (T X

) = 0, for j ≥ 2.

In particular,

(4.35) dim(Coker(α)) = e − 1,

where α is the map in (4.34).

Proof. From Lemma 3.4 and Remark 4.2(3), notice that dim(Ext ¹ (B _e , A _e )) = 0 occurs when, either b e = 3e + 1, 3e + 2 and for any b e − e < k _e < 2b e − 4e, or for b _e ≥ 3e + 3 and b _e − e < k _e < ^3b

^+2−5e ₂ < 2b _e − 4e.

Now h ^j (T _X

) = 0, for j ≥ 2, is (4.29) which more generally holds for any b _e , k _e as in (4.1).

We thus concentrate on h ^j (T X

), for j = 0, 1. Since h ¹ (A e − B _e ) = dim(Ext ¹ (B e , A e )) = 0, from (4.26) one has

h ⁰ (F ¹ (−3C e − b _e f )) = h ⁰ (A e − B _e ) + 1 = 6b e − 4k _e − 9e + 3, h ¹ (F ¹ (−3C e − b _e f )) = 0.

Passing to (4.25), from (3.10) and Leray’s isomorphism, one has h ⁱ (B _e − A _e ) = 0 for any i ≥ 0.

Thus

h ⁱ (Sym ² E e ⊗ (−3C _e − b _e f )) = h ⁱ (F ¹ (−3C e − b _e f )), for 0 ≤ i ≤ 2, and thus

h ⁰ (Sym ² E e ⊗ (−3C _e − b _e f )) = 6b _e − 4k _e − 9e + 3, h ¹ (Sym ² E e ⊗ (−3C _e − b _e f )) = 0.

The cohomology sequence associated to (4.16) along with (4.20) and (4.19) gives the first part of the statement.

Finally, for (4.35), it suffices to notice that the map β in (4.34) is surjective.  5. The general point of the component X e

In this section a description of the general point of X e , determined in Theorem 4.5, is pre- sented. The following preliminary result shows that in general scrolls arising from Proposition 4.4 do not fill up X e .

THEOREM 5.1. Let Y e be the locus in X e filled up by threefold scrolls X _e as in Proposition 4.4. Then

(i) if b _e − e < k _e < ^3b

^+2−5e ₂ , one has codim _X

( Y e ) = e − 1, (ii) if ^3b

^+2−5e ₂ ≤ k _e ≤ 2b _e − 4e, one has codim _X

( Y e ) ≤ e − 1.

Proof. In case (i), from Lemma 3.4, dim(Ext ¹ (B _e , A _e )) = 0. Therefore X _e ∼ = P(A e ⊕ B _e ) is uniquely determined, so dim( Y e ) = dim(Im(α)), where α is the map in (4.34). Thus

codim _X

( Y e ) = dim(Coker(α)) = e − 1 where the last equality comes from (4.35).

In case (ii) we have dim(Ext ¹ (B e , A e )) > 0; consider the following quantities.

(a) Denote by τ e the number of parameters counting isomorphism classes of projective bundles P(E e ) as in Proposition 4.4. In other words, τ _e takes into account weak isomorphism classes of extensions, which are parametrized by P(Ext ¹ (B e , A e )) (cf.

[22, p. 31]), see Lemma 3.4 for the calculation of Ext ¹ (B e , A e ). In particular, τ e = dim(Ext ¹ (B _e , A _e )) − 1 and, from Lemma 3.4, this number is as follows:

(5.1) τ _e :=

 5e + 2k _e − 3b _e − 2 ^3b

^+2−5e ₂ ≤ k _e < ^3b

^+2−4e ₂ 9e + 4k e − 6b _e − 3 ^3b

^+2−4e ₂ ≤ k _e < 2b e − 4e

(more precisely, note that if ^3b

^+2−5e ₂ ≤ k _e < 2b _e − 4e ≤ ^3b

^+2−4e ₂ , that is, when 3e + 3 ≤ b e ≤ 4e + 2, then (5.1) simply reads τ _e := 5e + 2k e − 3b _e − 2).

(b) G X

⊂ P GL(n _e + 1, C) denotes the projective stabilizer of X e ⊂ P ⁿ

, i.e. the subgroup of projectivities of P ⁿ

fixing X _e . In particular (cf. (4.13))

(5.2) dim(P GL(n e + 1, C)) − dim(G X

) = n e (n e + 2) − h ⁰ (T X

)

is the dimension of the full orbit of X _e ⊂ P ⁿ

under the action of all the projective

transformations of P ⁿ

. This equals dim(Im(α)), where α is the map in (4.34).

The rest of the proof now reduces to a parameter computation to obtain a lower bound for the dimension of Y e . From the exact sequence (3.1), we observe that:

(*) the line bundle A e is uniquely determined on F e , since A e ∼ = O _F

(2C e )⊗π e ∗ O _P

(2b e −k _e −2e);

(**) the line bundle B _e is uniquely determined on F e , similarly.

Let us compute how many parameters are needed to describe Y e . To do this, we have to add up the following quantities:

1) 0 parameters for A e on F e , by (*);

2) 0 parameters for B e , by (**);

3) τ _e as in (5.1), for isomorphism classes of P(E e );

4) n _e (n _e + 2) − h ⁰ (T _X

), as in (5.2), for the dimension of the full orbit of X _e ⊂ P ⁿ

chosen.

Thus,

(5.3) dim( Y e ) = τ _e + n _e (n _e + 2) − dim(G _X

)

The next step is to find an upper bound for dim(G X

). It is clear that there is an obvious inclusion

(5.4) G X

,→ Aut(X e ),

where Aut(X _e ) denotes the algebraic group of abstract automorphisms of X _e . Since X _e , as an abstract variety, is isomorphic to P(E e ) over F e , then

dim(Aut(X _e )) = dim(Aut(F e )) + dim(Aut _F

(P(E e ))),

where Aut _F

(P(E e )) denotes the group of automorphisms of P(E e ) fixing the base (cf. e.g.

[34]). From the fact that Aut(F e ) is an algebraic group, in particular smooth, it follows that dim(Aut(F e )) = h ⁰ (F e , T _F

) = e + 5

since e ≥ 2 (cf. [34, Lemma 10, p. 105]). On the other hand, dim(Aut _F

(P(E e ))) = h ⁰ (E e ⊗ E ^∨ e ) − 1, since Aut _F

(P(E e )) are given by endomorphisms of the projective bundle.

To sum up,

dim(Aut(X _e )) = h ⁰ ( E e ⊗ E ^∨ e ) + 4 + e.

From (5.4), dim(G _X

) ≤ dim(Aut(X _e )), then from (5.3) we deduce (5.5) dim( Y e ) ≥ τ _e + n _e (n _e + 2) − h ⁰ ( E e ⊗ E ^∨ e ) − 4 − e.

According to Lemma 3.6, one has

h ⁰ ( E e ⊗ E ^∨ e ) =







3b _e − 2k _e − 4e + 2 for ^3b

^+2−5e ₂ ≤ k _e < 2b _e − 4e ≤ ^3b

₂ ^−4e 1 for ^3b

₂ ^−4e ≤ k _e < 2b _e − 4e,

As for τ _e , we use (5.1) and hence we get

(a) for ^3b

^+2−5e ₂ ≤ k _e < ^3b

₂ ^−4e , τ e = 5e + 2k e − 3b _e − 2 and h ⁰ (E ⊗ E ^∨ ) = 3b e − 2k _e − 4e + 2, (b) for ^3b

₂ ^−4e ≤ k _e < ^3b

^+2−4e ₂ , τ e = 5e + 2k e − 3b _e − 2 and h ⁰ (E ⊗ E ^∨ ) = 1;

(c) for ^3b

^+2−4e ₂ ≤ k _e < 2b e − 4e, τ _e = 9e + 4k e − 6b _e − 3 and h ⁰ (E ⊗ E ^∨ ) = 1.

In all cases, from (5.5) we get dim( Y e ) ≥ n _e (n _e + 2) − 6b _e + 4k _e + 8e − 8. From (4.9), we get codim _X

( Y e ) = dim( X e ) − dim( Y e )

≤ n _e (n _e + 1) + 3k _e − 2b _e + 3e − 5 − (n _e (n _e + 2) − 6b _e + 4k _e + 8e − 8) = e − 1.



5.1. A candidate for the general point of X e . From Theorem 5.1, we need to exhibit a smooth variety in P ⁿ

which is a candidate to represent the general point of X e as in Theorem 4.5. In other words, this candidate must flatly degenerate in P ⁿ

to the threefold scroll X _e , corresponding to [X _e ] ∈ Y e general, in such a way that the base-scheme of this flat, embedded degeneration is contained in X e .

In this section we first construct this candidate and analyze some of its properties similar to those investigated for X e in §’s 3, 4. In § 5.2, we show that this candidate actually corresponds to the general point of X e .

For e ≥ 2 integer, consider

(5.6)  = 0, 1 according to  ≡ e (mod 2).

Consider the Hirzebruch surface F  , let π _ : F  → P ¹ be the natural projection and let C _ be the unique section of F  corresponding to O _P

⊕ O _P

(−) → → O _P

(−) on P ¹ . Thus C _ ² = −.

With notation as in Assumptions 4.3, consider

(5.7) b _ := b _e − 3(e − )

2 and k _ := k _e .

This choice of b _ is needed in order to ensure that the Hilbert polynomial data (in particular the degree) of X  are the same as those of X e , as it will become clear in (5.14).

Lemma 5.2. With (5.7) above, conditions (4.1) on b e and k e read as

(5.8) b _ ≥ 3

2 (e + ) + 1 ≥ 3

2 + 4 and b _ −  < b _ + (e − 3)

2 < k _ < 2b _ − 3 − e.

Proof. The proof is given by straightforward computations using (4.1) and (5.7). Indeed, by (5.7), b _e ≥ 3e + 1 in (4.1) reads as b _ + ^3(e−) ₂ ≥ 3e + 1 which is b _ ≥ ³ ₂ e + 1 + ^3 ₂ ; the latter is greater than or equal to ^3 ₂ + 4 since e ≥ 2 and from hypotheses on . Similarly, one has b  + ^(e−3) ₂ = b  −  + ^(e−) ₂ > b  −  for the same reasons.

Using b _ = b _e − ^3(e−) ₂ , one finds

(5.9) b e − e = b _ + 1

2 (e − 3).

Using (5.9), one gets

(5.10) 2b _e − 4e = 2(b _e − e) − 2e = 2b _ − 3 − e.

Since from (5.7) one has k  = k e , then one concludes by (4.1).  Consider now the following line bundles on F  (cf. (3.2)):

(5.11) A  ≡ 2C _ + (2b  − k _ − 2)f and

(5.12) B _ ≡ C _ + (k _ − b _ + 2)f.

Remark 5.3. Notice that, with these choices, both A  and B  are very-ample. Indeed, from [29, V Cor. 2.18], B  is very-ample if and only if k  > b  − , wheras A _ is very-ample if and only if k _ < 2b _ − 4. Both conditions are implied by (5.8), since e ≥ 2.

As in (3.1), we consider E  a rank–two vector bundle on F  fitting in the exact sequence

(5.13) 0 → A _ → E  → B _ → 0.

Thus

c 1 ( E  ) = A  + B  ≡ 3C _ + b  f and c 2 ( E  ) = A  B  = k  = k e . From (2.1) one has deg( E  ) = (3C  + b  f ) ² − k _ = −9 + 6b  − k _ . Thus (5.7) gives (5.14) deg( E  ) = 6b _e − 9e − k _e = d _e ,

where d e = deg( E e ) is as in (4.3).