---
abstract: |
    The paper investigates the non-vanishing of $H^1({\mathcal{E}}(n))$, where ${\mathcal{E}}$ is a (normalized) rank two vector bundle over any smooth irreducible threefold $X$ of degree $d$ such that ${\mathrm{Pic}}(X) \cong {\mathbb{Z}}$. If $\epsilon$ is the integer defined by the equality $\omega_X = {\mathcal{O}}_X(\epsilon)$, and $\alpha$ is the least integer $t$ such that $H^0({\mathcal{E}}(t)) \ne 0$, then, for a non-stable ${\mathcal{E}}$ ($\alpha \le 0)$ the first cohomology module does not vanish at least between the endpoints $\frac{\epsilon-c_1}{2}$ and $-\alpha-c_1-1$. The paper also shows that there are other non-vanishing intervals, whose endpoints depend on $\alpha$ and also on the second Chern class $c_2$ of ${\mathcal{E}}$. If ${\mathcal{E}}$ is stable the first cohomology module does not vanish at least between the endpoints $\frac{\epsilon-c_1}{2}$ and $\alpha-2$. The paper considers also the case of a threefold $X$ with ${\mathrm{Pic}}(X) \ne {\mathbb{Z}}$ but ${\mathrm{Num}}(X) \cong {\mathbb{Z}}$ and gives similar non-vanishing results.\
    **Keyword:** rank two vector bundles, smooth threefolds, non-vanishing of 1-cohomology.\
    **MSC 2010:** 14J60, 14F05.
author:
- 'E. Ballico'
- 'P. Valabrega'
- 'M. Valenzano'
title: |
    Non-vanishing theorems for rank two vector\
    bundles on threefolds [^1]
---

Introduction
============

In 1942 G. Gherardelli ([@Ghe]) proved that, if $C$ is a smooth irreducible curve in ${\mathbb{P}}^3$ whose canonical divisors are cut out by the surfaces of some degree $e$ and moreover all linear series cut out by the surfaces in ${\mathbb{P}}^3$ are complete, then $C$ is the complete intersection of two surfaces. Shortly and in the language of modern algebraic geometry: every $e$-subcanonical smooth curve $C$ in ${\mathbb{P}}^3$ such that $h^1({\mathcal{I}}_C(n)) = 0$ for all $n$ is the complete intersection of two surfaces.

Thanks to the Serre correspondence between curves and vector bundles (see [@Hvb], [@H1], [@H2]) the above statement is equivalent to the following one: if ${\mathcal{E}}$ is a rank two vector bundle on ${\mathbb{P}}^3$ such that $h^1({\mathcal{E}}(n)) = 0$ for all $n$, then ${\mathcal{E}}$ splits.

There are many improvements of the above result with a variety of different approaches (see for instance [@CV1], [@CV2], [@Ellia], [@P], [@RV]): it comes out that a rank two vector bundle ${\mathcal{E}}$ on ${\mathbb{P}}^3$ is forced to split if $h^1({\mathcal{E}}(n))$ vanishes for just one strategic $n$, and such a value $n$ can be chosen arbitrarily within a suitable interval, whose endpoints depend on the Chern classes and the least number $\alpha$ such that $h^0({\mathcal{E}}(\alpha)) \ne 0$.

When rank two vector bundles on a smooth threefold $X$ of degree $d$ in ${\mathbb{P}}^4$ are concerned, similar results can be obtained, with some interesting difference.

In 1998 Madonna ([@Madonna]) proved that on a smooth threefold $X$ of degree $d$ in ${\mathbb{P}}^4$ there are ACM rank two vector bundles (i.e. whose 1-cohomology vanishes for all twists) that do not split. And this can happen, for a normalized vector bundle ${\mathcal{E}}$ ($c_1\in\{0,-1\})$, only when $ 1-\frac{d+c_1}{2} < \alpha < \frac{d-c_1}{2}$, while an ACM rank two vector bundle on $X$ whose $\alpha$ lies outside of the interval is forced to split.

The following non-vanishing results for a normalized non-split rank two vector bundle on a smooth irreducible thereefold of degree $d$ in ${\mathbb{P}}^4$ are proved in [@Madonna]:

if $\alpha \le 1-\frac{d+c_1}{2}$, then $h^1({\mathcal{E}}(\frac{d-3-c_1}{2}))\ne 0$ if $d+c_1$ is odd, $h^1({\mathcal{E}}(\frac{d-4-c_1}{2}))\ne 0, h^1({\mathcal{E}}(\frac{d-2-c_1}{2}))\ne 0$ if $d+c_1$ is even, while $h^1({\mathcal{E}}(\frac{d-c_1}{2}))\ne 0$ if $d+c_1$ is even and moreover $\alpha \le -\frac{d+c_1}{2}$;

if $\alpha\ge \frac{d-c_1}{2}$, then $h^1({\mathcal{E}}(\frac{d-3-c_1}{2}))\ne 0$ if $d+c_1$ is odd, while $h^1({\mathcal{E}}(\frac{d-4-c_1}{2}))\ne 0$ if $d+c_1$ is even.

In [@Madonna] it is also claimed that the same techniques work to obtain similar non-vanishing results on any smooth threefold $X$ with ${\mathrm{Pic}}(X) \cong {\mathbb{Z}}$ and $h^1({\mathcal{O}}_X(n)) = 0$, for every $n$.

The present paper investigates the non-vanishing of $H^1({\mathcal{E}}(n))$, where ${\mathcal{E}}$ is a rank two vector bundle over any smooth irreducible threefold $X$ of degree $d$ such that ${\mathrm{Pic}}(X) \cong {\mathbb{Z}}$ and $H^1({\mathcal{O}}_X(n)) = 0, \forall n$. Actually we can prove that for such an ${\mathcal{E}}$ there is a wider range of non-vanishing for $h^1({\mathcal{E}}(n))$, so improving the above results.

More precisely, when ${\mathcal{E}}$ is (normalized and) non-stable ($\alpha \le 0$) the first cohomology module does not vanish at least between the endpoints $\frac{\epsilon-c_1}{2}$ and $-\alpha-c_1-1$, where $\epsilon$ is defined by the equality $\omega(X) = {\mathcal{O}}_X(\epsilon)$ (and is $d-5$ if $X \subset {\mathbb{P}}^4$). But we can show that there are other non-vanishing intervals, whose endpoints depend on $\alpha$ and also on the second Chern class $c_2$ of ${\mathcal{E}}$.

If on the contrary ${\mathcal{E}}$ is stable the first cohomology module does not vanish at least between the endpoints $\frac{\epsilon-c_1}{2}$ and $\alpha-2$, but other ranges of non-vanishing can be produced.

We give a few examples obtained by pull-back from vector bundles on ${\mathbb{P}}^3$.

We must remark that most of our non-vanishing results do not exclude the range for $\alpha$ between the endpoints $1-\frac{d+c_1}{2}$ and $\frac{d-c_1}{2}$ (for a general threefold it becomes $-\frac{\epsilon+3+c_1}{2} < \alpha < \frac{\epsilon+5-c_1}{2})$. Actually [@Madonna] produces some examples of nonsplit ACM rank two vector bundles on smooth hypersurfaces in ${\mathbb{P}}^4$, but it can be seen that they do not conflict with our theorems.

As to threefolds with ${\mathrm{Pic}}(X) \ne {\mathbb{Z}}$, we need to observe that a key point is a good definition of the integer $\alpha$. We are able to prove, by using a boundedness argument, that $\alpha$ exists when ${\mathrm{Pic}}(X) \ne {\mathbb{Z}}$ but ${\mathrm{Num}}(X) \cong {\mathbb{Z}}$. In this event the correspondence between rank two vector bundles and two-codimensional subschemes can be proved to hold. In order to obtain non-vanishing results that are similar to the results proved when ${\mathrm{Pic}}(X) \cong {\mathbb{Z}}$, we need also use the Kodaira vanishing theorem, which holds in characteristic 0. We can extend the results to characteristic $p > 0$ if we assume a Kodaira-type vanishing condition.

Notation
========

We work over an algebraically closed field ${\mathbf{k}}$ of any characteristic.\
Let $X$ be a non-singular irreducible projective algebraic variety of dimension 3, for short a smooth threefold.\
We fix an ample divisor $H$ on $X$, so we consider the polarized threefold $(X,H)$.\
We denote with ${\mathcal{O}_{X}}(n)$, instead of ${\mathcal{O}_{X}}(nH)$, the invertible sheaf corresponding to the divisor $nH$, for each $n\in{\mathbb{Z}}$.\
For every cycle $Z$ on $X$ of codimension $i$ it is defined its degree with respect to $H$, i.e. $\deg(Z;H) := Z \cdot H^{3-i}$, having identified a codimension 3 cycle on $X$, i.e. a $0$-dimensional cycle, with its degree, which is an integer.\
From now on (with the exception of section 7) we consider a smooth polarized threefold $(X,{\mathcal{O}}_{X}(1))=(X,H)$ that satifies the following conditions:

$\mathbf{(C1)}$

:   ${\mathrm{Pic}}(X)\cong{\mathbb{Z}}$ generated by $[H]$,

$\mathbf{(C2)}$

:   $H^1(X,{\mathcal{O}_{X}}(n)) = 0$ for every $n\in{\mathbb{Z}}$,

$\mathbf{(C3)}$

:   $h^0(O_X(1)) \ne 0$.

By condition $\mathbf{(C1)}$ every divisor on $X$ is linearly equivalent to $aH$ for some integer $a\in{\mathbb{Z}}$, i.e. every invertible sheaf on $X$ is (up to an isomorphism) of type ${\mathcal{O}_{X}}(a)$ for some $a\in{\mathbb{Z}}$, in particular we have for the canonical divisor $K_X \sim \epsilon H$, or equivalently $\omega_{X}\simeq{\mathcal{O}}_{X}(\epsilon)$, for a suitable integer $\epsilon$. Furthermore, by Serre duality condition $\mathbf{(C2)}$ implies that $H^2(X,{\mathcal{O}_{X}}(n)) = 0$ for all $n\in{\mathbb{Z}}$.\
Since by assumption $A^{1}(X)={\mathrm{Pic}}(X)$ is isomorphic to ${\mathbb{Z}}$ through the map $[H]\mapsto 1$, where $[H]=c_{1}({\mathcal{O}}_{X}(1))$, we identify the first Chern class $c_1({\mathcal{F}})$ of a coherent sheaf with a whole number $c_1$, where $c_1({\mathcal{F}}) = c_1 H$.\
The second Chern class $c_2({\mathcal{F}})$ gives the integer $c_2 = c_2({\mathcal{F}})\cdot H$ and we will call this integer the second Chern number or the second Chern class of ${\mathcal{F}}$.\
We set $$d := \deg(X;H) = H^3,$$ so $d$ is the degree of the threefold $X$ with respect to the ample divisor $H$.\
Let $c_1(X)$ and $c_2(X)$ be the first and second Chern classes of $X$, that is of its tangent bundle $TX$ (which is a locally free sheaf of rank 3); then we have $$c_1(X) = [-K_X] = -\epsilon [H],$$ so we identify the first Chern class of $X$ with the integer $-\epsilon$. Moreover we set $$\tau := \deg(c_2(X);H) = c_2(X) \cdot H,$$ i.e. $\tau$ is the degree of the second Chern class of the threefold $X$.\
In the following we will call the triple of integers $(d,\epsilon,\tau)$ the **characteristic numbers** of the polarized threefold $(X,{\mathcal{O}_{X}}(1))$.

We recall the well-known Riemann-Roch formula on the threefold $X$ (see [@Valenzano], proposition 4).

\[gRR\] Let ${\mathcal{F}}$ be a rank $r$ coherent sheaf on $X$ with Chern classes $c_{1}({\mathcal{F}})$, $c_{2}({\mathcal{F}})$ and $c_{3}({\mathcal{F}})$. Then the Euler-Poincaré characteristic of ${\mathcal{F}}$ is $$\begin{aligned}
\chi({\mathcal{F}}) = & \frac{1}{6}\Big(c_{1}({\mathcal{F}})^{3} - 3 c_{1}({\mathcal{F}})\cdot 
c_{2}({\mathcal{F}}) + 3 c_{3}({\mathcal{F}})\Big) + 
\frac{1}{4}\Big(c_{1}({\mathcal{F}})^{2} - 2 c_{2}({\mathcal{F}})\Big)\cdot 
c_{1}(X) + \\
& + \frac{1}{12} c_{1}({\mathcal{F}})\cdot\Big(c_{1}(X)^{2} + c_{2}(X)\Big) +
\frac{r}{24} c_{1}(X)\cdot c_{2}(X)\end{aligned}$$ where $c_{1}(X)$ and $c_{2}(X)$ are the Chern classes of $X$, that is the Chern classes of the tangent bundle $TX$ of $X$.

So applying the Riemann-Roch Theorem to the invertible sheaf ${\mathcal{O}_{X}}(n)$, for each $n\in{\mathbb{Z}}$, we get the Hilbert polynomial of the sheaf ${\mathcal{O}_{X}}(1)$ $$\chi({\mathcal{O}_{X}}(n)) = \frac{d}{6} \left(n - \frac{\epsilon}{2}\right) \left[ \left(n - \frac{\epsilon}{2}\right)^2 + \frac{\tau}{2d} - \frac{\epsilon^2}{4} \right]\!.$$

Let ${\mathcal{E}}$ be a rank 2 vector bundle on the threefold $X$ with Chern classes $c_1({\mathcal{E}})$ and $c_2({\mathcal{E}})$, so with Chern numbers $c_1$ and $c_2$. We assume that ${\mathcal{E}}$ is normalized, i.e. that $c_1 \in\{0,-1\}$. It is defined the integer $\alpha$, the so called first relevant level, such that $h^0({\mathcal{E}}(\alpha)) \ne 0, h^0({\mathcal{E}}(\alpha-1)) = 0$. If $\alpha > 0$, ${\mathcal{E}}$ is called stable, non-stable otherwise.

We set $$\vartheta = \frac{3c_2}{d} - \frac{\tau}{2d} + \frac{\epsilon^2}{4}-\frac{3c_1^2}{4},  \qquad
\zeta_0 = \frac{\epsilon-c_1}{2}, \quad \text{and} \quad w_0 = [\zeta_0]+1,$$ where $[\zeta_0]=$ integer part of $\zeta_0$, so the Hilbert polynomial of ${\mathcal{E}}$ can be written as $$\chi({\mathcal{E}}(n)) = \frac{d}{3}\big(n - \zeta_0\big)\Big[\big(n - \zeta_0\big)^2 - \vartheta\Big].$$ If $\vartheta\ge0$ we set $$\zeta = \zeta_0 + \sqrt{\vartheta}$$ so in this case the Hilbert polynomial of ${\mathcal{E}}$ has the three real roots $\zeta' \le \zeta_0 \le \zeta$ where $\zeta' =  \zeta_0 - \sqrt{\vartheta}, \zeta =  \zeta_0 + \sqrt{\vartheta}$. We also define $\bar\alpha = [\zeta]+1$.\
The polinomial $\chi({\mathcal{E}}(n))$, as a rational polynomial, has three real roots if and only if $\vartheta\ge0$, and it has only one real root if and only if $\vartheta<0$.\
If ${\mathcal{E}}$ is normalized, we set $$\delta = c_2+d\alpha^2+c_1d\alpha.$$

We have $\delta = 0$ if and only if ${\mathcal{E}}$ splits (see [@VV], Lemma 3.13: the proof works in general).

Unless stated otherwise, we work over the smooth polarized threefold $X$ and *${\mathcal{E}}$ is a normalized non-split rank two vector bundle on $X$*.

About the characteristic numbers $\epsilon$ and $\tau$
======================================================

In this section we want to recall some essentially known properties of the characteristic numbers of the threefold $X$ (see also [@SB] for more general statements). We start with the following remark.

\[OX\] 1. For the fixed ample invertible sheaf ${\mathcal{O}_{X}}(1)$ we have $$h^0({\mathcal{O}_{X}}(n)) \begin{cases} = 0 & \quad\text{for } n < 0 \\
= 1 & \quad \text{for } n = 0 \\
\ne 0 & \quad\text{for } n > 0 \end{cases}$$ and also $h^0({\mathcal{O}_{X}}(m)) - h^0({\mathcal{O}_{X}}(n)) > 0$ for all $n,m\in{\mathbb{Z}}$ with $m > n \ge 0$.\
2. It holds $$\chi({\mathcal{O}_{X}}) = h^0({\mathcal{O}_{X}}) - h^3({\mathcal{O}_{X}}) = 1 - h^0({\mathcal{O}_{X}}(\epsilon)),$$ so we have: $$\chi({\mathcal{O}_{X}}) = 1 \iff \epsilon < 0, \ \ \
\chi({\mathcal{O}_{X}}) = 0 \iff \epsilon = 0, \ \ \
\chi({\mathcal{O}_{X}}) < 0 \iff \epsilon > 0.$$

Let $(X,{\mathcal{O}_{X}}(1))$ be a smooth polarized threefold with [characteristic numbers $(d,\epsilon,\tau)$]{}. Then it holds:

1)

:   $\epsilon \ge -4$,

2)

:   $\epsilon = -4$ if and only if $X = {\mathbb{P}}^3$, i.e. $(d,\epsilon,\tau)=(1,-4,6)$ and so $\frac{\tau}{2d} - \frac{\epsilon^2}{4} = -1$,

3)

:   if $\epsilon = -3$, then $X$ is a hyperquadric in ${\mathbb{P}}^4$, so $(d,\epsilon,\tau)=(2,-3,8)$ and $\frac{\tau}{2d} - \frac{\epsilon^2}{4} = -\frac{1}{4}$,

4)

:   $\epsilon\tau$ is a multiple of $24$, in particular if $\epsilon < 0$ then $\epsilon \tau = -24$,

5)

:   if $\epsilon\ne 0$, then $\tau > 0$,

6)

:   if $\epsilon = 0$, then $\tau > -2d$,

7)

:   $\tau$ is always even,

8)

:   if $\epsilon$ is even, then $\frac{\tau}{2d} - \frac{\epsilon^2}{4} \ge -1$,

9)

:   if $\epsilon$ is odd, then $\frac{\tau}{2d} - \frac{\epsilon^2}{4} \ge -\frac{1}{4}$,

10)

:   if $\epsilon < 0$, then the only possibilities for $(\epsilon,\tau)$ are the following $$(\epsilon,\tau) \in \{(-4,6), \, (-3,8), \, (-2,12), \, (-1,24) \},$$

For statements **1)**, **2)**, **3)** see [@SB].\
**4)** Observe that $\chi({\mathcal{O}_{X}}) = - \frac{1}{24} \epsilon \tau$ is an integer, and moreover, if $\epsilon < 0$, then $\chi({\mathcal{O}_{X}})=1$.\
**5)** By Remark \[OX\] we have: if $\epsilon > 0$ then $- \frac{1}{24} \epsilon \tau < 0$, while if $\epsilon < 0$ then $- \frac{1}{24} \epsilon \tau > 0$. In both cases we deduce $\tau > 0$.\
**6)** If $\epsilon = 0$, then we have $$\chi({\mathcal{O}_{X}}(n)) = \frac{d}{6} n  \left(n^2 + \frac{\tau}{2d} \right),$$ and also $$\chi({\mathcal{O}_{X}}(n)) = h^0({\mathcal{O}_{X}}(n)) > 0 \quad \forall n > 0,$$ therefore we must have $\frac{2d+\tau}{12} > 0$, so $\tau > -2d$.\
**7)** Assume that $\epsilon$ is even, then we have $$d\left(1- \frac{\epsilon}{2}\right)\left(1+ \frac{\epsilon}{2}\right) +  \frac{\tau}{2} = 
d\left(1-\frac{\epsilon^2}{4}+\frac{\tau}{2d}\right) = 6\, \chi\left({\mathcal{O}_{X}}\left(\frac{\epsilon}{2}+1\right)\right) \in{\mathbb{Z}}$$ and moreover $d\left(1- \frac{\epsilon}{2}\right)\left(1+ \frac{\epsilon}{2}\right)\in{\mathbb{Z}}$, so $\tau$ must be even.\
If $\epsilon$ is odd, the proof is quite similar.\
**8)** Let $\epsilon$ be even. If it holds $$h^0\left({\mathcal{O}_{X}}\!\left(\frac{\epsilon}{2}+1\right)\right) - h^0\left({\mathcal{O}_{X}}\!\left(\frac{\epsilon}{2}-1\right)\right) = \chi\left({\mathcal{O}_{X}}\left(\frac{\epsilon}{2}+1\right)\right) < 0,$$ then we must have $h^0\left({\mathcal{O}_{X}}\left(\frac{\epsilon}{2}-1 \right)\right) \ne 0$, which implies $$h^0\left({\mathcal{O}_{X}}\!\left(\frac{\epsilon}{2}+1\right)\right) - h^0\left({\mathcal{O}_{X}}\!\left(\frac{\epsilon}{2}-1\right)\right) \ge 0,$$ a contradiction. So we must have:$$\chi\left({\mathcal{O}_{X}}\!\left(\frac{\epsilon}{2}+1\right)\right) =
 \frac{d}{6}\left(1 + \frac{\tau}{2d} - \frac{\epsilon^2}{4}\right) \ge 0,$$ therefore $$\frac{\tau}{2d} - \frac{\epsilon^2}{4} \ge -1.$$\
**9)** The proof is quite similar to the proof of **8**).\
**10)** If $\epsilon < 0$, then by **1)** we have $\epsilon\in\{-4,-3,-2,-1\}$, and moreover $\epsilon \tau = -24$ by **4)**, so we obtain the thesis.

Non-stable vector bundles ($\alpha \le 0$)
==========================================

**Case $\epsilon \ge 1$.**\
In this subsection we make the following assumptions:

*${\mathcal{E}}$ is a normalized non-split rank two vector bundle with $\alpha \le 0$ and $\epsilon \ge 1$*.\
The case $\epsilon \le 0$ is investigated in the next subsection.

\[nonstable0\] Assume that $\zeta_0 < -\alpha-c_1-1$. Then it holds: $$h^1({\mathcal{E}}(n))-h^2({\mathcal{E}}(n)) = (n-\zeta_0)\delta$$ for every $n$ such that $\zeta_0 < n \le -\alpha-c_1-1$.

First we assume $c_1 = 0$. It is enough to observe that, from the inequality $n+\alpha \le 0$ and the exact sequence $$0 \to {\mathcal{O}}_X(n-\alpha) \to {\mathcal{E}}(n) \to {\mathcal{I}}(n+\alpha) \to 0$$ we obtain: $h^0({\mathcal{E}}(n)) = h^0({\mathcal{O}}_X(n-\alpha)) = \chi({\mathcal{O}}_X(n-\alpha))+h^0({\mathcal{O}}_X(\epsilon-n+\alpha)) =  \chi({\mathcal{O}}_X(n-\alpha))$ since $\epsilon-n+\alpha \le -1$. We also have: $h^0({\mathcal{E}}(\epsilon-n)) = h^0({\mathcal{O}}_X(\epsilon-n-\alpha)) = \chi({\mathcal{O}}_X(\epsilon-n-\alpha))+h^0({\mathcal{O}}_X(n+\alpha)) =  \chi({\mathcal{O}}_X(\epsilon-n-\alpha))$.\
Now it is enough to observe that $h^1({\mathcal{E}}(n))-h^2({\mathcal{E}}(n)) = h^0({\mathcal{E}}(n))-h^0({\mathcal{E}}(\epsilon-n)) -\chi({\mathcal{E}}(n)) =  \chi({\mathcal{O}}_X(n-\alpha))- \chi({\mathcal{O}}_X(\epsilon-n-\alpha))-\chi({\mathcal{E}}(n))$. If we use the Riemann-Roch formulas for the Euler functions we obtain the required equality.\
Now we assume $c_1 = -1$. We recall that $h^3({\mathcal{E}}(n)) = h^0({\mathcal{E}}(\epsilon-n+1))$. As before we have $h^1({\mathcal{E}}(n))-h^2({\mathcal{E}}(n)) =  \chi({\mathcal{O}}_X(n-\alpha))- \chi({\mathcal{O}}_X(\epsilon-n-\alpha+1))-\chi({\mathcal{E}}(n))$ and the computation is very similar.

Observe that the statement of Proposition \[nonstable0\] still holds when $n = \zeta_0$, because the two sides of the equality vanish.

\[nonstable1\] Let us assume that $\zeta_0 < -\alpha-c_1-1$ and let $n$ be such that $\zeta_0 < n \le -\alpha-1-c_1$. Then $h^1({\mathcal{E}}(n)) \ge (n-\zeta_0)\delta$. In particular $h^1({\mathcal{E}}(n)) \ne 0$.

It is enough to observe that $$h^1({\mathcal{E}}(n))-h^2({\mathcal{E}}(n)) = (n-\zeta_0)\delta$$ and that the right side of the equality is strictly positive for a non-split vector bundle.

Observe that the above theorem describes a non-empty set of integers if and only if $-\alpha -1 -c_ 1 > \zeta_0$; this means $\alpha < -\frac{\epsilon+2+c_1}{2}$, i.e. $\alpha \le -\frac{\epsilon+3+c_1}{2}$. So our assumption on $\alpha$ agrees with the bound of [@Madonna].\
Observe that the inequality on $\alpha$ implies that $\alpha \le -2$ if $\epsilon \ge 1$.

\[nonstable2\] Assume that $\frac{6\delta}{d}-\frac{\tau}{2d}+\frac{\epsilon^2}{4}-\frac{3c_{1}^2}{4}\ge 0$. Let $n > \zeta_0$ be such that $ \epsilon-\alpha-c_1+1 \le n < \zeta_0+\sqrt{\frac{6\delta}{d}-\frac{\tau}{2d}+\frac{\epsilon^2}{4}-\frac{3c_{1}^2}{4}}$ and put $$S(n) =\frac{d}{6}\left(n-\frac{\epsilon-c_1}{2}\right)\!\left[(n-\frac{\epsilon-c_1}{2})^2-6\frac{c_2+d \alpha^2+c_1d\alpha}{d}+\frac{\tau}{2d}-\frac{\epsilon^2}{4}+\frac{3c_1^2}{4}\right]\!.$$ Then $h^1({\mathcal{E}}(n))\ge -S(n) > 0$. In particular $h^1({\mathcal{E}}(n))\ne 0$.

Assume $c_1 = 0$. Under our hypothesis $h^0({\mathcal{E}}(\epsilon-n)) = 0$ and so $-h^1({\mathcal{E}}(n))+h^2({\mathcal{E}}(n)) = \chi({\mathcal{E}}(n)) -  h^0({\mathcal{O}}_X(n-\alpha))$. Observe that $\chi({\mathcal{E}}(n)) - h^0({\mathcal{O}}_X(n-\alpha)) -S(n) = \frac{1}{2}nd\alpha(-\epsilon+n+\alpha))+\frac{1}{12}d\alpha(-3\epsilon \alpha+2\alpha^2+\epsilon^2+\frac{\tau}{d})  \le 0$. Therefore we have: $h^1({\mathcal{E}}(n)) \ge h^2({\mathcal{E}}(n))-S(n)$. Hence $h^1({\mathcal{E}}(n))$ may possibly vanish when $$\left(n-\frac{\epsilon}{2}\right)^2-6\frac{c_2+d \alpha^2}{d}+\frac{\tau}{2d}-\frac{\epsilon^2}{4} \ge 0.$$ When $S(n) < 0$, so $-S(n) > 0$, $h^1({\mathcal{E}}(n)) \ge -S(n) > 0$ and in particular it cannot vanish.\
If $c_1 = -1$ the proof is quite similar.

Now we put $\frac{\tau}{2d}-\frac{\epsilon^2}{4} =\lambda$ and consider the following degree $3$ polynomial: $$F(X) = X^3+\left(\lambda-\frac{6\delta}{d}\right)X+\frac{6\alpha \delta}{d}.$$ It is easy to see that, if $\frac{6\delta}{d}-\frac{\tau}{2d}+\frac{\epsilon^2}{4} \le 0$, $F(X)$ is strictly increasing and so it has only one real root $X_0$.

\[nonstable3\] Assume that $\frac{6\delta}{d}-\frac{\tau}{2d}+\frac{\epsilon^2}{4} \le 0$. Let $n$ be such that $ \epsilon-\alpha-c_1+1 \le n < -\alpha+X_0+\zeta_0$, where $X_0$ = unique real root of $F(X)$ . Then $h^1({\mathcal{E}}(n)) \ge -\frac{d}{6}F(n+\alpha-\zeta_0) > -\frac{d}{6}F(X_0) = 0$. In particular $h^1({\mathcal{E}}(n)) \ne 0$.

Assume $c_1 = 0$, the proof being quite similar if $c_1 = -1$.\
It holds (see proposition \[nonstable0\]): $$\begin{aligned}
h^1({\mathcal{E}}(n)) & - h^2({\mathcal{E}}(n)) = \chi({\mathcal{O}}_X(n-\alpha))-\chi({\mathcal{E}}(n)) = \\
& = \left(n-\frac{\epsilon}{2}\right)(c_2+d\alpha^2)+\chi({\mathcal{O}}_X(\epsilon-n-\alpha)) = \\
& = \left(n-\frac{\epsilon}{2}\right)(c_2+d\alpha^2)-\frac{d}{6}\left(n+\alpha-\frac{\epsilon}{2}\right)\left(\left(n+\alpha-\frac{\epsilon}{2}\right)^2+\lambda\right).\end{aligned}$$ If we put: $X = n+\alpha-\frac{\epsilon}{2}$, we obtain: $\frac{d}{6} F(X) =  \frac{d}{6}(X^3+(\lambda-\frac{6\delta}{d})X+\frac{6\alpha \delta}{d}) = h^2({\mathcal{E}}(n))-h^1({\mathcal{E}}(n))$. Therefore $h^1({\mathcal{E}}(n)) > -\frac{d}{6}F(n+\alpha-\zeta_0) > -\frac{d}{6}F(X_0) = 0$.

Observe that in Theorems \[nonstable2\] and \[nonstable3\] $\alpha$ can be $0$.

**Case $\epsilon \le 0$.**

In the event that $\epsilon \le -2$, we have $\epsilon-\alpha-c_1+1 \le -\alpha-c_1-1$. Therefore Theorems \[nonstable2\], \[nonstable3\] give something new only beyond $-\alpha-c_1-1.$

First of all we observe that Theorems \[nonstable1\], \[nonstable3\] obviously hold as they are stated also when $\epsilon \le 0$. So we discuss Theorem \[nonstable2\].\
A. $\epsilon \le -2$.\
In theorem \[nonstable2\] we need to know that $$\frac{1}{2}nd\alpha(-\epsilon+n+\alpha)+\frac{1}{12}d\alpha(\epsilon^2+\frac{\tau}{d}-3\epsilon \alpha+2\alpha^2)  \le 0.$$ The first term of the sum is for sure negative; as for $$\frac{1}{12}d\alpha\left(\epsilon^2+\frac{\tau}{d}\right)+\frac{1}{12}d\alpha^2(-3\epsilon+2\alpha)$$ we observe that the quantity in brackets has discriminant $$\Delta = \epsilon^2-8\frac{\tau}{d} = 4\left(\frac{\epsilon^2}{4}-\frac{\tau}{2d}+\frac{\tau}{2d}-8\frac{\tau}{d}\right) \le 4(1-15) < 0.$$ Therefore it is positive for all $\alpha \le 0$ and the product is negative.\
B. $\epsilon = -1$.\
In theorem \[nonstable2\] we need to know that $$\frac{1}{2}nd\alpha(1+n+\alpha)+\frac{1}{12}d\alpha\left(1+\frac{\tau}{d}\right)+\frac{1}{12}d\alpha^2(3+2\alpha)  \le 0.$$ If $\alpha \le -2$, then it is enough to observe that $\frac{\tau}{d}+3\alpha+2\alpha^2 \ge 0$. If $\alpha = -1$ we have to consider $-\frac{1}{2}n^2d+\frac{1}{12}d\frac{\tau}{d}$ and then we observe that $6n^2+\frac{\tau}{d} > 0$. If $\alpha = 0$ obviously the quantity is $0$.\
C. $\epsilon = 0$.\
In theorem \[nonstable2\] we need to know that $$\frac{1}{2}nd\alpha(n+\alpha)+\frac{1}{12}d\alpha\left(\frac{\tau}{d}\right)+\frac{1}{12}d\alpha^2(2\alpha)  \le 0.$$\
It is enough to observe that $2\alpha^2+\frac{\tau}{d} > 0$ by Proposition 3.2, $\mathbf{6)}$ if $\alpha < 0$; otherwise we have a $0$ quantity, and that $n+\alpha \le 0$.

Observe that the case $\alpha = 0$ in Theorem \[nonstable1\] can occur only if $\epsilon \le -c_1-3$.

In theorem \[nonstable2\] we do not use the hypothesis $-\frac{\epsilon+3}{2} \ge \alpha$, but we assume that $6\frac{c_2+d \alpha^2}{d}-\frac{\tau}{2d}+\frac{\epsilon^2}{4}-1\ge 0$. In theorem \[nonstable3\] we do not use the hypothesis $-\frac{\epsilon+3}{2} \ge \alpha$, but we assume that $6\frac{c_2+d \alpha^2}{d}-\frac{\tau}{2d}+\frac{\epsilon^2}{4} < 0$. Moreover in both theorems there is a range for $n$, the left endpoint being $\epsilon-\alpha-c_1+1$ and the right endpoint being either $\zeta_0+\sqrt{6\frac{c_2+d \alpha^2}{d}-\frac{\tau}{2d}+\frac{\epsilon^2}{4}-1}$ (\[nonstable2\]) or $\zeta_0-\alpha+X_0$ (\[nonstable3\]).\
In [@Madonna] there are examples of ACM nonsplit vector bundles on smooth threefolds in ${\mathbb{P}}^4$, with $-\frac{\epsilon+3+c_1}{2} < \alpha < \frac{\epsilon+5-c_1}{2}$. We want to emphasize that our theorems do not conflict with the examples of [@Madonna]: if $C$ is any curve described in [@Madonna] and lying on a smooth threefold of degree $d$, then our numerical constraints cannot be satisfied (we have checked it directly in many but not all cases).

Let us consider a smooth degree $d$ threefold $X \subset {\mathbb{P}}^4$.\
We have: $$\epsilon = d-5,\ \ \tau =  d(10-5d+d^2),\ \  \theta = \frac{3c_2}{d}-\frac{d^2-5+3c_1^2}{4}$$ (see [@Valenzano]). As to the characteristic function of $O_X$ and ${\mathcal{E}}$, it holds: $$\chi({\mathcal{O}}_X(n)) = \frac{d}{6}\left(n-\frac{d-5}{2}\right)\!\left[\left(n-\frac{d-5}{2}\right)^2+\frac{d^2-5}{4}\right]\!,$$ $$\chi({\mathcal{E}}(n)) = \frac{d}{3} \left(n-\frac{d-5-c_1}{2}\right)\!\left[\left(n-\frac{d-5-c_1}{2}\right)^2+\frac{d^2}{4}-\frac{5}{4}+\frac{3c_1^2}{4}-\frac{3c_2}{d}\right]\!.$$ Then it is easy to see that the hypothesis of Theorem \[nonstable2\], i.e. $6\frac{\delta}{d}-\frac{d^2-5+3c_1^2}{4}\ge 0$ is for sure fulfilled if $c_2 \ge 0, \alpha \le -\frac{d-2+c_1}{2}$. In fact we have (for the sake of simplicity when $c_1 = 0)$: $-6\frac{6c_2+d\alpha^2}{d}+\frac{d^2-5}{4} \le \frac{d^2-5}{4}-6\frac{d^2-2d+1}{4} = -\frac{5d^2-12d+11}{4} < 0$.

Condition $\mathbf{(C2)}$ holds for sure if $X$ is a smooth hypersurface of $ {\mathbb{P}}^4$. In general, for a characteristic $0$ base field, only the Kodaira vanishing holds ([@HAL], remark 7.15) and so, unless we work over a threefold $X$ having some stronger vanishing, we need assume, in Theorems \[nonstable1\], \[nonstable2\], \[nonstable3\] that $n-\alpha \notin \{0,...,\epsilon\}$ (which implies, by duality, that also $\epsilon -n+\alpha \notin \{0,...,\epsilon\}$).

Observe that the first assumption ($n-\alpha \notin \{0,...,\epsilon\})$ in the case of Theorem \[nonstable1\] is automatically fulfilled because of the hypothesis $\zeta_0 < -\alpha-c_1-1$, and in Theorems \[nonstable2\] and \[nonstable3\] because of the hypothesis $\epsilon-\alpha-c_1+1 \le n$. In fact $n-\alpha$ is greater than $\epsilon$. But this implies that $\epsilon-n+\alpha < 0$ and so also the second condition is fulfilled, at least when $\epsilon \ge 0$. For the case $\epsilon < 0$ in positive characteristic see [@SB].

Observe that, if $\epsilon < 0$, Kodaira (and so $\mathbf{(C2)}$) holds for every $n$.

For a general discussion, also in characteristic $p > 0$, of this question, see section 7, remark 7.8.

In the above theorems we assume that ${\mathcal{E}}$ is a nonsplit bundle. If ${\mathcal{E}}$ splits, then (see section 2) $\delta = 0$. In Theorem \[nonstable1\] this implies $h^1({\mathcal{E}}(n))-h^2({\mathcal{E}}(n)) = 0$ and so nothing can be said on the non-vanishing.

Let us now consider Theorem \[nonstable2\]. If $\delta = 0$, then we must have: $\zeta_0 < n < \zeta_0+\sqrt{-\frac{\tau}{2d}+\frac{\epsilon^2}{4}-\frac{3c_1^2}{4}} \le \zeta_0+1$ (the last inequality depending on Proposition 3.2, $\mathbf{8),9)})$. As a consequence $\zeta_0$ cannot be a whole number. Moreover, since we have $2\zeta_0-\alpha+1 \le n < \zeta_0+\sqrt{-\frac{\tau}{2d}+\frac{\epsilon^2}{4}-\frac{3c_1^2}{4}}$, we obtain that $\zeta_0 < \alpha \le 0$, hence $\epsilon-c_1 \le -1$. If $c_1 = 0$, $\epsilon \in \{-1, -3\}$. If $\epsilon = -3$, then $n$ must satisfy (see Proposition 3.2, $\mathbf{8})$ the following inequalities: $-\frac{3}{2} < n < -1$, which is a contradiction. If $\epsilon = -1$, then, by Proposition 3.2, $\mathbf{8})$ we have $-1+\alpha+1 < -\frac{1}{2}+ \frac{1}{2} = 0$, which implies $\alpha > 0$, a contradiction. If $c_1 = -1$, then $\epsilon \in \{-2, -4\}$. If $\epsilon = -4$, we have $\sqrt{-\frac{\tau}{2d}+\frac{\epsilon^2}{4}-\frac{3c_1^2}{4}} = \frac{1}{2}$, and so we must have: $-\frac{3}{2} < n < -1$, which is impossible. If $\epsilon = -2$, then $\zeta_0 = -\frac{1}{2}$ and so $-2-\alpha+2 < -\frac{1}{2}+\sqrt{1-\frac{3}{4}}$, which implies $-\alpha < 0$ hence $\alpha > 0$, a contradiction with the non-stability of ${\mathcal{E}}$.

Then we consider Theorem \[nonstable3\]. The vanishing of $\delta$ on the one hand implies $\lambda > 0$ and $X_0 = 0$. But on the other hand from our hypothesis on the range of $n$ we see that $\zeta_0 \le -2$, hence $\epsilon = -4, c_1 = 0$. But this contradicts Proposition 3.2, $\mathbf{2)}$.

Stable vector bundles
=====================

In the present section we assume that $\alpha \ge \frac{\epsilon-c_1+5}{2}$, or equivalently that $c_1+2\alpha\ge\epsilon+5$. This means that $\alpha \ge 1$ in any event, so ${\mathcal{E}}$ is stable.

The following lemma holds both in the stable and in the non-stable case.

\[leftvanishing\] If $h^1({\mathcal{E}}(m)) = 0$ for some integer $m \le \alpha-2$, then $h^1({\mathcal{E}}(n)) = 0$ for all $n \le m$.

First of all observe that, by our condition $\mathbf{(C3)}$, from the restriction exact sequence we can obtain in cohomology the exact sequence $$0 \to H^0({\mathcal{E}}(t-1) \to H^0({\mathcal{E}}(t)) \to H^0({\mathcal{E}}_H(t)) \to 0.$$ Then we can follow the proof given in [@VV] for ${\mathbb{P}}^3$ (where condition $\mathbf{(C3)}$ is automatically fulfilled).

\[stable1\] Let ${\mathcal{E}}$ be a rank 2 vector bundle on the threefold $X$ with first relevant level $\alpha$. If $\alpha\ge\frac{\epsilon+5-c_1}{2}$, then $h^1({\mathcal{E}}(n))\ne 0$ for $w_0\le n\le \alpha-2$.

By the hypothesis it holds $w_0 \le \alpha-2$, so we have $h^0({\mathcal{E}}(n))=0$ for all $n\le w_0+1$. Assume $h^1({\mathcal{E}}(w_0))=0$, then by Lemma \[leftvanishing\] it holds $h^1({\mathcal{E}}(n))=0$ for every $n\le w_0$. Therefore we have $$\chi({\mathcal{E}}(w_0)) = h^0({\mathcal{E}}(w_0)) + h^1({\mathcal{E}}(-w_0+\epsilon-c_1)) - h^0({\mathcal{E}}(-w_0+\epsilon-c_1)) = 0.$$ Now observe that the characteristic function has at most three real roots, that are symmetric with respect to $\zeta_0$. Therefore, if $w_0$ is a root, then $w_0 = \zeta_0+\sqrt{\theta}$ and the other roots are $\zeta_0$ and $ \zeta_0-\sqrt{\theta}$. This implies that $\chi({\mathcal{E}}(w_0+1)) > 0$. On the other hand $$\chi({\mathcal{E}}(w_0+1)) =  - h^1({\mathcal{E}}(w_0+1)) \le 0,$$ a contradiction. So we must have $h^1({\mathcal{E}}(w_0))\ne0$, then by Lemma \[leftvanishing\] we obtain the thesis.

If ${\mathcal{E}}$ is *ACM*, then $\alpha<\frac{\epsilon+5-c_1}{2}$.

\[stable2\] Let ${\mathcal{E}}$ be a normalized rank 2 vector bundle on the threefold $X$ with $\vartheta\ge0$, then the following hold:

1)

:   $h^1({\mathcal{E}}(n))\ne 0$ for $\zeta_0< n < \zeta$.

2)

:   $h^1({\mathcal{E}}(n))\ne 0$ for $w_0\le n \le \bar\alpha-2$, and also for $n=\bar\alpha-1$ if $\zeta\notin{\mathbb{Z}}$.

3)

:   If $\zeta\in{\mathbb{Z}}$ and $\alpha<\bar\alpha$, then $h^1({\mathcal{E}}(\bar\alpha-1))\ne 0$.

**1)** The Hilbert polynomial of the bundle ${\mathcal{E}}$ is strictly negative for each integer such that $w_0\le n < \zeta$, but for such an integer $n$ we have $h^2({\mathcal{E}}(n))\ge0$ and $h^0({\mathcal{E}}(n))-h^0({\mathcal{E}}(-n+\epsilon-c_1)) \ge 0$ since $n\ge-n+\epsilon-c_1$ for every $n\ge w_0$, therefore we must have $h^1({\mathcal{E}}(n))\ne 0$.\
**2)** It is simply a restatement of 1) in term of $\bar\alpha$, which is, by definition, the integral part of $\zeta+1$.\
**3)** If $\zeta\in{\mathbb{Z}}$, then $\zeta=\bar\alpha-1$, so we have $\chi({\mathcal{E}}(\bar\alpha-1))=\chi({\mathcal{E}}(\zeta))=0$. Moreover $h^0({\mathcal{E}}(\bar\alpha-1))\ne0$ since $\alpha<\bar\alpha$, therefore $h^0({\mathcal{E}}(\bar\alpha-1))-h^3({\mathcal{E}}(\bar\alpha-1)) > 0$, and $h^1({\mathcal{E}}(n)) = 0$ implies $h^1({\mathcal{E}}(m)), \forall m \le n$; hence we must have $h^1({\mathcal{E}}(\bar\alpha-1))\ne 0$ to obtain the vanishing of $\chi({\mathcal{E}}(\bar\alpha-1))$.

If ${\mathcal{E}}$ is *ACM*, then $\vartheta<0$.

Observe that in this section we assume $\alpha \ge \frac{\epsilon-c_1+5}{2}$, in order to have $w_0 \le \alpha-2$ and so to have a non-empty range for $n$ in Theorem \[stable1\].

Observe that in the stable case we need not assume any vanishing of $h^1({\mathcal{O}}_X(n))$.

Observe that split bundles are excluded in this section because they cannot be stable.

Examples
========

We need the following

Let $X \subset {\mathbb{P}}^4$ be a smooth threefold of degree $d$ and let $f$ be the projection onto ${\mathbb{P}}^3$ from a general point of ${\mathbb{P}}^4$ not on $X$, and consider a normalized rank two vector bundle ${\mathcal{E}}$ on ${\mathbb{P}}^3$ which gives rise to the pull-back ${\mathcal{F}}= f^*({\mathcal{E}})$. We want to check that $f_\ast (\mathcal {O}_X) \cong \oplus _{i=0}^{d-1} \mathcal {O}_{\mathbb {P}^3}(-i)$.\
Since $f$ is flat and $\deg (f)=d$, $f_\ast (\mathcal {O}_X)$ is a rank $d$ vector bundle. The projection formula and the cohomology of the hypersurface $X$ shows that $f_\ast (\mathcal {O}_X)$ is ACM. Thus there are integers $a_0\ge \cdots \ge a_{d-1}$ such that $f_\ast (\mathcal {O}_X) \cong \oplus _{i=0}^{d-1} \mathcal {O}_{\mathbb {P}^3}(a_i)$. Since $h^0(X,\mathcal {O}_X)=1$, the projection formula gives $a_0 = 0$ and $a_i<0$ for all $i>0$. Since $h^0(X,\mathcal {O}_X(1)) = 5 = h^0(\mathbb {P}^3,\mathcal {O}_{\mathbb {P}^3}(1))+h^0(\mathbb {P}^3,\mathcal
{O}_{\mathbb {P}^3})$, the projection formula gives $a_1=-1$ and $a_i \le -2$ for all $i \ge 2$. Fix an integer $t \le d-2$ and assume proved $a_i = -i$ for all $i\le t$ and $a_i < -t$ for all $i>t$. Since $h^0(X,\mathcal {O}_X(t+1))= \binom{t+5}{4}
= 
\sum _{i=0}^{t} \binom{t+4-i}{3}$, we get $a_{t+1} = -t-1$ and, if $t+1 \le d-2$, $a_i<-t-1$ for all $i>t+1$. Since $f_\ast (\mathcal {O}_X) \cong \oplus _{i=0}^{d-1} \mathcal {O}_{\mathbb {P}^3}(-i)$, the projection formula gives the following formula for the first cohomology module: $$H^i({\mathcal{F}}(n)) = H^i({\mathcal{E}}(n)) \oplus H^i({\mathcal{E}}(n-1)) \oplus ...\oplus H^i({\mathcal{E}}(n-d+1))$$ all $i$. Observe that, as a consequence of the above equalitiy for $ i = 0$, we obtain that ${\mathcal{F}}$ has the same $\alpha$ as ${\mathcal{E}}$. Moreover the pull-back ${\mathcal{F}}= f^*({\mathcal{E}})$ and ${\mathcal{E}}$ have the same Chern class $c_1$, while $c_2({\mathcal{F}}) = dc_2({\mathcal{E}})$ and therefore $\delta({\mathcal{F}}) = d\delta({\mathcal{E}})$.

**Examples**

**1.** (a stable vector bundle with $c_1 = 0$, $c_2 = 4$ on a quadric hypersurface $X$).\
Choose $d = 2$ and take the pull-back ${\mathcal{F}}$ of the stable vector bundle ${\mathcal{E}}$ on ${\mathbb{P}}^3$ of [@VV], example 4.1. Then the numbers of ${\mathcal{F}}$ (see Notation) are: $c_1 = 0$, $c_2 = 4$, $\alpha = 1$, $\bar\alpha = 2$, $\zeta_0 = -\frac{3}{2}$, $w_0 = -1$, $\theta = \frac{25}{4}$, $\zeta = -\frac{3}{2}+\sqrt{\frac{25}{4}} = 1 \in {\mathbb{Z}}$. From [@VV], example 4.1, we know that $h^1({\mathcal{E}}) \ne 0$. Since $H^1({\mathcal{F}}(1)) = H^1({\mathcal{E}}(1)) \oplus H^1({\mathcal{E}})$, we have: $ h^1({\mathcal{F}}(1))\ne 0$, one shift higher than it is stated in Theorem \[stable2\], 2.\
**2.** (a non-stable vector bundle with $c_1 = 0$, $c_2 = 45$ on a hypersurface of degree $5$). Choose $d = 5$ and take the pull-back ${\mathcal{F}}$ of the stable vector bundle ${\mathcal{E}}$ on ${\mathbb{P}}^3$ of [@VV], example 4.5. Then the numbers of ${\mathcal{F}}$ (see Notation) are: $c_1 = 0$, $c_2 = 45$, $\alpha = -3$, $\delta = 90$, $\zeta_0 = 0$. From [@VV], theorem 3.8, we know that $h^1({\mathcal{E}}(12)) \ne 0$. Since $H^1({\mathcal{F}}(16)) = H^1({\mathcal{E}}(16)) \oplus \dots \oplus H^1({\mathcal{E}}(12))$, we have: $ h^1({\mathcal{F}}(16))\ne 0$ (Theorem \[nonstable2\] states that $h^1({\mathcal{F}}(10) \ne 0$).\
**3.** (a stable vector bundle with $c_1 = -1$, $c_2 = 2$ on a quadric hypersurface).\
Let ${\mathcal{E}}$ be the rank two vector bundle corresponding to the union of two skew lines on a smooth quadric hypersurface $Q \subset {\mathbb{P}}^4$. Then its numbers are : $c_1 = -1$, $c_2 = 2$, $\alpha = 1$ and it is known that $h^1({\mathcal{E}}(n)) \ne 0$ if and only if $n = 0$.\
Observe that in this case $\theta = \frac{5}{2} \ge 0, \zeta_0 = -1$, $\bar\alpha = 1$. Therefore theorem \[stable2\] states exactly that $h^1({\mathcal{E}}(0)) \ne 0$, hence this example is sharp.\
**4.** (a non-stable vector bundle with $c_1 = 0$, $c_2 = 8$ on a quadric hypersurface).\
Choose $d = 2$ and take the pull-back ${\mathcal{F}}$ of the non-stable vector bundle ${\mathcal{E}}$ on ${\mathbb{P}}^3$ of [@VV], example 4.10. Then the numbers of ${\mathcal{F}}$ (see Notation) are: $c_1 = 0$, $c_2 = 8$, $\alpha = 0$, $\zeta_0 = -\frac{3}{2}$, $\delta = 8$. We know (see [@VV], example 4.10) that $h^1({\mathcal{E}}(2)) \ne 0, h^1({\mathcal{E}}(3)) = 0$. Since $H^1({\mathcal{F}}(3)) = H^1({\mathcal{E}}(3)) \oplus H^1({\mathcal{E}}(2))$, we have: $ h^1({\mathcal{F}}(3)) \ne 0$, exactly the bound of Theorem \[nonstable2\].

The bounds for a degree $d$ threefold in ${\mathbb{P}}^4$ agree with [@VV], where ${\mathbb{P}}^3$ is considered.

Threefolds with ${\mathrm{Pic}}(X) \ne {\mathbb{Z}}$
====================================================

Let $X$ be a smooth and connected projective threefold defined over an algebraically closed field $\mathbb {K}$. Let ${\mathrm{Num}}(X)$ denote the quotient of ${\mathrm{Pic}}(X)$ by numerical equivalence. Numerical classes are denoted by square brackets $[\,\,]$. We assume ${\mathrm{Num}}(X) \cong \mathbb {Z}$ and take the unique isomorphism $\eta : {\mathrm{Num}}(X)\to \mathbb {Z}$ such that $1$ is the image of a fixed ample line bundle. Notice that $M\in {\mathrm{Pic}}(X)$ is ample if and only if $\eta ([M])>0$.

\[boundedness\] Let $\eta : {\mathrm{Num}}(X) \to \mathbb {Z}$ be as before. Notice that every effective divisor on $X$ is ample and hence its $\eta$ is strictly positive. For any $t\in \mathbb {Z}$ set ${\mathrm{Pic}}_t(X):= \{L\in {\mathrm{Pic}}(X): \eta ([L])=t\}$. Hence ${\mathrm{Pic}}_0(X)$ is the set of all isomorphism classes of numerically trivial line bundles on $X$. The set ${\mathrm{Pic}}_0(X)$ is parametrized by a scheme of finite type ([@La], Proposition 1.4.37). Hence for each $t\in \mathbb {Z}$ the set ${\mathrm{Pic}}_t(X)$ is bounded. Let now ${\mathcal{E}}$ be a rank $2$ vector bundle on $X$. Since ${\mathrm{Pic}}_1(X)$ is bounded there is a minimal integer $t$ such that there is $B\in {\mathrm{Pic}}_t(X)$ and $h^0(E\otimes B) >0$. Call it $\alpha (E)$ or just $\alpha$. By the definition of $\alpha$ there is $B\in {\mathrm{Pic}}_{\alpha}(X)$ such that $h^0(X,{\mathcal{E}}\otimes B) >0$. Hence there is a non-zero map $j: B^\ast \to E$. Since $B^\ast$ is a line bundle and $j\ne 0$, $j$ is injective. The definition of $\alpha$ gives the non-existence of a non-zero effective divisor $D$ such that $j$ factors through an inclusion $B^\ast \to B^\ast (D)$, because $\eta ([D]) >0$. Thus the inclusion $j$ induces an exact sequence $$\label{eqa1}
0 \to B^\ast \to {\mathcal{E}}\to \mathcal {I}_Z\otimes B \otimes \det ({\mathcal{E}}) \to 0$$ in which $Z$ is a closed subscheme of $X$ with pure codimension $2$.\
Observe that $\eta([B]) = \alpha, \eta([B^*]) = -\alpha, \eta([B \otimes det({\mathcal{E}})]) =  \alpha +c_1$, hence the exact sequence is quite similar to the usual exact sequence that holds true in the case ${\mathrm{Pic}}(X) \cong {\mathbb{Z}}$.

NOTATION:\
We set $\epsilon:= \eta ([\omega _X])$, $\alpha := \alpha ({\mathcal{E}})$ and $c_1:= \eta ([\det ({\mathcal{E}})])$. So we can speak of a normalized vector bundle ${\mathcal{E}}$, with $c_1 \in \{ 0,-1\}$. Moreover we say that ${\mathcal{E}}$ is stable if $\alpha > 0$, nonstable if $\alpha \le 0$. Moreover ª$\zeta_0, \zeta, w_0, \bar \alpha, \theta$ are defined as in section 2.

Fix any $L \in {\mathrm{Pic}}_1(X)$ and set: $d = L^3 = $ degree of $X$.The degree $d$ does not depend on the numerical equivalence class. In fact, if $R$ is numerically equivalent to $0$, then $(L+R)^3 = L^3+R^3+3L^2R+3LR^2 = L^3+0+0+0 = L^3$. Then it is easy to see that the formulas for $\chi({\mathcal{O}}_X(n))$ and $\chi({\mathcal{E}}(n))$ given in section 2 still hold if we consider ${\mathcal{O}}_X \otimes L^{\otimes n}$ and ${\mathcal{E}}\otimes L^{\otimes n}$ (see [@Valenzano]).

\[a1\] (a) Assume the existence of $L\in {\mathrm{Pic}}(X)$ such that $\eta ([L]) =1$ and $h^0(X,L)>0$. Then for every integer $t>\alpha $ there is $M\in {\mathrm{Pic}}(X)$ such that $\eta ([M])=t$ and $h^0(X,E\otimes M)>0$.

\(b) Assume $h^0(X,L)>0$ for every $L\in {\mathrm{Pic}}(X)$ such that $\eta ([L]) =1$. Then $h^0(X,E\otimes M)>0$ for every $M\in {\mathrm{Pic}}(X)$ such that $\eta ([M]) > \alpha$.

\[a2\] Let ${\mathcal{E}}$ be a normalized rank two vector bundle and assume the existence of a spanned $R\in {\mathrm{Pic}}(X)$ such that $\eta ([R])=1$. If char $K > 0$, assume that $\vert R \vert$ induces an embedding of $X$ outside finitely many points. Assume $$\label{eqb1}
2\alpha \le -\epsilon-3-c_1$$ and $h^1(X,{\mathcal{E}}\otimes N)=0$ for every $N\in {\mathrm{Pic}}(X)$ such that $\eta ([N]) \in \{-\alpha -c_1-1,\alpha +2+e\}$. If $h^1(X,B)=0$ for every $B\in {\mathrm{Pic}}(X)$ such that $\eta
([B])=-2\alpha -c_1$, then ${\mathcal{E}}$ splits.

If moreover $h^1(X,M)=0$ for every $M\in {\mathrm{Pic}}(X)$ then it is enough to assume that $h^1(X,{\mathcal{E}}\otimes N)=0$ for every $N\in {\mathrm{Pic}}(X)$ such that $\eta ([N]) = -\alpha -c_1-1$.

By assumption there is $M\in {\mathrm{Pic}}(X)$ such that $\eta ([M]) = \alpha$ and $h^0(X,{\mathcal{E}}\otimes M)>0$. Set $A:= M^\ast$. We have seen in remark \[boundedness\] that ${\mathcal{E}}$ fits into an extension of the following type: $$\label{b2}
0 \to A \to {\mathcal{E}}\to \mathcal {I}_C\otimes \det ({\mathcal{E}})\otimes A^\ast \to 0$$ with $C$ a locally complete intersection closed subscheme with pure dimension $1$. Let $H$ be a general element of $\vert R\vert$ and $T$ the intersection of $H$ with another general element of $\vert R\vert$. Observe that $T$, under our assumptions, is generically reduced by Bertini’s theorem-see [@HAL], Theorem II, 8.18 and Remark II, 8.18.1. Since $R$ is spanned, $T$ is a locally complete intersection curve and $C\cap T=\emptyset$. Hence ${\mathcal{E}}\vert T$ is an extension of $\det ({\mathcal{E}})\otimes A^\ast \vert T$ by $A\vert T$. Since T is generically reduced and locally a complete intersection, it is reduced. Hence $h^0(T,M^\ast ) = 0$ for every ample line bundle $M$ on $T$. Since $\omega _T \cong (\omega _X\otimes R^{\otimes 2})\vert T$, we have $\dim (\mbox{Ext}^1(T,\det ({\mathcal{E}})\otimes A^\ast,A)
=h^0(T,\det ({\mathcal{E}})\otimes (A^\ast)^{\otimes 2} \otimes \omega _\otimes R^{\otimes 2}))\vert T)= 0$ (indeed $\eta ([\det ({\mathcal{E}})\otimes (A^\ast )^{\otimes 2}\otimes \omega _X\otimes R^{\otimes 2}])
= 2\alpha +c_1+e+2 <0$). Hence ${\mathcal{E}}\vert T \cong A\vert T\oplus (\det ({\mathcal{E}})\otimes A^\ast )\vert T$. Let $\sigma$ be the non-zero section of $({\mathcal{E}}\otimes (A\otimes \det ({\mathcal{E}})^\ast )\vert T$ coming from the projection onto the second factor of the decomposition just given. The vector bundle ${\mathcal{E}}\vert H$ is an extension of $\det ({\mathcal{E}})\otimes A^\ast \vert H$ by $A\vert H$ if and only if $C\cap H =\emptyset$. Since $R$ is ample, $C\cap H = \emptyset$ if and only if $C=\emptyset$. Hence we get simultaneously $C\cap H=\emptyset$ and ${\mathcal{E}}\vert H \cong A\vert H\oplus \det ({\mathcal{E}})\otimes A^\ast \vert H$ if we prove the existence of $\tau \in H^0(H,({\mathcal{E}}\otimes (A\otimes \det ({\mathcal{E}})^\ast )\vert H)$ such that $\tau \vert T = \sigma$. To get $\tau$ it is sufficient to have $H^1(H,(E\otimes (A\otimes \det ({\mathcal{E}})^\ast \otimes R^\ast )\vert H) =0$. A standard exact sequence shows that $H^1(H,({\mathcal{E}}\otimes (A\otimes \det ({\mathcal{E}})^\ast \otimes R^\ast )\vert H) =0$ if $h^1(X,({\mathcal{E}}\otimes (A\otimes \det ({\mathcal{E}})^\ast \otimes R^\ast ) =0$ and $h^2(X,({\mathcal{E}}\otimes (A\otimes \det ({\mathcal{E}})^\ast \otimes R^\ast
\otimes R^\ast) =0$. Since ${\mathcal{E}}^\ast \cong {\mathcal{E}}\otimes \det ({\mathcal{E}})^\ast$, Serre duality gives $h^2(X,(E\otimes (A\otimes \det ({\mathcal{E}})^\ast \otimes R^\ast
\otimes R^\ast) =h^1(X,{\mathcal{E}}\otimes A\otimes R^{\otimes 2}\otimes \omega _X)$. Since $\eta ([A\otimes \det ({\mathcal{E}})^\ast \otimes R^\ast ])=-\alpha -c_1-1$ and $\eta ([A\otimes R^{\otimes 2}\otimes \omega _X] )= \alpha +e+2$, we get that $C=\emptyset$. The last sentence follows because $\eta ([A^{\otimes 2}\otimes \det ({\mathcal{E}})^\ast] )= -2\alpha -c_1$.

\[a3\] Instead of the smoothness of $X$ we may assume that $X$ is locally algebraic factorial, i.e. that all local rings $\mathcal {O}_{X,P}$ are factorial. This assumption seems to be essential, because without it a non zero section of $E\otimes M$ with $\eta ([M]) = \alpha (E)$ could vanish on an effective Weil divisor and hence we could not claim the existence of the exact sequence (\[b2\]).

\[a4\] Fix integers $t < z \le \alpha -2$. Assume the existence of $L \in {\mathrm{Pic}}(X)$ such that $\eta ([L])=z$ and $h^1(X,E\otimes L)=0$. If there is $R\in {\mathrm{Pic}}(X)$ such that $\eta ([R])=1$ and $h^0(X,R)>0$, then there exists $M\in {\mathrm{Pic}}(X)$ such that $\eta ([M]) =t$ and $h^1(X,E\otimes M)=0$. If $h^0(X,R)>0$ for every $R\in {\mathrm{Pic}}(X)$ such that $\eta ([R])=1$, then $h^1(X,E\otimes M)=0$ for every $M\in
{\mathrm{Pic}}(X)$ such that $\eta ([M]) =t$.

The proof can follow the lines of Lemma \[leftvanishing\]. In fact consider a line bundle $R$ with $\eta([R]) = 1$ and let $H$ be the zero-locus of a non-zero section of $R$; then we have the following exact sequence: $$0 \to {\mathcal{E}}\otimes L \to {\mathcal{E}}\otimes L \otimes R \to ({\mathcal{E}}\otimes L\otimes R)_H \to 0.$$ Now observe that the vanishing of $h^1(X,{\mathcal{E}}\otimes L)$ implies that $h^0({\mathcal{E}}\otimes L \otimes R)_H = 0$. And now we can argue as in Lemma \[leftvanishing\] (see also [@VV]).

\[a6\] (a) Assume the existence of $L\in {\mathrm{Pic}}(X)$ such that $\eta ([L]) =1$ and $h^0(X,L)>0$. Then for every integer $t>\alpha $ there is $M\in {\mathrm{Pic}}(X)$ such that $\eta ([M])=t$ and $h^0(X,E\otimes M)>0$.

\(b) Assume $h^0(X,L)>0$ for every $L\in {\mathrm{Pic}}(X)$ such that $\eta ([L]) =1$. Then $h^0(X,E\otimes M)>0$ for every $M\in {\mathrm{Pic}}(X)$ such that $\eta ([M]) > \alpha$.

In all our results of sections 4 and 5 we use the vanishing of $h^1({\mathcal{O}}_X(n))$ (and by Serre duality of $h^2({\mathcal{O}}_X(n))$), $\forall n$ (or, at least, $\forall n\notin \{0,\cdots,\epsilon\}$), see Remark 4.12.

From now on we need to use similar vanishing conditions and so we introduce the following condition:

$\mathbf{(C4)}$ $h^1(X,L) =0$ for all ${\mathrm{Pic}}(X)$ such that either $\eta ([L]) <0$ or $\eta ([L]) >\epsilon$.\
Observe that $\mathbf{(C4)}$ is always satisfied in characteristic 0 (by the Kodaira vanishing theorem). In positive characteristic it is often satisfied. This is always the case if $X$ is an abelian variety ([@Mumford] p. 150).\
Observe also that, if $\epsilon \le -1$, the Kodaira vanishing and our condition put no restriction on $n$ (see also Remark 4.12).

**Example**. If (\[eqb1\]) holds, then $-2\alpha -c_1 > \epsilon$. Hence we may apply Proposition \[a2\] to $X$. In particular observe that, in the case of an abelian variety with ${\mathrm{Num}}(X) \cong \mathbb {Z}$ or in the case of a Calabi-Yau threefold with ${\mathrm{Num}}(X) \cong \mathbb {Z}$, we have $\epsilon = 0$. Notice that Proposition \[a2\] also applies to any threefold $X$ whose $\omega _X$ is a torsion sheaf.

With the assumption of condition $\mathbf{(C4)}$ the proofs of Theorems \[nonstable1\], \[nonstable2\], \[nonstable3\] can be easily modified in order to obtain the statements below (${\mathcal{E}}$ is normalized, i.e. $\eta ([\det ({\mathcal{E}})]) \in \{-1,0\}$), where, by the sake of simplicity, we assume $\epsilon \ge 0$ (if $\epsilon < 0$, $\mathbf{(C4)}$, which holds by [@SB], implies that all the vanishing of $h^1$ and $h^2$ for all $L\in {\mathrm{Pic}}(X)$ hold).

\[v1\] Assume $\mathbf{(C4)}$, $\alpha \le 0$, the existence of $R\in {\mathrm{Pic}}(X)$ such that $\eta ([R])=1$ and $\zeta _0 < -\alpha -c_1-1$. Fix an integer $n$ such that $\zeta _0 < n \le -\alpha - 1 -c_1$. Fix $L\in {\mathrm{Pic}}(X)$ such that $\eta ([L]) = n$. Then $h^1({\mathcal{E}}\otimes L) \ge (n-\zeta _0)\delta >0$.

Observe that we should require the following conditions: $n-\alpha \notin \{0,\dots ,\epsilon \}, \epsilon-n+\alpha  \notin \{0,\dots ,\epsilon \}$. But they are automatically fulfiled under the assumption that $\zeta_0 < -\alpha-c_1-1$.

\[v2\] Assume $\mathbf{(C4)}$, $\alpha \le 0$, the existence of $R\in {\mathrm{Pic}}(X)$ such that $\eta ([R])=1$ and the same hypotheses of Theorem \[nonstable2\]. Fix $L\in {\mathrm{Pic}}(X)$ such that $\eta ([L]) = n$. Then $h^1({\mathcal{E}}\otimes L) \ge
-S(n) >0$ ($S(n)$ being defined as in Theorem \[nonstable2\]).

\[v3\] Assumption as in Theorem \[nonstable3\]. Moreover assume $\mathbf{(C4)}$ and $n -\alpha \notin \{0,\dots ,\epsilon \}$. Fix $L\in {\mathrm{Pic}}(X)$ such that $\eta ([L]) = n$. Then $h^1({\mathcal{E}}\otimes L) \ge
-\frac{d}{6}F(n+\alpha -\zeta _0)>0$ ($F$ being defined as in Theorem \[nonstable3\]).

Observe that in Theorems \[v2\] and \[v3\] we should require $n -\alpha \notin \{0,\dots ,\epsilon \}$, but the assumption $\epsilon-\alpha-c_1+1 \le n$ implies that it is automatically fulfilled. Observe that in Theorems \[v2\] and \[v3\] we require $n -\alpha \notin \{0,\dots ,\epsilon \}$, but the assumption $\epsilon-\alpha-c_1+1 \le n$ implies that the requirement is automatically fulfilled.

The proofs of the above theorems are based on the existence of the exact sequence (\[eqa1\]) and on the properties of $\alpha$. They follow the lines of the proofs given in the case ${\mathrm{Pic}}(X) \cong {\mathbb{Z}}$. Here and in section 4 we actually need only the Kodaira vanishing (true in characteristic 0 and assumed in characteristic $p > 0$) and no further vanishing of the first cohomology.

Also the stable case can be extended to a smooth threefold with ${\mathrm{Num}}(X) \cong {\mathbb{Z}}$. Observe that the proofs can follow the lines of the proofs given in the case ${\mathrm{Pic}}(X) \cong {\mathbb{Z}}$ and make use of Remark 7.6 (which extends \[leftvanishing\]).

More precisely we have:

\[v4\] Assumptions as in \[stable1\] and fix $L\in {\mathrm{Pic}}(X)$ such that $\eta ([L]) = n$. Then, if $\alpha\ge\frac{\epsilon+5-c_1}{2}$, then $h^1({\mathcal{E}}\otimes L)\ne 0$ for $w_0\le n\le \alpha-2$.

\[v4\] Assumptions as in \[stable2\] and fix $L\in {\mathrm{Pic}}(X)$ such that $\eta ([L]) = n$. Then the following hold:

1)

:   $h^1({\mathcal{E}}\otimes L)\ne 0$ for $\zeta_0< n < \zeta$.

2)

:   $h^1({\mathcal{E}}\otimes L)\ne 0$ for $w_0\le n \le \bar\alpha-2$, and also for $n=\bar\alpha-1$ if $\zeta\notin{\mathbb{Z}}$.

3)

:   If $\zeta\in{\mathbb{Z}}$ and $\alpha<\bar\alpha$, then $h^1({\mathcal{E}}\otimes N) \ne 0$, for every $N$ such that $\eta([N]) = \bar\alpha-1$.

\[a5\] The above theorems can be applied to any $X$ such that ${\mathrm{Num}}(X)$ $\cong \mathbb {Z}$, $\epsilon =0$ and $h^1(X,L)=0$ for all $L\in {\mathrm{Pic}}(X)$ such that $\eta ([L]) \ne 0$, for instance to $X = $ an abelian threefold with ${\mathrm{Num}}(X) \cong \mathbb {Z}$.

If $X$ is any threefold (in characteristic $0$ or positive) such that $h^1(X,L) = 0, \forall L \in {\mathrm{Pic}}(X)$, then we can avoid the restriction $n-\alpha \notin \{0,...,\epsilon\}$. Not many threefolds, beside any $X \subset {\mathbb{P}}^4$, fulfil these conditions.

Observe that in Theorem \[v4\] we do not assume $\mathbf{(C4)}$ (see also remark 5.8)

Observe that also in the present case (${\mathrm{Num}}(X) \cong {\mathbb{Z}}$), we have: $\delta = 0$ if and only if ${\mathcal{E}}$ splits. Therefore Remarks 4.13 and 5.9 apply here.

[00]{} L. Chiantini, P. Valabrega, Subcanonical curves and complete intersections in projective 3-space, Ann. Mat. Pura Appl. **138** (4) (1984) 309–330. L. Chiantini, P. Valabrega, On some properties of subcanonical curves and unstable bundles, Comm. Algebra **15** (1987) 1877–1887. PH. Ellia, Sur la cohomologie de certains fibr' es de rang deux sur ${\mathbb{P}}^3$, Ann. Univ. Ferrara **38** (1992) 217–227. G. Gherardelli, Sulle curve sghembe algebriche intersezioni complete di due superficie, Atti Reale Accademia d’Italia **4** (1943) 128–132. R. Hartshorne, Algebraic Geometry, GTM 52, Springer Verlag, New York, 1977. R. Hartshorne, Stable vector bundles of rank 2 on ${\mathbb{P}}^3$, Math. Ann. **238** (1978) 229–280. R. Hartshorne, Stable reflexive sheaves, Math. Ann. **254** (1980) 121–176. R. Hartshorne, Stable Reflexive Sheaves II, Invent. Math. **66** (1982) 165–190. R. Lazarsfeld, Positivity in Algebraic Geometry I, Springer, Berlin, 2004. C. Madonna, A Splitting Criterion for Rank 2 Vector Bundles on Hypersurfaces in ${\mathbb{P}}^4$, Rend. Sem. Mat. Univ. Pol. Torino **56** (2) (1998) 43–54. D. Mumford, Abelian Varieties, Oxford University Press, 1974. S. Popescu, On the splitting criterion of Chiantini and Valabrega, Rev. Roumaine Math. Pures Appl. **33** (10) (1988) 883–887. M. Roggero, P. Valabrega, Some vanishing properties of the intermediate cohomology of a reflexive sheaf on ${\mathbb{P}}^n$, J. Algebra **170** (1) (1994) 307–321. N. Shepherd-Barron, Fano Threefolds in positive characteristic, Compositio Math. **105** (1997) 237–265. P. Valabrega, M. Valenzano, Non-vanishing theorems for non-split rank $2$ bundles on ${\mathbb{P}}^3$: a simple approach, Atti Acc. Peloritana **87** (2009) 1–18 \[DOI:10.1478/C1A0901002\]. M. Valenzano, Rank $2$ reflexive sheaves on a smooth threefold, Rend. Sem. Mat. Univ. Pol. Torino **62** (2004) 235–254.

<span style="font-variant:small-caps;">Edoardo BALLICO</span>\
Dipartimento di Matematica, Università di Trento\
via Sommarive 14, 38050 Povo (TN), Italy\
e-mail: `ballico@science.unitn.it`\
<span style="font-variant:small-caps;">Paolo VALABREGA</span>\
Dipartimento di Matematica, Politecnico di Torino\
Corso Duca degli Abruzzi 24, 10129 Torino, Italy\
e-mail: `paolo.valabrega@polito.it`\
<span style="font-variant:small-caps;">Mario VALENZANO</span>\
Dipartimento di Matematica, Università di Torino\
via Carlo Alberto 10, 10123, Torino, Italy\
e-mail: `mario.valenzano@unito.it`

[^1]: The paper was written while all authors were supported by MIUR (PRIN grant) and by local funds of their Universities, and were members of INdAM-GNSAGA.
