---
abstract: 'Here we study vector bundles $E$ on the Hirzebruch surface $F_e$ such that their twists by a spanned, but not ample, line bundle $M = \mathcal {O}_{F_e}(h+ef)$ have natural cohomology, i.e. $h^0(F_e,E(tM)) >0$ implies $h^1(F_e,E(tM)) = 0$.'
address:
- |
    Dept. of Mathematics\
    University of Trento\
    38050 Povo (TN), Italy
- |
    Dip. Matematica, Università di Torino\
    via Carlo Alberto 10, 10123 Torino, Italy
author:
- 'E. Ballico'
- 'F. Malaspina'
title: 'Vector bundles on Hirzebruch surfaces whose twists by a non-ample line bundle have natural cohomology'
---

[^1]

Introduction {#S1}
============

Let $F_e$, $e>0$, denote the Hirzebruch surface with a section with self-intersection $-e$. For any $L\in \mbox{Pic}(F_e)$ and any vector bundle $E$ on $F_e$ we will say that $E$ has property ££  (resp. £ ) with respect to $L$ if $h^1(F_e,E\otimes L^{\otimes m})=0$ for all $m \in \mathbb 
{Z}$ (resp. for all $m\in \mathbb {Z}$ such that $h^0(F_e,E\otimes L^{\otimes m}) \ne 0$). We think that property £  is nicer for reasonable $L$. We take as a basis of $\mbox{Pic}(F_e) \cong \mathbb 
{Z}^2$ a fiber $f$ of the ruling $\pi : F_e \to {\bf {P}}^1$ and the section $h$ of $\pi$ with negative self-intersection. Thus $h^2 = -e$, $h\cdot f = 1$ and $f^2 = 0$. We have $\omega _{F_e} 
\cong \mathcal {O}_{F_e}(-2h-(e+2)f)$. $\mathcal {O}_{F_e}(\alpha h+\beta f)$ is spanned (resp. ample) if and only if $\alpha \ge 0$ and $\beta \ge \alpha e$ (resp. $\alpha > 0$ and $\beta > e\alpha$). The Leray spectral sequence of $\pi$ and Serre duality give that $h^1(F_e,\mathcal {O}_{F_e}(\gamma h+\delta f))=0$ if and only if either $\gamma \ge 0$ and $\delta \ge e\gamma -1$ or $\gamma =-1$ or $\gamma \le -2$ and $-\delta - e-2 \ge e(-\gamma -2)-1$ (i.e. $\delta \le e\gamma +e-1$). We consider as the test line bundle the spanned, but not ample, line bundle $M:= \mathcal {O}_{F_e}(h+ef)$. Notice that the linear system $\vert 
M^{\otimes 2}\vert$ contains the sum of the effective divisor $h$ and the ample divisor $h+2ef$. Thus for every vector bundle $E$ on $F_e$ there is an integer $m_0(E)$ such that $h^0(F_e,E\otimes M^{\otimes m}) \ne 0$ for all $m \ge 
m_0(E)$. We will see that property £ £  is too strong and not interesting (see Remarks \[a0\] and \[a1\]). We stress the property £  with respect to $M$ is quite different from similar looking properties (e.g. natural cohomology) with respect to an ample line bundle.(see Remarks \[a1\] and \[a2\] for the rank $1$ case). Obviously, properties £  and £ £  may be stated for arbitrary projective varieties. In dimension $n \ge 
3$, one need to choose between vanishing of $h^1$ or vanishing of all $h^i$, $1 \le i \le n-1$. We considered here the example $(F_e,M)$, because it is geometrically significant. Indeed, let $\phi _M$ denote the morphism associated to the base point free linear system $\vert M\vert$. If $e=1$ the morphism $\phi _M$ is the blowing up $F_1 \to {\bf {P}}^2$. If $e\ge 2$, then $\phi _M : F_e \to {\bf {P}}^{e+1}$ contracts $h$ and its image is a cone over the rational normal curve of ${\bf {P}}^e$. Moreover, for any spanned and non-trivial line bundle $L$ on $F_e$ there is an effective divisor $D$ such that $L \cong M(D)$. For any spanned, but not ample line bundle $A$ on $F_e$ there is an integer $c \ge 0$ such that $A \cong M^{\otimes c}$. We prove the following results.

\[o4\] Fix integers $e \ge 1$, $r \ge 1$, $u, v$ such that $v \le e(u-r+1)-2$. Then there is no rank $r$ vector bundle $E$ on $F_e$ with property £  with respect to $M$ such that $c_1(E) = \mathcal {O}_{F_e}(uh+vf)$.

\[o2\] Fix integers $e, m, u, v$ such that $e\ge 1$, and $v \ge e(u-1)-1$ and $m \ge 0$. Set $\widetilde{a}:=
\sum _{i=0}^{u+2m-2} v+2m-1-ie$ and $\widetilde{b}:= \sum _{i=0}^{u+2m-1} v+2m -ie$. Fix any integer $s$ such that $\widetilde{a}
\le s \le \widetilde{b}$. Then there exists a rank $2$ vector bundle $E$ on $F_e$ with property £  with respect to $M$ such that $c_1(E) = \mathcal {O}_{F_e}(uh+vf)$ and $c_2(E):=s -e(u+m-1) + (1-m)(v+em)$. Set $R:= \mathcal 
{O}_{F_e}(h+(e+1)f)$ and assume $m=0$, $u \ge 3$, and $v < 2eu$. Then we may find $E$ as above which is $R$-stable in the sense of Mumford-Takemoto and (under the additional condition $v \le 2eu -3$) such that $N\cdot M < c_1(E)\cdot M/2$ for all rank $1$ subsheaves $N$ of $E$.

The case “ $r=1$ ” of Theorem \[o4\] is obviosly true (use the cohomology of line bundles on $F_e$, i.e. Remark \[a1\] below). In this case the converse is true, i.e. $\mathcal 
{O}_{F_e}(uh+vf)$ has property £ with respect to $M$ if and only if $v \ge eu-1$ (Remark \[a1\]). We were surprised thar for $r \ge 2$ there is no way to overcome this $c_1$-obstruction.

The assumptions of the last part of Theorem \[o2\] may be relaxed and instead of $R$ we may take an arbitrary ample divisor $H$. An interesting offshot of our proof of Theorem \[o2\] is that our examples are given by an extension (\[eqo9\]) and all locally free sheaves fitting in (\[eqo9\]) have property £  with respect to $M$ and (under the additional conditions listed in Theorem \[o2\]) are $R$-stable and $N\cdot M < c_1(E)\cdot M/2$ for all rank $1$ subsheaves $N$ of $E$.

In the case of direct sums of line bundles we will prove the following result.

\[o3\] Fix integers $e \ge 1$, $r \ge 2$ and $L_i\in \mbox{Pic}(F_e)$, $1 \le 
i \le r$, say $L_i \cong
\mathcal {O}_{F_e}(u_ih+v_if)$. Set $E:= L_1\oplus \cdots \oplus L_r$. Up to a permutation of the factors of $E$ we may assume $u_1 \ge \cdots \ge u_r$ and that if $u_i = u_j$ for some $i<j$, then $v_i \ge v_j$. Set $m:= -u_1$ if $v_1 \ge eu_1$ and $m:= -u_1+1$ if $v_1 = eu_1-1$. The vector bundle $E$ has property £  with respect to $M$ if and only if $v_i \ge 
eu_i-1$ for all $i$, and for each $i \in \{2,\dots ,r\}$ either $u_i-m \ge -1$ or $-1 \le v_i-eu_i \le e-1$.

We raise the following question.

\[o3.1\] Assume $e=1$ or $e=2$. Is it possible to describe all invariants $r, 
c_1, c_2$ of vector bundles on $F_e$ with property £ with respect to $M$?

The proofs {#S2}
==========

For any sheaf $F$ we will often write $F(mM)$ instead of $F\otimes 
M^{\otimes m}$.

\[a0\] The line bundle $\mathcal {O}_{F_e}(ch+df)$ is ample if and only if $c>0$ and $d > ec$. Hence any ample line bundle is spanned. Assume that $H:= \mathcal {O}_{F_e}(ch+df)$ is ample. The cohomology of line bundles on $F_e$ shows that for every $t\in \mathbb {Z}$ the line bundle $H^{\otimes t}$ has property £ £  with respect to $H$. Hence $\mathcal {O}_{F_e}$ has property £ £  with respect to any ample line bundle. Set $H':= \mathcal {O}_{F_e}(ch+(d+2)f)$. Taking $m_t:= -tc$ we see that no $H^{\otimes t}$, $t > 0$, has property £ £  with respect to the ample line bundle $H'$. Taking $m_t:= -tc$ we see that no $H^{\otimes t}$, $t 
> 0$, has property £ £  with respect to $M$.

\[a1\] Here we study properties £  and £ £  with respect to $M$ for line bundles on $F_e$. Fix $L\in \mbox{Pic}(F_e)$, say $L \cong \mathcal {O}_{F_e}(u h + vf)$. First assume $v \ge eu$. We have $h^0(F_e,L(xM)) >0$ if and only if $x \ge -u$. Since $h^1(F_e,\mathcal {O}_{F_e}(ch+df)) = 0$ if $c \ge 0$ and $d \ge ec$, $L$ has property £  with respect to $M$. Now assume $v < eu$. We have $h^0(F_e,L(xM)) > 0$ if and only if $ex 
\ge -v$. Since $h^1(F_e,\mathcal {O}_{F_e}(ch+df)) = 0$ if $c \ge 0$ if and only if $d \ge ec-1$, we get that $L$ has property £  with respect to $M$ if and only if $v \ge eu-1$. Take $m:= -u$. If $v=eu$, then we saw in the introduction that $L$ has property £ £  with respect to $M$ if and only if $e=1$. Notice that $h^1(F_e,\mathcal {O}_{F_e}((u-x)h + (v-ex)f)) = h^1(F_e,\mathcal 
{O}_{F_e}((x-u-2)h+(ex-v-e-2)) >0$ when $x \ge -u-2$ if and only if $-eu -2e \le -v-e-1$, i.e. if and only if $v \le eu+e-1$. Notice that $h^1(F_e,\mathcal {O}_{F_e}((u+x)h + (v+ex)f)) =0$ for $x \ge -u $ if and only if $v \ge eu-1$. Notice that $h^1(F_e,\mathcal {O}_{F_e}(-h + (v-eu-e)f)) = 0$ for every $v\in 
\mathbb {Z}$. Hence $L$ has property £ £  with respect to $M$ if and only if $eu - 1 \le v \le eu+e-1$.

\[a2\] Here we look at property £  with respect to the ample line bundle $R:= \mathcal {O}_{F_e}(h+(e+1)f)$ for line bundles on $F_e$. Fix $L\in \mbox{Pic}(F_e)$, say $L \cong \mathcal {O}_{F_e}(u h + vf)$. We heve $h^0(F_e,L(xR)) = 0$ if and only if $x \ge -u$ and $x(e+1) \ge -v$. We immediately see that if $v \ge (e+1)u$, then $L$ has property £  with respect to $M$. Now assume $v < (e+1)u$. Set $y:= \lceil -v/(e+1)\rceil$. We have $h^0(F_e,L(xR)) >0$ if and only if $x \ge y$. Fix an integer $x 
\ge y$. Since $u+x \ge u+y \ge 0$, $h^1(F_e,\mathcal {O}_{F_e}(u+x)h+(v+(e+1)x)f)) >0$ if and only if $v+(e+1)x \le eu+ ex-2$. The strongest condition is obtained when $x=y$. We get that $L$ has property $\alpha $ with respect to $R$ if and only if either $v \ge (e+1)u$ or $v+ey \ge eu-1$, where $y:= 
\lceil -v/(e+1)\rceil$.

\[a3\] $E_1\oplus E_2$ has property £ £  with respect to $L$ if and only if both $E_1$ and $E_2$ have property £ £  with respect to $L$. If $E_1\oplus E_2$ has property £  with respect to $L$, then the same is true for $E_1$ and $E_2$. Now we check that the converse is not true. Both $\mathcal {O}_{F_e}$ and $\mathcal {O}_{F_e}(-2h+(-e+4)f)$ have property £  with respect to $M$ (Remark \[a1\]). Since $h^1(F_e,\mathcal {O}_{F_e}(-2h+(-e+4)f)) = 
h^1(F_e,\mathcal {O}_{F_e}(-2f)) = 1$, $\mathcal {O}_{F_e}\oplus \mathcal {O}_{F_e}(-2h+(-e+4)f)$ has not property £  with respect to $M$.

\[a3.0\] The definition of property £  may be given for an arbitrary torsion free sheaf, but not much may be said in the general case. Here we look at the rank $1$ case, because we will need it in the proofs of Theorems \[o4\] and \[o2\]. Let $A$ be a rank $1$ torsion free sheaf on $F_e$. Hence $A \cong \mathcal {I}_Z(uh+vf)$ for some zero-dimensional scheme $Z$ and some integers $u,v$. Since $Z$ is zero-dimensional, $h^1(F_e,\mathcal {O}_{F_e}((u+t)h+(v+et)f)) \le h^1(F_e,\mathcal 
{I}_Z((u+t)h+(v+et)f))$ for all $t\in \mathbb {Z}$. Taking $t \gg 0$ we see that if $A$ has property £  with respect to $M$, then $v \ge eu-1$. When $v \ge eu-1$, for a general $Z$ (in the following sense) $A$ has property £  with respect to $M$ for the following reason. Fix an integer $z>0$. Since $F_e$ is a smooth surface, the Hilbert scheme $\mbox{Hilb}^z(F_e)$ of all length $z$ zero-dimensional subschemes of $F_e$ is irreducible and of dimension $2z$ ([@f]). Take a general $S\in \mbox{Hilb}^z(F_e)$, i.e. take $z$ general points of $F_e$. Since $h^0(F_e,\mathcal {I}_S\otimes 
L) =
\max \{0,h^0(F_e,L)-z\}$ for every $L\in \mbox{Pic}(F_e)$, it is easy to check that if $v \ge eu-1$, then $\mathcal {I}_S(uh+vf)$ has property £  with respect to $M$. Now take $v = eu-1$, any integer $z>0$ and any zero-dimensional length $z$ subscheme $B$ of $h$. Twisting with $(-u+1)M$ we see that $\mathcal {I}_B(uh,(eu-1)f)$ has not property £  with respect to $M$. Now assume $v > eu$. Take a zero-dimensional length $z\ge 2$ scheme $W$ of a fiber of $\pi$. Twisting with $-uM$. We see that $\mathcal 
{I}_W(uh+vf)$ has not property £  with respect to $M$. If $z \ge 3$ and $v = eu$, twisting with $(-u+1)M$ and using the same $W$ we get a sheaf without property £  with respect to $M$.

Property £  with respect to $M$ has the following open property.

\[a4\] Let $\{E_t\}_{t\in T}$ be a flat family of vector bundles on $F_e$ parametrized by an integral variety $T$. Assume the existence of $s\in T$ such that $E_s$ has property £  with respect to $M$. Then there exists an open neighborhood $U$ of $s$ in $T$ such that $E_t$ has property £  with respect to $M$ for all $t\in U$.

Let $m$ be the minimal integer such that $h^0(F,e,E_s(mM)) >0$. Thus $h^1(F_e,E_s(xM)) = 0$ for all $x \ge m$. By semicontinuity there is an open neighborhood $V$ of $s$ in $T$ such that $h^0(F_e,E_t((m-1)M)) = 0$ for all $t\in V$. By semicontinuity for every integer $x \ge m$ there is an open neighborhhod $V_x$ of $s$ in $T$ such that $h^1(F_e,E_t(xM))$ for all $t\in V_x$. Fix an irreducible $D\in \vert M\vert$. Hence $D \cong {\bf {P}}^1$. Since $D^2 >0$, there is an integer $a$ such that $h^1(D,E_s(aM)\vert D) = 0$. By semicontinuity there is an open neighborhood $V$ of $s$ in $T$ such that $h^1(D,E_t(aM)\vert D) = 0$ for every $t\in V$. Since $D^2>0$, $h^1(D,E_t(xM)\vert D) = 0$ for every $t\in V$ and every integer $x \ge a$. Fix an integer $x \ge a$. From the exact sequence $$0 \to E_t((x-1)M) \to E_t(xM) \to E_t(xM)\vert D \to 0$$ we get that if $h^1(F_e,E_t((x-1)M)) = 0$, then $h^1(F_e,E_t(xM))$. Hence we may take $U:= V\cap \bigcap _{x=m}^{\max \{a,m\}} V_x$.

If $E$ has property £  with respect to $M$, then each $L_i$ has property £  with respect to $M$ (Remark \[a3\]) and hence $v_i \ge eu_i-1$ for all $i$. Now we assume $v_i \ge eu_i-1$ for all $i$. Notice that $m$ is the minimal integer $t$ such that $h^0(F_e,E(t)) \ne 0$. Since $L_1$ has property £  with respect to $M$, $E$ has property £  with respect to $E$ if and only if $h^1(F_e,L_i(tM)) =0$ for all $t \ge m$ and all $i=2,\cdots ,r$. If $u_i-m \ge -1$, then $h^1(F_e,L_i(tM)) =0$ for all $t \ge 
m$ because $v_i \ge eu_i-1$. Now assume $u_i-m \le -2$. We get $h^1(F_e,L_i(tM)) =0$ for any $t \ge m$ if and only if $-1 \le v_i-eu_i \le e-1$.

Here we discuss the set-up for the rank $2$ case. Consider an exact sequence $$\label{eqo1}
0 \to \mathcal {O}_{F_e}(D) \to E(mM) \to \mathcal {I}_Z(c_1+2mM-D) \to 
0$$ in which $Z$ is a zero-dimensional scheme with length $s$ and either $D=0$ or $D = h$ or $D \in \vert zf\vert$ for some $1 \le z 
\le e$ or $e \ge 2$ and $D \in \vert h+wf\vert$ for some $1 \le w \le e-1$. We have $c_1(E(mM)) = c_1+2nm$ and $c_2(E(mM))
= s + D\cdot c_1 +2mM\cdot D -D^2$. Thus $c_1(E) = c_1$ and $c_2(E) = 
c_2$ by the choice of $s$ ([@h2], Lemma 2.1). Each $E$ fitting in (\[eqo1\]) is torsion free. To have some locally free $E$ fitting in (\[eqo1\]) a necessary condition is that $Z$ is a locally complete intersection. Notice that $h^1(F_e,\mathcal {O}_{F_e}(D)(-M))
= 0$ if $h$ is not a component of $D$. Hence a sufficient condition to have $h^0(F_e,E(mM)) >0$ and $h^0(F_e,E((m-1)M))=0$ is the equality $$\label{eqo2}
h^0(F_e, \mathcal {I}_Z(c_1+(2m-1)M-D))=0$$ and (\[eqo2\]) is a necessary condition if $h$ is not a component of $D$. Assume that $Z$ is a locally a complete intersection. The Cayley-Bacharach condition associated to (\[eqo1\]) is satisfied if $$\label{eqo3}
h^0(F_e,\mathcal {I}_{Z'}(c_1+2mM-2D-2h -(e+2)f)) =0$$ for every length $s-1$ closed subscheme of $s$ ([@c]). This condition is satisfied if $h^0(F_e, \mathcal {I}_Z(c_1+(2m-1)M-D))=0$, $Z_{red}\cap h = 
\emptyset$ and no connected component of $Z$ is tangent to a fiber of the fiber of $\pi$, because the line bundle $\omega _{F_e}^\ast (-M)
= \mathcal {O}_{F_e}(h+2f)$ is base point free outside $h$ and the morphism associated to $\vert f\vert$ is the ruling; if $e =12$, then (\[eqo3\]) is satisfied if (\[eqo2\]) is satified, because $\mathcal {O}_{F_1}(h+2f)$ is very ample; if $e=2$ it is sufficient to assume $Z_{red}\cap h = 
\emptyset$, because the morphism associated to $\mathcal {O}_{F_2}(h+2f)$ is an embedding outside $h$.

If $r=1$, then use Remark \[a1\]. Assume the existence of a rank two vector bundle $E$ with property £  with respect to $M$ and $c_1(E) = \mathcal 
{O}_{F_e}(uh+vf)$. Let $m$ be the first integer such that $h^0(F_e,E(mM)) >0$. We get an exact sequence (\[eqo1\]) with $D\in \vert \mathcal {O}_{F_e}(xh+yf)$ with the convention $(x,y)=(0,0)$ if $D = \emptyset$. Hence either $(x,y) = (0,0)$ or $(x,y) = (1,0)$ or $x=0$ and $1 \le y \le e$ or $e \ge 2$, $x=1$, and $1 \le y \le 
e-1$. Since $h^2(F_e,M^{\otimes z}(D)) = 0$ for all $z \ge 0$, (\[eqo1\]) and property £  for $E$ imply $h^1(F_e,\mathcal {I}_Z((u+2m-x+z)h+(v+2me-y+ze)f)=0$ for all $z \ge 0$. As in Remark \[a3.0\] we see that when $z \gg 0$ the last equality implies $v-y \ge e(u-x)-1$. If $v \le e(u-1)-2$ the last inequality is not satisfied for any choice of the pair $(x,y)$ in the previous list.

Fix a general $S \subset 
F_e$ such that $\sharp (S)=s$. Let $E$ be any torsion free sheaf fitting in the following exact sequence: $$\label{eqo9}
0 \to \mathcal {O}_{F_e}((1-m)h-emf) \to E \to \mathcal 
{I}_S((u+m-1)h+(v+em)f) \to 0$$ We have $c_1(E) = \mathcal {O}_{F_e}(uh+vf)$ and $c_2(E) = s -e(u+m-1) 
+ (1-m)(v+em)$. By construction $h^0(F_e,E(mM)) \ne 0$. We have $h^0(F_e,E((m-1)M)) = 0$. If $h^0(F_e,\mathcal {I}_S((u+2m-2)h+(v+2em-e)f)=0$. Since $S$ is general, $h^0(F_e,\mathcal {I}_S((u+2m-2)h+(v+2em-e)f)=0$ if and only if $$\label{eqo10}
h^0(F_e,\mathcal {O}_{F_e}((u+2m-2)h+(v+2em-e)f)\le s$$ Since $S$ is general, every subset of it is general. Hence to check the Cayley-Bacharach condition and hence show the local freeness of a general $E$ given by the extension (\[eqo10\]) it is sufficient to prove check the following inequality: $$\label{eqo11}
h^0(F_e,\mathcal {O}_{F_e}((u+2m-5)h+(v+2em-2e-2)f))\le s-1$$ This is true, because we assumed $s \ge \widetilde{a}$ and $\widetilde{a} >
h^0(F_e,\mathcal {O}_{F_e}((u+2m-5)h+(v+2em-2e-2)f))$. Hence a general $E$ fitting in the extension (\[eqo10\]) is locally free. Since $\mathcal {O}_{F_e}(3h+e+2)$ has a a subsheaf the very ample line bundle $\mathcal {O}_{F_e}(h+e+2)$, (\[eqo11\]) is satisfied if (\[eqo10\]) is satisfied. The generality of $S$ implies that $h^1(F_e,\mathcal {I}_S((u+m-1+t)h+(v+em+et)f)=
0$ if and only if $h^1(F_e,\mathcal {O}_{F_e}((u+m-1+t)h+(v+em+et)f)=0$ and $h^0(F_e,\mathcal {O}_{F_e}((u+m-1+t)h+(v+em+et)f)
\ge s$. Notice that $\widetilde{a} = h^0(F_e,\mathcal 
{O}_{F_e}((u+2m-2)h
+ (v+2me -e)f))$ and $\widetilde{b} = h^0(F_e,\mathcal 
{O}_{F_e}((u+2m-1)h
+ (v+2me )f))$. Since $h^1(F_e,(u+m-1+t)u + (v+me+te)f)) = 0$ for all $t \ge 0$, $\widetilde{a} \le s \le \widetilde{b}$ and $S$ is general, any sheaf $E$ in (\[eqo9\]) has property £  with respect to $M$. Since a general extension (\[eqo9\]) has locally free middle term $E$, the proof of the first part of Theorem \[o2\] is over. Now assume $m=0$, $u \ge 3$,$v < 2eu$, and that $E$ is not $R$-stable, i.e. assume the existence of $N\in \mbox{Pic}(F_e)$ such that $N\cdot R \ge 
c_1(E)\cdot R/2$ and an inclusion $j:
N \to E$; here to have $N$ locally free we use that $E$ is reflexive. Since $m=0$ and $u \ge 3$, $c_1(E)\cdot R > 
2(\mathcal {O}_{F_e}(h))\cdot R$. Hence $j$ induces a non-zero map $N \to \mathcal {I}_S((u-1)h+vf)$. Any non-zero map $N \to \mathcal 
{O}_{F_e}((u-1)h + vf)$ is associated to a unique non-negative divisor $\Delta \in \vert
\mathcal {O}_{F_e}((u-1)h + v)f)\otimes N^\ast \vert$. Since $j$ factors through $\mathcal {I}_S((u-1)h+vf)$, $h^0(F_e,\mathcal 
{I}_S(\Delta ) >0$. We fixed $R$ and the integers $m, u, v$. There are only finitely many possibilities for the line bundle $\mathcal {O}_{F_e}(\Delta )$. Since $S$ is general, we get $h^0(F_e,\mathcal {O}_{F_e}(\Delta )) \ge s$. Write $N = \mathcal 
{O}_{F_e}(\gamma h+\delta f)$ for some integers $\gamma ,\delta$. The inequality $N\cdot R \ge 
c_1(E)\cdot R/2$ is equivalent to the inequality $$\label{eqo12}
2\gamma + 2\delta \ge u+v$$ We have $\mathcal {O}_{F_e}(\Delta ) = \mathcal {O}_{F_e}((u-1 -\gamma 
)h+(v -\delta )f)$. Since $h^0(F_e,\mathcal {O}_{F_e}(\Delta )) \ge s $ and $s \le 
\widetilde{b} = h^0(F_e,\mathcal {O}_{F_e}((u-1)h
+ (v )f))$, either $\gamma \le 0$ or $\delta \le 0$. Since $\Delta$ is effective, we also have $\gamma \le u-1$ and $\delta \le v$. First assume $\delta \le 0$. Hence $\gamma \ge (u+v)/2$. Since $ \gamma \le u-1$, we get $v \le u-2$. Since $v \ge eu -e$, we get a contradiction. Now assume $\gamma \le 0$. We get $\delta \ge (u+v)/2$. Consider the exact sequence $$\label{eqo13}
0 \to N \to E \to \mbox{Coker}(j) \to 0$$ Notice that $\mbox{Coker}(j)^{\ast \ast }\cong \mathcal 
{O}_{F_e}((u-\gamma )h+(v -\delta )f)$. Since $\gamma \le 0$, $\delta \ge (u+v)/2$, and $v < 2eu$, we have $v- 
\delta \le e(u -\gamma )-2$. In Remark \[a3.0\] we checked that $h^1(F_e,\mbox{Coker}(j)(tM)) > 0$ for $t \gg 0$. Since $h^2(F_e,L(tM))
=0$ for $t \gg 0$ and any $L\in \mbox{Pic}(F_e)$, the exact sequence (\[eqo13\]) gives that $E$ has not property £  with respect to $M$, contradicting the already proved part of Theorem \[o2\]. If instead of $R$ we use $M$ for the intersection product, instead of (\[eqo12\]) we only have the inequality $2 \delta \ge v$. Everything works in the same way with only minor numerical modifications.

\[o5\] There are at least $2$ well-known and related ways to obtain rank $r 
\ge 3$ vector bundles as extensions. Instead of (\[eqo1\]) we may take the exact sequence $$\label{eqo4}
0 \to \oplus _{i=1}^{r-1} \mathcal {O}_{F_e}(D_i-m_iM) \to \mathcal 
{I}_Z(uh+vf) \to 0$$ In [@hl], proof of Theorem 5.1.6, the following extension is used: $$\label{eqo5}
0 \to L_1 \to E \to \oplus _{i=2}^{r} \mathcal {I}_{Z_i}(u_ih+v_if) \to 
0$$ The latter extension was behind the proof of Proposition \[o3\]. Both extensions can give several examples of vector bundles with or without property £  with respect to $M$. To prove Theorem \[o4\] we will use iterated extensions, i.e. increasing filtrations $E_i$, $1 \le i 
\le r$, of $E$ such that $E_1$ is a line bundle, $E_r = E$ and each $E_i/E_{i-1}$ is a rank $1$ torsion free sheaf.

Assume the existence of a rank $r$ vector bundle $E$ with property £  with respect to $M$ and $c_1(E) = \mathcal 
{O}_{F_e}(uh+vf)$. Let $m_1$ be the first integer such that $h^0(F_e,E(m_1M)) >0$. Fix a general $\sigma \in 
H^0(F_e,E(m_1M))$. Since $h^0(F_e,E((m_1-1)M)) =0$, $\sigma$ induces an exact sequence $$\label{eqo6}
0 \to \mathcal {O}_{F_e}(-m_1M+D_1) \to E \to G_1 \to 0$$ with $F_1$ torsion free, $D_1$ of type $(x_1,y_1)$ and either $(x_1,y_1) = (0,0)$ or $(x_1,y_1) = (1,0)$ or $x_1=0$ and $1 \le y_1 \le e$ or $e \ge 2$, $x_1=1$, and $1 \le y_1 
\le e-1$. Notice that $c_1(G_1) = \mathcal {O}_{F_e}((u+m_1-x_1)h + (v+em_1-y_1)f)$. Set $E_1:= \mathcal {O}_{F_e}(-m_1M+D_1)$. Since $h^2(F_e,\mathcal {O}_{F_e}((t-m_1)D+D_1)) = 0$ for all $t \ge 
m_1$, property £for $m$ with respect to $M$ implies $h^1(F_e,F_1(tM)) = 0$ for all $t 
\ge m_1$. Let $m_2$ be the first integer such that $m_2 \ge m_1$ and $h^0(F_e,F_1(m_2M)) >0$. A non-zero section of $H^0(F_e,G_1(m_2M))$ induces an exact sequence $$\label{eqo7}
0 \to \mathcal {I}_{Z_1}(-m_2M+D_2) \to G_1 \to G_2 \to 0$$ with $\mathcal {I}_{Z_1}$ zero-dimensional, $G_2$ torsion free and $D_2$ an effective divisor of type $(x_2,y_2)$ and either $(x_2,y_2) = 
(0,0)$ or $(x_2,y_2) = (1,0)$ or $x_2=0$ and $1 \le y_2 \le e$ or $e \ge 2$, $x_2=1$, and $1 \le y_2 
\le e-1$. Here we cannot claim that $Z_1 = \emptyset$, because $G_1$ is not assumed to be locally free. Notice that $c_1(G_2) = \mathcal {O}_{F_e}((u+m_1+m_2-x_1-x_2)h + 
(v+em_1+em_2-y_1-y_2)f)$. Since $Z_1$ is zero-dimensional, $h^2(F_e,\mathcal {I}_{Z_1}\otimes L)= 
h^2(F_e,L)$ for every $L\in \mbox{Pic}(F_e)$. Hence as in the first step we get $h^1(F_e,G_2(tM)) = 0$ for all $t \ge m_2$. If $r=3$, we are done as im the proof of the case $r=2$. If $r \ge 4$, we iterate the last step $r-3$ times.

[99]{}

F. Catanese, Footnotes to a theorem of Reider, in: Algebraic Geometry Proceedings, L’Aquila 1988 (ed. by A. J. Sommese, A. Biancofiore, E. L. Livorni), 64–74, Lecture Notes in Math. 1417, Springer, Berlin, 1990.

J. Fogarty, Algebraic families on an algebraic surface, Amer. J. Math. 90 (1968), 511–521.

R. Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), no. 2, 121–176.

D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves, Friedr. Vieweg & Sohn, Braunschweig, 1997.

[^1]: The author was partially supported by MIUR and GNSAGA of INdAM (Italy).
