not an end of $R$, and let $p\in C_1$ be the node. The existence of such a component is easy to prove: since $R$ is a proper subcurve of $C$, there must be a node of $C$ where $R$ meets a component not in $R$. Removing this node disconnects $C$ into a connected component containing $R$ and a connected component $K$ disjoint from $R$. Clearly an end $C_1$ of the component $K$ (different from the one meeting $R$ if there is more than one end) is then also an end of $C$ not contained in $R$. Since $C_1$ is a smooth rational curve, ${\mathcal F}| _{C_1} \simeq \bigoplus {\mathcal O}(a_j)$, with $a_j \geq 0$, thanks to the fact that ${\mathcal F}| _{C_1}$ is globally generated. In particular $h^1 (C_1, {\mathcal F}(-p)) = 1$, and it is clear that we can now apply Lemma \[pieces\] to remove the component $C_1$ without changing $h^1$ and conclude using induction. [ $\Box $ ]{}

The last lemma of this section is an explicit computation of the cohomology of a locally free sheaf on a curve which will be extremely useful in the later sections.

\[tretre\] Let $C$ be a rational tree and $f: C 