On a degree $n$ Hirzebruch surface $F\_n$, suppose we have a very ample linear system. It is known that its generic smooth irreducible members give a Lefschetz pencil on $F\_n$. Let us take a member, $G$, in this pencil. And suppose we know that $G$ intersects the fiber, $F$, of $F\_n$ $m$ times. Generically they intersect at $m$ points. 


Is it possible to move $G$ in a way that $G$ and $F$ intersect at 1 point with multiplicity $m$? I don't want to change the genus of $G$ while moving. Or is it possible to find $G'$ in the pencil which intersects $F$ in the desired way?


I am not an algebro-geometer so, sorry if this question is trivial. I'd appreciate any suggestions. Thanks. 



Very ample linear systems - intersections with multiplicity >1
A partial answer (too long for a comment): suppose $G$ is sufficiently ample so that $H^1(\mathbb{F}\_{n},\mathcal{O}(G-S))=0$. Then the restriction map $H^0(\mathbb{F}\_{n},\mathcal{O}(G))\rightarrow H^0(S,\mathcal{O}(G)\_{|S})$ is surjective. This shows that:


1) There exists $G'\in \lvert G \rvert$ such that $G\cdot S=4s$, for any $s\in S$;


2) There will be no such $G'$ in a general pencil $P\subset \lvert G \rvert$.


