The setting is as follows: Let $X$ be an algebraic surface, $\mathcal{F}$ a locally free sheaf of rank 2 on $X$ contained in $\Omega^1_X$, and $D$ a divisor on $X$ such that $\mathcal{F}\otimes\mathcal{O}_X(-D)$ has a non-zero global section. Then there is a non-zero divisor $S$ on $X$ such that $\mathcal{F}\otimes\mathcal{O}_X(-D-S)$ admits a global section with at most isolated zeros.
Why is the last statement true? This is the first sentence in the proof of Proposition VII.4.3 in the book Compact Complex Surfaces by Barth, Hulek, Peters, van de Ven. Any help is appreciated, thanks!
Best Answer
This a general fact and nothing to do with subsheaf of $\Omega^1_X$. If $E$ is a rank two vector bundle with a section, we have an inclusion $O_X\to E$. If you pull back the torsion subsheaf of $E/O_X$, we get an exact sequence, $0\to L\to E\to G\to 0\to 0$, with $L=O_X(S)$, $S$ an effective divisor and $G$ torsion free. One easily checks that the section $O_X\to E(-S)$ vanish only at isolated points.