Let $\pi:E\to X$ be a holomorphic vector bundle on a complex algebraic variety $X$, and assume $E$ has nonzero global sections; fix a divisor $D\subset E$ and a point $P\in X$.
I have the vague feeling that the following question has affirmative answer, but I am not able to prove it.
Q. Does the generic section $s\in H^0(X,E)$ satisfy $s(P)\notin D$?
In other words: if we fix the point and the divisor, we should be able to send the point outside the divisor with "almost all" the sections.
I just tried to reason in the opposite direction by showing that $$V:=\{s\in H^0(X,E)\,|\,s(P)\in D\}$$ is not the whole of $H^0(X,E)$. But how to compute $\dim V$, for instance?
Thanks for any suggestions.
Edit. Leu us assume, also (see comments below), that $E$ is base-point free and no fiber of $\pi$ is contained in a divisor.