$\underline {Background}$: Let $X$ be a nonsingular projective variety over $\mathbb C$.
Let,$\mathcal F$ be a vector bundle on $X$ and $L$ be a line bundle on $X$
$\underline {Question}$:What is the meaning of the statement $\phi:\mathcal F \to \mathcal F \otimes L$ is a nonzero nilpotent matrix at the general point of $X$( what is "the" general point?)
$\underline { Guess}$: I have following two possible interpretations but they seem in contradiction with each other.
(1)$\phi$ is nilpotent means $\exists n$ such that the natural composition map
$\phi^n:\mathcal F \to \mathcal F\otimes L^{\otimes n}$ is $0$.
Then for all point $p\in X$ the corresponding stalk map is $0$ (but then the special case of"the" general point is lost).
(2) OR we can simply say $\exists U$ an open dense neighbourhood in $X$ such that $\forall p\in U$ , $\phi_{p}$ is a nonzero nilpotent matrix .(so in particular also for the generic point)
Is it true that we can modify possibilities (1) and (2) and make a relation between them?
any help from anyone is welcome.