Let X/k be a projective integral variety over a field, with fixed $\mathcal{O}(1)$, $\mathcal{E}$ be a rank two vector bundle, for simplicity assume it is stable. Regarding the collection of sub line bundles $\mathcal{L}\subset \mathcal{E}$ up to isomorphism,

*EDIT:* and moreover $\mathcal{L}$ is the line bundle given by an effective divisor,

denoted this collection by $Sub(\mathcal{E})$, I have the following questions:

(1) When is this collection finite?

(2) Take another scheme $T/k$, and base change $X_T/T$, when is $Sub(\mathcal{E}_T)=Sub(\mathcal{E})_T$ ?

(3) If the above equality doesn't hold, when does $Sub(\mathcal{E}_T)=Sub(\mathcal{E})_T$ up to tensoring pull back of a line bundle on T?

(4) If the answer to (2) and (3) are not very trivial, then consider the functor $Sub(\mathcal{E}_T\otimes f_T^*(\mathcal{N}))$, where $f:X\rightarrow \mathcal{S}pec\ k$ is the structure morphism, $\mathcal{N}$ is a line bundle on T, and we identify two sub line bundles of $\mathcal{E}$ up to tensoring with $f^*_T(\mathcal{N})$. When is this functor representable?

*Edit*: Here by a sub line bundle I just mean a rank one subsheaf which is itself a line bundle; i.e. in an exact sequence like $0\rightarrow \mathcal{L}\rightarrow \mathcal{E}\rightarrow \mathcal{Q}\rightarrow 0$, I didn't mean both $\mathcal{L}$ and the "quotient" $\mathcal{Q}$ are line bundles.