Computing the normal bundle via rank 2 bundles

Question

Probably related to this and this.

Suppose $X$ is a smooth projective variety, and let $E$ be a rank two vector bundle over $X$. Let $E_1,E_2$ be rank 1 subbundles.

Let $Y\subset X$ be the subvariety defined by the condition that the vanishing of the composition

$$E_1\to E\to E/E_2,$$

that is to say that $y\in Y$ if and only if $E_{1,y}=E_{2,y}$ in $E_y$.

Can we use this data to compute the normal bundle of $Y$ in $X$? It seems possible that we could somehow use the linked questions to do so, but it is not clear to me if $X=\mathbb{P}_Y(E)$ or something similar that would allow us to apply these results.

Answer 1

The composition $E_1 \to E/E_2$ gives a global section of the line bundle $$ L = E_1^\vee \otimes (E/E_2). $$ The subscheme $Y \subset X$ is the zero locus of this section of $L$, hence (unless this section is a zero divisor) $$ N_{Y/X} \cong L\vert_Y. $$

Answer 2

If $E_1=E_2$, there is not much to say, since then $Y=X$. So, assume $E_1\neq E_2$. Then, the map $E_1\to E/E_2$ is non-zero and $Y$ is a codimension one subscheme. Its normal bundle is just $E_1^{\vee}\otimes (E/E_2)_{|Y}$.