I am studying about projective bundles now. And I have the following doubts.
1) If we have an exact sequence of vector bundles over a scheme $X$, $0\longrightarrow E'\longrightarrow E\longrightarrow E''\longrightarrow 0$, then what happens to the corresponding projective bundles. Is there some injective/surjective morphism between them
2) Also assuming that my guess is right, if $E''$ is a quotient of $E$, that is, $E\longrightarrow E''\longrightarrow 0$. Then $Y=P(E'')$ is a closed subscheme of $X=P(E)$. Then what is the ideal of $Y$?
I would be very thankful if someone clarifies these doubts. Thanks in advance!
For 2), let $\pi:X=\mathbb{P}(E)\to S$ be the natural map. We have a canonical surjection $\pi^*E\to\mathcal{O}_X(1)$ and thus a map by composition $\pi^*E'(-1)\to\mathcal{O}_X$. The image is the ideal sheaf defining $Y=\mathbb{P}(E")$.