Take the 2-minute tour ×  Let $E$ a rank $r\geq 3$ vector bundle
over a curve $C$ and let $E'$ a rank $r-1$ subbundle of $E$.
Thus we have $\mathbb{P} (E') \subset \mathbb{P} (E)$; what can
be said about $ \xi|_{\mathbb{P}(E')}$, where $\xi$ is the
tautological class of $\mathbb P (E)$?  If $\xi'$, $F'$ are
resp. the tautological class of $\mathbb P (E')$ and the class
of a fibre in $\mathbb P (E')$, is it possible to compute the
components of $ \xi|_{\mathbb{P}(E')}$ with respetc to $\xi'$,
$F'$?  share|improve this question  1 Answer 1  up vote 1 down
vote accepted  I personally prefer to work with sheaves which
ultimately gives you the same thing, but sometimes you need to
work in a dual setting. So, I will use sheaves below, feel free
to rewrite this for yourself in the language of bundles.  Let
$\alpha: \mathscr E_1\to \mathscr E_2$ be a surjective morphism
of sheaves and a scheme $X$ and let $\pi_i:\mathbb P(\mathscr
E_i)\to X$ be the associated projective bundle for $i=1,2$.  By
construction there exis