Question

Hello,

I have a question about certain immersions. If $X$ is a base scheme, and $E$ is a loccally free quasi coherent sheaf over X, then we can build $P(E)$ the projective bundle associated to E.

Now if $0\to L \to E \to F\to 0$ is an exact sequence of vector bundles, with L invertible. We get a closed immersion form $P(F)$ to $P(E)$. Supposedly this closed immersion is given by the zero scheme of the section $p^*L^{-1}\otimes O(1)$ induced by $L\to E$.

And i fail to produce a proof of this fact. Even geometrically, I'm so sure I get a clear picture of what's going on. I presume that $P(F)$ will be defined as the locus where $p^*L_x$ and $\gamma_x$ "coincide" (where $\gamma$ is the tautological bundle on $P(E)$).

Answer 1

For the fact that $\mathbf P(F)\to\mathbf P(E)$ is a closed embedding, see EGA II, 4.1.2. By 4.2.9, an $X$-morphism $f\colon T\to\mathbf P(E)$ factors over $\mathbf P(F)$ iff $(pf)^*E\to f^*O(1)$ factors over $(pf)^*F$. This is equivalent to $(pf)^*L\to f^*O(1)$ being zero, and this again is equivalent to $f$ factoring over the zero locus of $p^*L\to O(1)$.