In Nitsure's Cohomology of the moduli of parabolic vector bundles, we have the following Remark on page $62$:
([Remark 1.2) Let $E$ be a vector bundle on a scheme $S$ and let $F_1$ and $F_2$ be subbundles. Let $\Phi:F_1\to E/F_2$ be the natural map. Then the function $s\to \operatorname{rank}_{\kappa(s)}\Phi(s) $ is lower semicontinuous on $S$. (Equivalently the function $s\to \operatorname{dim}_{\kappa(s)}F_1(s)\cap F_2(s) $ is upper semicontinuous).
- What is exactly the natural map $\Phi$?
- How can we prove this remark? Is there any reference?
Thank you.
The first question has already been answered in the comments: take the composition of the inclusion $F_1\to E$ and the projection $E\to E/F_2$.
The second question can be dealt with locally. Choose an affine open $\operatorname{Spec} R\subset S$ where $F_1$ and $E/F_2$ are free, so that $\Phi$ is locally represented by a matrix $M$ with entries in $R$. Then the condition that $\operatorname{rank} \Phi(s) < r$ is that all the determinants of the $r\times r$ minors of $M$ vanish at $s$. This is a closed condition, which exactly implies lower semicontinuity of rank. (See for instance characterization 3 on wikipedia.)