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The following is, amongst others, a Hartshorne exercise:

Let $V$ be a $k$-variety of dimension $n$ and $\mathcal{E}$ a vector bundle of rank greater than $n$, then, generically, a generating section is nowhere vanishing.

One could prove this using an argument similar to Bertini's theorem as suggested in the book.

Now a similar result is true in topology and one way to prove this is to compute the connectivity of the map $BO(k-1) \rightarrow BO(k)$ where $k$ is the rank of our bundle. Having a rank $k$-bundle is a map $X \rightarrow BO(k)$ and having a nonvanishing section would be equivalent to splitting an $k-1$ summand and thus equivalent to lifting this map to a map to $BO(k-1)$ and so the problem is now about computing the connectivity of the map above.

My question is: can I use a similar method in algebraic geometry by considering the "connectivity" of the map of stacks $BGL(k-1) \rightarrow BGL(k)$ ? I am novice so I don't actually know what the right notion of connectivity should be so I am interested in a reference for that as well!

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