Nowhere-vanishing sections of vector bundles

Let be a proper variety over some field, and let be a vector bundle on .  The functor of global sections of , i.e. the functor sending a scheme to the set , is (representable by) a nice affine -scheme, namely the scheme . Let denote the subfunctor corresponding to nowhere-vanishing sections . We’d like this subfunctor to be representable by an open subscheme. How should we prove this?

Let be the structure map. The identity map corresponds to a universal section . Let denote the zero locus of . This is a closed subset. But now we observe that the projection is proper, hence universally closed, and so is a closed subset of .  One then checks directly that is the open subscheme corresponding to the open subset , so we win.

I guess this sort of thing is child’s play for an experienced algebraic geometer, and indeed it took Johan about 0.026 seconds to suggest that one should try to argue using the universal section.  I only cared about the above problem, though, as a toy model for the same question in the setting of a vector bundle over a relative Fargues-Fontaine curve . In this situation, is a diamond over , cf. Theorem 22.5 here, but it turns out the above argument still works after some minor changes.

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