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It is claimed here that there exist proper schemes (probably over a field but not explicitly stated) with trivial Picard group. This means that every locally free $O_X$-module of rank 1 is trivial.

Do there exist proper schemes over a field such that every locally free $O_X$-module of finite rank is trivial?

Maybe it is easier to give some examples with algebraic spaces, but the accepted answer should give a scheme.

EDIT: examples should be positive-dimensional.

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According to this paper it is not known if any proper algebraic scheme admits nontrivial vector bundles. Partial results can be found here.

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  • $\begingroup$ great! the paper looks like it is pretty old. Hopefully, @SándorKovács could add something useful. $\endgroup$ – schematic_boi Apr 24 at 13:53

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