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It is well-known by Grothendieck (or earlier by Dedekind-Weber) that every vector bundle on $\mathbb{P}^1_k$ for $k$ a field decomposes into a sum of the line bundles $\mathcal{O}(k)$.

As investigated by Hübl and Sun, this fails if we replace $k$ by a discrete valuation ring $D$: There are vector bundles on $\mathbb{P}^1_D$ that do not decompose into line bundles and the whole situation is considerably more complicated than over a field. But if we pass to the open subset $\mathbb{A}^1_D$, life is easy again and every vector bundle on $\mathbb{A}^1_D$ is even trivial. A possible argument is the following: By Horrock's theorem, it is enough to extend the vector bundle on $\mathbb{A}^1_D$ to $\mathbb{P}^1_D$. In general, we can extend a vector bundle defined on an open subset only to a reflexive sheaf on the whole scheme; but on a regular scheme of dimension $2$ all reflexive sheaves are vector bundles.

If we pass to complete curves of genus $\geq 1$, the situation is complicated even over an (algebraically closed) field. Atiyah could classify vector bundles on elliptic curves, but on higher genus curves the classification problem is wild. Replacing the field by a discrete valuation ring certainly makes the situation even worse.

If we pass to an open subset of a complete smooth curve, the situation certainly cannot be worse than that: We still can extend every vector bundle defined on an open subset of a complete smooth curve (defined over a field or a DVR) to the whole curve; on the other hand, many vector bundles on the complete curve might be isomorphic over the given open subset. My question is:

Are there known simplification of the problem of classifying vector bundles on curves if we pass to open (affine) subsets? Is there maybe even a number of points I can delete from a complete smooth curve so that every vector bundle decomposes into line bundles?

Again, I am interested not only in the case of a base field, but also in the case of a base DVR.

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By a theorem of Serre, see Theorem 1 in this paper, vector bundles over smooth affine curves over fields are direct sum of a line bundle and a trivial bundle. Then the classification problem is given by the line bundle classification. In particular, both questions have positive answers over fields.

For smooth affine curves over a discrete valuation ring, the same result allows to reduce to the classification of rank two bundles. However, it will probably not be possible to remove a fixed number of points to split all vector bundles; only the other way round will work. Obstruction theory in $\mathbb{A}^1$-homotopy theory will probably allow to get a vector bundle classification in terms of the first two Chern classes, but I would have to check the details (I only know this for surfaces over fields...)

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