Assume we have a smooth projective variety $X$ over a field (finite field if matters). Consider a smooth hypersurface section $H$ of $X$. Let $Spec(A)=U=X \setminus H$ be an affine variety. I want to know whether there are any nice description of vector bundles on $X$ such that they are trivial on $U$ and $H$? My suspicion was that they are all direct sums of line bundles of the form $O(nH)$ is that correct?

Edit: Let's assume that we are talking about rank $m$ vector bundles that are subsheaves of the constant sheaf given by the vector space $F(X)^{\oplus m}$ which $F(X)$ is the function field or equivalently field of fractions of $A$. Does that change anything?

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    $\begingroup$ This is not true. Try the projective plane and $H$ a hyeprplane. $\endgroup$ – Mohan Apr 10 '19 at 1:33
  • $\begingroup$ @Mohan Is this true if we assume the variety is 1 dimensional? And assume the $H$ is just 1 point. $\endgroup$ – user127776 Apr 10 '19 at 3:29
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    $\begingroup$ This is definitely false. Take a smooth projective curve $X$ of genus $\geq 2$. There are of course many nontrivial vector bundles on $X$ with trivial determinant. Their restriction to the complement of a point is trivial, because a projective module over a Dedekind ring with trivial determinant is free. $\endgroup$ – abx Apr 10 '19 at 5:24
  • $\begingroup$ I made an edit, I'd really appreciate your opinions on it. $\endgroup$ – user127776 Apr 10 '19 at 6:55
  • $\begingroup$ All rank $m$ vector bundles embed into $F(X)^{\oplus m}$ (by restriction to the generic fiber). $\endgroup$ – abx Apr 10 '19 at 7:07

In many cases, $\mathcal{O}_X(n \cdot H)$ itself will not be trivial on both $H$ and $U$, because it will be nontrivial on $H$. For example, if $H \subset \mathbb{P}^n$ is a degree $d$ hypersurface, then $ \mathcal{O}_{\mathbb{P}^n}(H) \simeq \mathcal{O}_{\mathbb{P}^n}(d)$ and so $\mathcal{O}_{\mathbb{P}^n}(H)|_H \simeq \mathcal{O}_H(d)$ which is non-trivial as long as $n> 1$, for instance because it is ample.

EDIT: I realize what this shows is: not all sums of line bundles of the form $\cal{O}_X(n \cdot H)$ are trivial on both $U$ and $H$. All the same it seemed relevant.


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