Let $C$ be a nonsingular complete algebraic curve over $\mathbb C$. The projective curve $\mathbb P^1$ is an example with property that it is covered by affine opens which are spectra of PIDs. I wonder if there are other examples with this property, i.e.
Are there nonsingular complete algebraic curves other than $\mathbb P^1$ which have affine open covers by spectra of PIDs?
The motivation is calculations of vector bundles over a curve. For example given ranks and degrees, what is the dimension of $\mathrm{Ext}^1(E_1,E_2)$? I know finite locally free sheaves over PIDs behave well, indeed finite free. Any reference of calculations about vector bundles are welcome, too.
No, this is the only example. As any PID is a UFD and any smooth curve is normal, the class group of any affine open subset of a curve which is the spectrum of a PID must vanish. But the class group of the entire curve is an extension of the class group of this open subset and a finitely generated abelian group by repeated applications of the exact sequence $\Bbb Z\cdot p\to \operatorname{Cl} X \to \operatorname{Cl} (X\setminus p)\to 0$. Identifying the Picard and class groups, this says that the Picard group of your variety is at most countable. But for any curve of positive genus, the Picard group is isomorphic to $\Bbb Z$ times the Jacobian, and the Jacobian is uncountable.