I have already asked similar questions before, but now I realized that there a nice general way to ask what I want. Namely let $X$ be a normal affine variety over a field $k$. Assume first that $k$ is finite. Then is it true that
1) IF $X$ is smooth, then the set of isomorphism classes of vector bundles of given rank on $X$ is finite?
2) More generally, is it true that for any $X$ the set of isomorphism classes of Cohen-Macaulay torsion free sheaves of fixed generic rank is finite?
When the field $k$ is arbitrary then I would expect that there exists a finite-dimensional (over $k$) family of vector bundles or Cohen-Macaulay sheaves which contains every isomorphism class. Is this true?