Vector bundles on affine scheme 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?
 A: As shown by David, the answer is no for line bundles. 
More generally, the answer in no even for higher dimensional vector bundles. In fact, in his paper "Vector bundle over affine surfaces birationally equivalent to ruled surfaces" Murthy proves the following results:
(1) Let $V$ be an irreducible affine non-singular surface defined over an algebraically closed field $k$ and such that $V$ is birational to $C \times \mathbb{P}^1$, vhere $C$ is a curve. Then any vector bundle over $V$ is a direct sum of a trivial bundle and a line bundle. In general, there are infinitely many equivalence classes of line bundles (as shown by David  for $V= \mathbb{P}^1 \times \mathbb{P}^1 \setminus \Delta$).
(2) There exist an affine, nonsingular rational variety of dimension $3$ over which there are infinitely many non-isomorphic indecomposable vector bundles of rank $>1$. 
A: Suppose that $X = \mathbb{P}^1 \times \mathbb{P}^1 \setminus \Delta$, where $\Delta$ is the diagonal. The Picard group of $\mathbb{P}^1 \times \mathbb{P}^1$ is $\mathbb{Z}^2$. Yanking out $\Delta$ should just kill the generator $(1,1)$ in $\mathbb{Z}^2$, so $\mathrm{Pic}(X)$ should be $\mathbb{Z}$. So there is an infinite discrete parameter describing line bundles on $X$. Unless I am missing something...
