Let $S$ be a smooth projective surface with an ample divisor $X\subset S$. Consider the moduli stack of vector bundles $F$ on $S$ such that
3) The restriction of $F$ to $X$ is fixed (say, isomorphic to a bundle $M$).
1) Is it true that for any $n$ and $M$ the above stack is of finite type?
2) Is it true, that for any given $M$ the above stack is empty for $n$ sufficiently small?