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
1) $c_1(F)=0$
2) $c_2(F)=n$
3) The restriction of $F$ to $X$ is fixed (say, isomorphic to a bundle $M$).
$\mathbf{Questions:}$
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?
Take Q 2). Assume that we are in characteristic zero, that $F$ has rank $2$ and that $X$ is smooth. If $n<0$ then $F$ is Bogomolov unstable: there is a short exact sequence $$0\to O(A)\to F\to I_Z.O(-A)\to 0$$ with $A^2>0$ and $A.X>0$ and $I_Z$ the ideal sheaf of a $0$-dimensional subscheme $Z$. In fact $c_2(F)=\deg Z-A^2$, so $A^2\ge -n$. Suppose that we can make $n$ arbitrarily small (i.e., large and negative). Then (index theorem) $A.X$ is arbitrarily large (and positive). On the other hand, restrict the inclusion $O(A)\to F$ to $X$; this gives a non-zero homomorphism $O_X(A)\to M$, which is impossible for fixed $M$.
So yes, your stack is empty in this case (rank $2$, char. $0$). In higher rank Bogomolov gives a destabilizing flag and maybe an elaboration of the argument above will then work. I don't know about char. $p$.
