Let $X$ be a smooth, projective, algebraic surface and $H$ be an ample divisor on $X$. Let $ M_H : = M_{X,H}(r;c_1,c_2)$ be the moduli space of rank $r$, $H$-stable vector bundles on $X$ with fixed Chern invariants $c_1,c_2$. Assume that $ M_H$ is a fine moduli space and $U$ be the universal bundle.
Then can we say that there exists an effective divisor $D$ on $X$ such that for every $p \in M_H$, $H^i(U|_{X \times \{p\}} \otimes D)=0$ for $i >0$?
In my situation there are more cohomological conditions on the members of $M_H$ but I guess that has nothing to do with existence of such a $D$.
Any suggestion regarding the existence of $D$ which is independent of $p$ is appreciated
