Stable bundles with $c_1=0$ on rational surfaces

For an arbitrary rational surface $X$the author proves the existence of a nonempty component of the moduli variety $M^0(X,n,r)$ of rank $r$ bundles with $c_1=0$ and $c_2=n\geqslant r$ in which the $\mathscr L$-stable bundles constitute a nonempty open subset for any ample $\mathscr L$. Moreover, any birational isomorphism $\varphi\colon X\to Y$ of surfaces gives rise to a birational isomorphism $\varphi_*\colon M^0(X)\to M^0(Y)$.