For any minimal compact complex surface $S$ with $b_2(S)>0$ containing global spherical shells (GSS) there exists a family of surfaces $\cal S\to B$ with GSS which contains as fibers $S$, some Inoue-Hirzebruch surface and non minimal surfaces, such that blown up points are generically effective parameters. These families are versal outside a non empty hypersurface $T\subset B$. In case of surfaces with a cycle and one tree of rational curves we give new normal forms of contracting germs in Cremona group $Bir(\bb P^2(\bb C))$ and show that they admit a birational structure. These families contain all possible surfaces, in particular all surfaces $S$ with GSS and $0