  In a holomorphic family $(X_b)_{b\in B}$ of non-K\"ahlerian compact
manifolds, the holomorphic curves representing a fixed 2-homology class do not
form a proper family in general. The deep source of this fundamental difficulty
in non-K\"ahler geometry is the {\it explosion of the area} phenomenon: the
area of a curve $C_b\subset X_b$ in a fixed 2-homology class can diverge as
$b\to b_0$. This phenomenon occurs frequently in the deformation theory of
class VII surfaces. For instance it is well known that any minimal GSS surface
$X_0$ is a degeneration of a 1-parameter family of simply blown up primary Hopf
surfaces $(X_z)_{z\in D\setminus\{0\}}$, so one obtains non-proper families of
exceptional divisors $E_z\subset X_z$ whose area diverge as $z\to 0$. Our main
goal is to study in detail this non-properness phenomenon in the case of class
VII surfaces. We will prove that, under certain technical assumptions, a lift
$\widetilde E_z$ of $E_z$ in the universal cover $\widetilde X_z$ does converge
to an effective divisor $\widetilde E_0$ in $\widetilde X_0$, but this limit
divisor is not compact. We prove that this limit divisor is always bounded
towards the pseudo-convex end of $\widetilde X_0$ and that, when $X_0$ is a a
minimal surface with global spherical shell, it is given by an infinite series
of {\it compact} rational curves, whose coefficients can be computed
explicitly. This phenomenon - degeneration of a family of compact curves to an
infinite union of compact curves - should be called {\it infinite bubbling}.
  We believe that such a decomposition result holds for any family of class VII
surfaces whose generic fiber is a blown up primary Hopf surface. This statement
would have important consequences for the classification of class VII surfaces.
