In terms of the Virasoro operator $L_{0}$ the collapse is described by a subset (cluster) $S_{1}$ in the set of eigenvalues of $L_{0}$ which approach to zero “with the same speed”, as $E_{min}\to 0$. The next level of the collapse is described by another subset $S_{2}$ of eigenvalues of $L_{0}$. Elements of $S_{2}$ approach to zero “modulo the first collapse” (i.e. at the same speed, but “much slower” than elements of $S_{1}$). One can continue to build a tower of degenerations. It leads to an hierarchy of boundary strata. Namely, if there are further degenerations of CFTs parametrized by $X$, one gets a fiber bundle over the space of triples $(X,{{\bf R}}_{+}^{\ast}\cdot g_{X},\phi_{X})$ with the fiber which is the space of triples of similar sort. Finally, we obtain the following qualitative geometric picture of the boundary $\partial\overline{{\cal M}}_{c\leq c_{0}}$.

A boundary point is given by the following data:

1) A finite tower of maps of compact topological spaces $p_{i}:\overline{X}_{i}\to\overline{X}_{i-1},0\leq i\leq k$, $\overline{X}_{0}=\{pt\}$.

2) A sequence of smooth manifolds $({X}_{i},g_{{X}_{i}}),0\leq i\leq k$, such that ${X}_{i}$ is a dense subspace of $\overline{X}_{i}$, and $dim\,X_{i}>dim\,X_{i-1}$, and $p_{i}$ defines a fiber bundle $p_{i}:X_{i}\to X_{i-1}$.

3) Riemannian metrics on the fibers of the restrictions of $p_{i}$ to $X_{i}$, such that the diameter of each fiber is finite. In particular the diameter of $X_{1}$ is finite, because it is the only fiber of the map $p_{1}:X_{1}\to\{pt\}$.

4) A map $X_{k}\to{\cal M}_{c\leq c_{0}-dim\,X_{k}}$.

The data above are considered up to the natural action of the group $({{\bf R}}_{+}^{\ast})^{k}$ (it rescales the metrics on fibers).

There are some additional data, like non-linear connections on the bundles $p_{i}:X_{i}\to X_{i-1}$. The set of data should satisfy some conditions, like differential equations on the metrics. We cannot formulate this portion of data more precisely in general case. It will be done below in the case of $N=2$ SCFTs corresponding to sigma models with Calabi-Yau target spaces.