  We construct examples of non-smoothable stable surfaces, that we call Corona
surfaces, with invariants in a wide range comprising all possible invariants of
smooth minimal surfaces of general type but non-standard second plurigenus.
  We deduce that the moduli space of stable surfaces $\overline{\mathfrak
M}_{k,l}$ has least $k-l+2$ connected components distinguished by the second
plurigenus and, in particular, it always has more irreducible components than
the Gieseker moduli space with the same invariants.
