  We identify the stable surfaces around the stable limit of the examples of Y.
Lee and J. Park [LP07], and H. Park, J. Park and D. Shin [PPS09] using the
explicit 3-fold Mori theory in [HTU13]. These surfaces belong to the
Koll\'ar--Shepherd-Barron--Alexeev compactification of the moduli space of
simply connected surfaces of general type with $p_g=0$ and $K^2=1,2,3$.

Algebraic Geometry