Exploiting the boundary of the moduli space of stable surfaces with $K^2=1$ and $p_g=2$
In this talk we will show two different methods of analyzing the boundary components of the Moduli space of stable surfaces with $K^2=1$ and $p_g=2$: the first is based on degenerations of the Hodge type and the second method relies on the study of the canonical ring and its algebraic degenerations.