The "easiest type" of curves of general type are those of genus two. In this case $\chi(X,\mathcal O_X) = -1$ and $\deg K = 2$, where $K$ is the canonical sheaf.
I'm a bit lost when it comes to surfaces of general type. It seems surfaces of special type (K3 surfaces, abelian surfaces, Enriques surfaces, elliptic fibrations, etc.) are well-studied, but I haven't seen surfaces of general type been "partially" classified.
A surface $S$ of general type with ample canonical sheaf $K$ has a Hilbert polynomial which is determined by $\chi(X,\mathcal O_X)$ and $K^2$.
What are the easiest values $\chi(X,\mathcal O_X)$ and $K^2$ take for a surface of general type with ample canonical sheaf? Are they $\chi(X,\mathcal O_X) = -1$ and $K^2= 1$ (or maybe $K^2= 2$)?
Have such easy surfaces of general type been studied somewhere?
