# What are the easiest surfaces of general type

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?

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This is a bit late, I know, but to my mind the easiest surfaces of general type are smooth quintics in $\mathbf P^3$. They have $K^2=1$, but $\chi>0$. To get $\chi<0$ you need a surface with nonvanishing $H_1$, which therefore cannot be a hypersurface or even a complete intersection; to me, this disqualifies such a surface from being the "easiest". –  user64687 May 12 '14 at 10:43