I am new to algebraic surface, so I have some questions maybe naive:
- How should I think of the Kodaira dimension of a surface?
- Why those with Kodaira dimension 2 are called "of general type"?
- What is the relation between surface being general type and canonical bundle being ample? I seem read from somewhere that ample canonical bundle implies general type. I tried to prove this, but I can only solve the special case of hypersurface. (Since the canonical bundle is $\mathcal O_S(d-4)$ and using the exact sequence $0\to \mathcal O(-d)\to \mathcal O \to \mathcal O_S\to 0$, twist $k(d-4)$ times and take cohomology. I don't know if this kind of argument apply to general case?)
- Why the surface with anti-ample canonical bundle should be think of opposite to of general type?
Sorry about put all these questions together :)