Enriques–Kodaira classification

Remarks

The reduction to minimal surfaces uses Castelnuovo's contraction theorem: a smooth rational curve with self-intersection $-1$ can be blown down to a smooth point, which lowers the second Betti number. For each value of $\kappa$, the finer classification uses the invariants $q = h^1(\mathcal{O}_S)$, $p_g = h^0(K_S)$, the plurigenera $P_n = h^0(nK_S)$, and the Chern numbers $K_S^2$ and $c_2(S)$. Castelnuovo's rationality criterion is the key step for $\kappa = -\infty$. Kodaira extended the classification to all compact complex surfaces, which adds non-algebraic classes such as surfaces of class VII and Kodaira surfaces. Bombieri and Mumford extended it to algebraically closed fields of positive characteristic, where new classes appear in characteristics $2$ and $3$. Example: a smooth quartic surface in $\mathbb{P}^3$ has $K_S = 0$ by adjunction and lands in case 2 as a K3 surface. A smooth quintic surface has $K_S$ ample and lands in case 4.

Going deeper

For a student

The classification is the two-dimensional model of the minimal model program. In dimension one, curves split by genus into $g = 0$, $g = 1$ and $g \ge 2$, which matches $\kappa = -\infty, 0, 1$; for surfaces there is one more class. The invariants that tell the classes apart are birational invariants, so they can be computed on any smooth model. Results like the Bogomolov–Miyaoka–Yau inequality constrain the invariants of the last class.