Castelnuovo's rationality criterion

Remarks

The converse holds as well, since $q$ and $P_2$ are birational invariants and both vanish for $\mathbb{P}^2$. Castelnuovo proved it over $\C$, and Zariski extended it to positive characteristic. The proof passes to a minimal model and uses Riemann–Roch on the surface to show that $K_X$ is not nef. A minimal surface with this property is $\mathbb{P}^2$ or a ruled surface, and $q = 0$ makes the base curve of the ruling rational. The plurigenus $P_2$ cannot be replaced by the geometric genus $p_g = h^0(K_X)$: an Enriques surface has $q = p_g = 0$ but $P_2 = 1$, and it is not rational. Example: over $\C$ a unirational surface $X$ is dominated by $\mathbb{P}^2$. Pulling back forms shows that $q$ and $P_2$ of $X$ are at most those of $\mathbb{P}^2$, which vanish. Hence every complex unirational surface is rational.

Going deeper

For a student

The criterion is the key step in the classification of surfaces of Kodaira dimension $-\infty$. It shows that, for surfaces over $\C$, the Lüroth problem has a positive answer: unirational implies rational. This fails in dimension three, where there are unirational threefolds that are not rational, and it fails for surfaces in positive characteristic. The Enriques surface shows that the weaker condition $q = p_g = 0$ is not enough.