Minimal algebraic surface

A smooth projective surface is minimal when it contains no exceptional curve of the first kind.

An abelian surface is a two-dimensional abelian variety. It contains no rational curve: a morphism from ${\mathbb{P}}^1$ to an abelian variety is constant. Consequently every abelian surface is a minimal algebraic surface.

The Hirzebruch surface ${\mathbb{F}}_n$ is minimal for $n=0$ and every $n\ge 2$; ${\mathbb{F}}_1$ is not minimal because its negative section is a $(-1)$-curve. For $n>0$, the negative section is the unique irreducible curve of negative self-intersection, so its self-intersection $-n$ distinguishes the isomorphism class. The surfaces ${\mathbb{F}}_n$ for $n\ge 2$ therefore give infinitely many pairwise nonisomorphic minimal rational surfaces.