Exceptional curve of the first kind

An exceptional curve of the first kind is a smooth rational curve $E$ on a smooth surface with $E^2=-1$. It can be contracted to a smooth point.

This is precisely the curve contracted by the Castelnuovo contraction criterion.

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

Minimality here is the smooth-surface condition, rather than a claim about every higher-dimensional minimal model.

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.

Let $C$ be a smooth connected ample curve in an abelian surface $A$. Translate its chosen basepoint to zero and apply the universal property of the Jacobian variety. The image of $J(C)\to A$ is a closed connected algebraic subgroup containing the translated curve. If it had dimension one, that curve would be the subgroup itself; a distinct coset is disjoint and numerically equivalent, forcing $C^2=0$. This contradicts ampleness, so the homomorphism is surjective.

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.