Abelian surface

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.