= Ample curve on an abelian surface generates the 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.
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