= Curve generation criterion for an abelian variety
A smooth projective curve $C\subset A$ generates $A$ by its point differences if and only if it meets every prime <Weil divisor>. If a prime <Weil divisor> avoids $C$, its restriction to $C$ has degree zero; <constancy of line bundle degree in a family> shows that every translate either contains $C$ or misses it. This forces invariance under $C-C$ and puts the generated subgroup inside the <line bundle translation stabilizer>. If this subgroup were all of $A$, the divisor would be algebraically trivial, contradicting its positive intersection with an <ample line bundle>. Conversely a proper closed generated subgroup admits a disjoint divisor by the <pole divisor avoiding a fiber> construction, after translating the fiber to contain $C$.
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