The minimal genus of a non-trivial curve on A1 × A2 for very general abelian surfaces A1 and A2 is 6.
1 [HM98] Joe Harris and Ian Morrison.Moduli of curves, volume 187 ofGraduate Texts in Mathematics
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Curves on the product of two $K-$trivial surfaces
The minimal genus of a non-trivial curve on A1 × A2 for very general abelian surfaces A1 and A2 is 6.