Let $C_1, C_2$ be two projective smooth curves over $\mathbb{C}$. Is it possible to say when $C_1 \times C_2$ a complete intersection in some projective space?
For three curves the answer is "it is never a complete intersection" since a product of three curves has too big Picard group and it would contradict the Lefschetz hyperplane theorem. However, for two curves it is possible: for example, if $C_1=C_2=\mathbb{P^1}$, then it is just a quadric. And, sometimes, it can't be a complete intersection, for example, if both $C_1$ and $C_2$ have genus 1.
Best Answer
You were on the right track with the Lefschetz Hyperplane Theorem. Here is a version for complete intersections, see Corollay $\mathrm{I}.20.5$ of Compact Complex Surfaces by Barth, Hulek, Peters, and Van de Ven.
A product of two curves has dimension $2$, so if it is a complete intersection, it must be simply connected. Therefore $\mathbb{CP}^1\times\mathbb{CP}^1$ is the only example.