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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.

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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.

Let $Y \subset \mathbb{CP}^N$ is a smooth complete intersection of dimension $m \geq 1$. Then the inclusion homomorphisms \begin{align*} H_i(Y; \mathbb{Z}) &\to H_i(\mathbb{CP}^N; \mathbb{Z})\\ \pi_i(Y; \mathbb{Z}) &\to \pi_i(\mathbb{CP}^N) \end{align*} are isomorphisms for $0 \leq i \leq m-1$. In particular, such a complete intersection is connected and if its dimension is at least $2$, also simply connected.

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.

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