# Is a product of curves a complete intersection?

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.

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.