Question: Is it true that for every smooth compact complex curve $C$ there exists a smooth curve $C'$ in $\mathbb CP^2$ that admits a non-trivial morphism (i.e. holomorphic map) $C'\to C$?
Motivation. Unfortunately, I don't know yet any application for a positive answer to this question. But a negative answer to this question would solve in negative a great question of Francesco Polizzi: http://mathoverflow.net/questions/32938/surfaces-in-mathbbp3-with-isolated-singularities
Indeed, here is a simple exercise:
Exercise. Suppose that $C$ is a smooth curve that is not covered by any smooth plane curve. Then the surface $C\times \mathbb CP^1$ is not birational to any surface in $\mathbb CP^3$ with isolated singularities.
