Complex curves covered by smooth plane curves

2013-01-06T21:17:52Z

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.

Answer by Michael Zieve for Complex curves covered by smooth plane curves

2013-01-06T23:05:59Z

Edit: I made a mistake, my attempted answer is wrong. It was just the same as @Felipe's comment, namely, that Jason Starr's answer to this question might apply here too. But it doesn't. Sorry for the mistake.

Complex curves covered by smooth plane curves - MathOverflow

Complex curves covered by smooth plane curves

Answer by Michael Zieve for Complex curves covered by smooth plane curves