Complex curves covered by smooth plane curves - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T14:58:45Z http://mathoverflow.net/feeds/question/118224 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118224/complex-curves-covered-by-smooth-plane-curves Complex curves covered by smooth plane curves aglearner 2013-01-06T21:17:52Z 2013-01-09T11:17:16Z <p><strong>Question:</strong> 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) $C'\to C$?</p> <p><strong>Motivation.</strong> 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: <a href="http://mathoverflow.net/questions/32938/surfaces-in-mathbbp3-with-isolated-singularities" rel="nofollow">http://mathoverflow.net/questions/32938/surfaces-in-mathbbp3-with-isolated-singularities</a></p> <p>Indeed, here is a simple exercise: </p> <p><strong>Exercise.</strong> 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.</p> http://mathoverflow.net/questions/118224/complex-curves-covered-by-smooth-plane-curves/118232#118232 Answer by Michael Zieve for Complex curves covered by smooth plane curves Michael Zieve 2013-01-06T23:05:59Z 2013-01-09T11:17:16Z <p>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 <a href="http://mathoverflow.net/questions/118117/can-every-curve-be-written-as-fxgy" rel="nofollow">this question</a> might apply here too. But it doesn't. Sorry for the mistake.</p>