Suppose $X$ is a smooth projective curve over an algebraically closed field $k$.

Is the morphism $ X \to \operatorname{Spec}(k) $ necessarily flat?

What kind of conditions on the above morphism are equivalent to the hypothesis of $X$ being smooth and projective?

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    $\begingroup$ To answer your second question, $X$ is smooth and projective iff the morphism is smooth and projective. I'm not sure what else you could expect. $\endgroup$ – user64687 Feb 3 '14 at 14:02

Over a field every module (hence every algebra, and hence every scheme) is flat.

By the way, this is a typical situation where various assumptions (smooth, projective, curve, algebraically closed) make the obvious hard to see ...

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    $\begingroup$ Yes thanks Martin for reminding me about this trivial fact: Modules over a field are nothing else than a vector spaces, and these are of course free hence flat. $\endgroup$ – Abramo Feb 3 '14 at 11:57

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