Today in a lecture it was claimed
Proposition Let $X$ be a proper scheme of finite type and dimension $1$ over a field $k$. Then $X$ is projective.
So, I already knew the above statement for integral curves - and I can see how to drop the condition of being reduced. However, I don't see how to reduce to the case where $X$ is irreducible. The proof of the Proposition was only sketched and this step was actually missing, so I wonder, how one might fix that (and, in particular, if the statement as given holds true).
On further note, what can one say in higher dimensions - is it true that (for finite type $k$-schemes), projectivity can be checked on irreducible components? If not, is there a handy counterexample?
My first attempt in proving such a statement would be looking at the Picard groups of $X$ and its normalization, and then try to argue using ample line bundles. Looking for counterexamples, I am at first sight rather hopeless (because if I recall correctly, counterexamples for proper varieties that are not projective are rather complicated in any case).