In Chap 0, B) Hilbert point of a curve of http://www.math.tifr.res.in/~publ/ln/tifr69.pdf, He consider a "complete" curve $X\subset \mathbb{P}^N$. What does the "complete" mean? Does not "projective" imply "complete"? Can you give exact meaning?
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1$\begingroup$ I'm reasonably sure it's just to indicate that $X$ is closed, so, not just quasiprojective. $\endgroup$– Allen KnutsonCommented Sep 9, 2013 at 13:55
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2 Answers
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Projectivity (over complex numbers) implies completeness of course, but $X\subseteq\mathbb{P}^{N}$ does not imply that $X$ is projective. For we have $\mathbb{A}^{1} \subseteq \mathbb{P}^{1}$. But $\mathbb{A}^{1}$ is neither projective nor complete. So his meaning is that $X$ is closed and hence projective and complete.
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In the case of curves, complete does imply projective, see for example Exercise III.5.8 of Hartshorne. But in general completeness means that your scheme is proper over the base field, i.e. it's separated, of finite type, and universally closed over the base field (see the definition on p. 100 of Hartshorne). Projective means that it's a closed subscheme of some projective space over the base field. As Darius Math pointed out, projectivity implies completeness.
As I said though, in the case of curves there is no difference.
