# What 's the meaning of complete curve in Gieseker's book?

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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I'm reasonably sure it's just to indicate that $X$ is closed, so, not just quasiprojective. –  Allen Knutson Sep 9 at 13:55

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.