Let $X$ be a proper scheme over a field $\Bbbk$. Let $\mathscr L\in\mathrm{Pic}(X)$ be a globally generated line bundle. If for some choice of global sections $V\subseteq\mathscr L(X)$, the induced morphism $\phi_V:X\to\mathbb P(V)$ is finite, then $\mathscr L$ is ample. The converse also holds. However, I have searched through Görtz-Wedhorn, Hartshorne, Liu, Mumford's red book, the stacks project and also EGA II without finding a reference for this statement. Does anyone know any?
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1$\begingroup$ One reason you won't find it in EGA II is because the proof rests on the fact that proper quasi-finite maps are finite, which wasn't proved until later volumes. In effect, the "shadow" of this result which you will find in EGA II is the fact that ampleness of the structure sheaf (for a separated finite type scheme over a ring) is among several equivalent definitions of "quasi-affineness" as developed in 5.1 of EGA II (this fact is applied to the fibers of $\phi_V$, which are then proper and quasi-affine, hence finite, etc.). $\endgroup$– MarguaxCommented Oct 22, 2013 at 15:37
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$\begingroup$ @Marguax: That makes perfect sense, thanks for the information. Can I find the statement in some later volume of EGA by any chance? $\endgroup$– Jesko HüttenhainCommented Oct 22, 2013 at 16:01
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$\begingroup$ I am pretty sure that it is not in later volumes (by which time they had bigger fish to fry than the basic lemmas about ampleness). $\endgroup$– MarguaxCommented Oct 22, 2013 at 16:53
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1 Answer
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Robert Lazarsfeld, Positivity in Algebraic Geometry I, Corollary 1.2.15 page 28.
answered Oct 22, 2013 at 8:52
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$\begingroup$ Lightning fast. Thanks! I am only allowed to accept in 2 minutes. $\endgroup$ Commented Oct 22, 2013 at 8:57