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?



Line bundle ample iff induced morphism finite, looking for reference
Robert Lazarsfeld, *Positivity in Algebraic Geometry I*, Corollary 1.2.15 page 28.


