Ample line bundle

Summary

A line bundle is ample when its high tensor powers have enough global sections to embed the scheme into projective space. Let $X$ be a scheme of finite type over a Noetherian ring. A line bundle $L$ on $X$ is ample exactly when some power $L^{\otimes m}$ is very ample. Very ample means that it is the pullback of $\mathcal{O}(1)$ under an immersion into projective space. Ampleness is the algebraic form of positivity: it plays the role of positive curvature in complex geometry. Many vanishing, finiteness and embedding theorems assume an ample line bundle.