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  "body": "\n## Summary\n\nA line bundle is ample when its high tensor powers have enough global sections to embed the scheme into projective space.\nLet $X$ be a scheme of finite type over a Noetherian ring.\nA line bundle $L$ on $X$ is ample exactly when some power $L^{\\otimes m}$ is very ample.\nVery ample means that it is the pullback of $\\mathcal{O}(1)$ under an immersion into projective space.\nAmpleness is the algebraic form of positivity: it plays the role of positive curvature in complex geometry.\nMany vanishing, finiteness and embedding theorems assume an ample line bundle.\n\n## Explanations\n\n### L1 human\n\nA line bundle attaches a line to each point of a space, and its sections are like functions with values in these lines.\nA line bundle is ample if, after taking a high enough power, its sections can tell all points apart and all tangent directions apart.\nThen the sections give coordinates that place the space inside a projective space without folding it.\nOn the projective line, the bundle $\\mathcal{O}(1)$, whose sections are the linear forms, is ample.\nIts dual $\\mathcal{O}(-1)$ has no nonzero sections in any positive power, so it is not ample.\n",
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