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ample
Idea: a notion of “positivity” for e.g. line bundles, related to having lots of global sections.
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\({\mathcal{L}}\) is basepoint free iff \({\mathcal{L}}\) admits enough sections to define a morphism to some \({\mathbf{P}}^N\).
- Equivalently, \({\mathcal{L}}\) is globally generated, i.e. ${\mathcal{L}}\in {}_{{\mathcal{O}}_X}{\mathsf{Mod}} $ admits a set of global sections \(\left\{{s_i \in {{\Gamma}\qty{X; {\mathcal{L}}} }}\right\}_{i\in I}\) such that \(\bigoplus_{i\in I} {\mathcal{O}}_X \twoheadrightarrow{\mathcal{L}}\).
- \({\mathcal{L}}\) is semi-ample iff some positive power \({\mathcal{L}}{ {}^{ \scriptscriptstyle\otimes_{k}^{n} } }\) is basepoint free (idea: non-negative).
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\({\mathcal{L}}\) is very ample iff \({\mathcal{L}}\) admits enough sections to define a closed immersion into some \({\mathbf{P}}^N\).
- Equivalently, \({\mathcal{L}}\) is basepoint free and the associated morphism \(f: X\to {\mathbf{P}}^N\) is a closed immersion.
- Equivalently, \(X\) can be embedded into some \({\mathbf{P}}^N\) such that \({\mathcal{L}}= { \left.{{ {\mathcal{O}}(1) }} \right|_{{X}} }\).
- \({\mathcal{L}}\) is ample iff some positive power \({\mathcal{L}}{ {}^{ \scriptscriptstyle\otimes_{k}^{n} } }\) is very ample.
Kleiman’s criteria: 
Pushforwards of ample divisors are nef: 