Ample line bundle

In algebraic geometry, a very ample line bundle is one with enough global sections to set up an embedding of its base variety or manifold ${\displaystyle M}$ into projective space. An ample line bundle is one such that some positive power is very ample. Globally generated sheaves are those with enough sections to define a morphism to projective space.

Given a morphism ${\displaystyle f\colon X\to Y}$, any vector bundle ${\displaystyle {\mathcal {F}}}$ on Y, or more generally any sheaf in ${\displaystyle {\mathcal {O}}_{Y}}$ modules, e.g. a coherent sheaf, can be pulled back to X, (see Inverse image functor). This construction preserves the condition of being a line bundle, and more generally the rank.

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