to f or f-ample if the following equivalent conditions are met: For each open affine subset , the restriction of L to is ample (in the usual sense). f is quasi-separated and there is an open immersion induced by the adjunction map: . The condition 2. without "open". The condition 2 says (roughly) that X can be openly compactified to a projective scheme with (not just to a proper scheme). See also General algebraic geometry Algebraic geometry of projective spaces Fano variety: a variety whose canonical bundle is anti-ample Matsusaka's big theorem Divisorial scheme: a scheme admitting an ample family of line bundles Ampleness in complex geometry Holomorphic vector bundle Kodaira embedding theorem: on a compact complex manifold, ampleness and positivity coincide. Kodaira vanishing theorem Lefschetz hyperplane theorem: an ample divisor in a complex projective variety X is topologically similar to X. Notes Sources . . . . . . External links The Stacks Project Algebraic geometry Geometry of divisors Vector bundles