Ample line bundle

A line bundle $L$ is ample when some positive tensor power $L^{\otimes m}$ is very ample.

On a projective scheme, a line bundle is ample exactly when its restriction to every reduced irreducible component is ample. The disjoint union of those components maps finitely and surjectively to the scheme, so finite-surjective descent of the finite pullback of an ample line bundle proves this. The same argument applies to restrictions to any closed subscheme. Thus nilpotent structure does not affect ampleness.

For a finite morphism $f:X\to Y$, the pullback of an ample line bundle is ample. If $f$ is also surjective, ampleness is equivalent in both directions. These results hold for proper schemes over a field, including nonreduced schemes. Cohomological ampleness and the exact finite pushforward prove the forward direction; coherent-sheaf devissage gives finite-surjective descent. In particular, a finite map to projective space makes the pulled-back hyperplane bundle ample.

If $A$ is an ample line bundle on a projective variety and $L$ is any line bundle, then $A^{\otimes m}\otimes L$ is very ample for all sufficiently large integers $m$. Taking $L$ to be trivial also makes $A^{\otimes m}$ very ample.