(Very ample)
Let \(f:X\to Y\) be a morphism of schemes and let \(\mcl\) be an invertible sheaf on \(X\). It is very ample over \(Y\) if there is a \(Y\)-immersion \(\iota:X\to\PP_Y^n\), for some finite \(n\), with \(\iota^*\OO(1)\cong\mcl\) [@Har10a, Chapter II, §7].
(Ample)
Let \(X\) be a noetherian scheme and let \(\mcl\) be an invertible sheaf on \(X\). It is ample if for every coherent \(\mcf\) there is an integer \(N_0\) such that \(\mcf\tensor\mcl^{\otimes n}\) is globally generated for every \(n\ge N_0\) [@Har10a, Chapter II, §7].
Let \(X\) be of finite type over a noetherian ring \(A\) and let \(\mcl\) be invertible. Then \(\mcl\) is ample if and only if \(\mcl^{\otimes m}\) is very ample over \(\Spec A\) for some \(m>0\) [@Har10a, Theorem II.7.6].
If \(f:X\to Y\) is proper and \(\mcl\) is very ample over \(Y\), its immersion into \(\PP_Y^n\) is proper, hence a closed immersion. Thus \(f\) is projective in the sense of admitting a closed immersion into some \(\PP_Y^n\). Conversely, such a closed immersion gives a very ample sheaf by pulling back \(\OO(1)\) [@Har10a, Chapter II, §§4 and 7].
(Ampleness on a complete curve)
Let \(X\) be a complete nonsingular curve over an algebraically closed field, and let \(\mcl\) be an invertible sheaf. Then \(\mcl\) is ample if and only if \(\deg\mcl>0\) [@Har10a, Corollary IV.3.3].