Lemma 29.39.4. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Assume $S$ affine and $f$ of finite type. The following are equivalent

$\mathcal{L}$ is ample on $X$,

$\mathcal{L}$ is $f$-ample,

$\mathcal{L}^{\otimes d}$ is $f$-very ample for some $d \geq 1$,

$\mathcal{L}^{\otimes d}$ is $f$-very ample for all $d \gg 1$,

for some $d \geq 1$ there exist $n \geq 1$ and an immersion $i : X \to \mathbf{P}^ n_ S$ such that $\mathcal{L}^{\otimes d} \cong i^*\mathcal{O}_{\mathbf{P}^ n_ S}(1)$, and

for all $d \gg 1$ there exist $n \geq 1$ and an immersion $i : X \to \mathbf{P}^ n_ S$ such that $\mathcal{L}^{\otimes d} \cong i^*\mathcal{O}_{\mathbf{P}^ n_ S}(1)$.

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