The Stacks project

Remark 33.38.13. In fact, if $X$ is a scheme whose reduction is a Noetherian separated scheme of dimension $1$, then $X$ has an ample invertible sheaf. The argument to prove this is the same as the proof of Proposition 33.38.12 except one uses Limits, Lemma 32.11.4 instead of Cohomology of Schemes, Lemma 30.17.5.