Proper curves over some field are projective

Question

I'm looking for a reference of the statement

Let $X$ be a proper curve (scheme of dimension one) over the field $k$. Then $X$ is projective.

There is a some kind of guided exercise in Liu's Algebraic Geometry and Arithmetic Curves (Ex. 7.5.4).

Does someone know a reference? Thank you very much in advance!

Answer 1

There is a reference in the Stacks-Project:

We have the statement with the tag 0A26, which says:

Let $X$ be a proper scheme over a field $k$. If $\dim(X) \leq 1$, then $X$ is H-projective over $k$.

Now use tag 0B45, which says that being H-projective and projective is equivalent if the base scheme has an ample invertible sheaf (true for the base $\operatorname{Spec(k)}$).

Answer 2

See Proposition 7.4.9 in EGAII.