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!

Assuming the curve is nonsingular, this is Proposition II.6.7 in Hartshorne, Algebraic Geometry. — User3773, Jan 31 '18 at 12:37
The only assumption should be being proper over the field $k$. — windsheaf, Jan 31 '18 at 13:08
Pass to the normalisation, which is projective, and pull back the polarisation. The pull back is still ample over the starting curve, which is then projective. — User3773, Jan 31 '18 at 13:32
@Cla Thank you, but I'm still looking for a reference, which I could cite. — windsheaf, Jan 31 '18 at 13:56

score 4 · Accepted Answer · answered Feb 10 '18 at 12:05

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 $\operatorname{Spec(k)}$).

score 1 · Answer 2 · answered Dec 25 '21 at 13:15

1

See Proposition 7.4.9 in EGAII.

answered Dec 25 '21 at 13:15

Bacon Lee

11

Proper curves over some field are projective

2 Answers2

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