I have a curious question about an argument/hint given in following thread:

The OP asked if there an irreducible smooth proper one-dimensional $\mathbb{Z}$-scheme that is not isomorphic to the projective line.

In the answer Ariyan Javanpeykar showed that the generic fiber $C := X_{\mathbb{Q}}:= X \times_{Spec\mathbb{Z}} Spec(\mathbb{Q})$ of the morphism$X \to Spec(\mathbb{Z}) $ is a ** smooth proper geometrically connected curve of genus zero**.

It stays to show that this implies $C_{\mathbb{Q}} =\mathbb{P}^1_{\mathbb{Q}}$ from what we can deduce $X = \mathbb{P}^1_{\mathbb{Z}}$. According to the hint one have to use that the sets of $\mathbb{F}_p$-valued points $X(\mathbb{F}_p)$ are non empty for all $p$.

Could anybody explain in what way this hint with non empty $X(\mathbb{F}_p)$ has to be used to acquire desired result?

My considerations base essensially on Ravi Vakhil's https://math.stanford.edu/~vakil/0708-216/216class41.pdf

using the "curve to projective" extension theorem (2.1; p. 5) and the embedding theorem for non singular curves (Thm 0.2; p. 7):

I think that for $C_{\mathbb{Q}} =\mathbb{P}^1_{\mathbb{Q}}$ as well for $X = \mathbb{P}^1_{\mathbb{Z}}$ we have essential problem is to find "comparing" morphisms $f: C \to \mathbb{P}^1$; the candidates for an isomorphism.

Firslty regarding $C_{\mathbb{Q}} =\mathbb{P}^1_{\mathbb{Q}}$:

Thm. 0.2 provides an open immersion $f: C_{\mathbb{Q}} \to \mathbb{P}^1_{\mathbb{Q}}$ which is proper, so closed and since both irreducible, a homeomorphism. I think that I can use a genus argument (interpreting it as dimension of first cohomology of $C$) on the resulting sequnce of the induced morphisms between structure sheaves to verify that $f$ is already an isomorphism of schemes.

I think that similar combination of thms 0.2 and 2.1 would provide an immersion $g:X \to \mathbb{P}^1_{\mathbb{Z}}$ and the verification this this is an isomorphism on the level of structure sheaves should work similar as in case above using genus property on resulting sheaf cohomology.

QUESTION:

The point that I really curious about is what argument is meant by the observation that the $\mathbb{F}_p$-valued points $X(\mathbb{F}_p)$ aren't empty?

How does it help here? Where does it flow in for the argumentation?

2more comments