irreducible smooth proper one-dimensional $\mathbb{Z}$-schemes

Question

Is there an irreducible smooth proper one-dimensional $\mathbb{Z}$-scheme not isomorphic to projective line?

Answer 1

The answer is no.

Let $X$ be an irreducible scheme and let $X\to \mathrm{Spec} \ \mathbb{Z} = S$ be a smooth proper morphism of schemes. Let $X\to T\to S$ be the Stein factorization of $X\to S$. Note that $T$ is irreducible and that $T\to S$ is smooth proper and finite. It follows from Hermite-Minkowski's theorem that $T =S$ (as $\mathrm{Spec} \ \mathbb{Z}$ is simply connected). Thus, $X\to S$ has geometrically connected fibres. Let $C := X_{\mathbb{Q}}$ be the generic fibre of $X\to \mathrm{Spec} \ \mathbb{Z}$. Then $C$ is a smooth proper geometrically connected curve over $\mathbb{Q}$ with good reduction everywhere. In particular, its Jacobian $Jac(C)$ has good reduction everywhere. By the theorem of Abrashkin-Fontaine (see https://link.springer.com/article/10.1007%2FBF01388584) it follows that $Jac(C_{\mathbb{Q}})=0$, so that $C_{\mathbb{Q}}$ is a smooth proper geometrically connected curve of genus zero. I leave it to you to conclude that $C_{\mathbb{Q}} = \mathbb{P}^1_{\mathbb{Q}}$ and that $X = \mathbb{P}^1_{\mathbb{Z}}$. (Hint: use that $X(\mathbb{F}_p)$ is non-empty for every $p$.)