# When does a smooth variety over $\mathbb{Q}_p$ have regular flat integral model?

For a smooth variety $$X$$ over $$\mathbb{Q}_p$$. is there always a scheme $$Y$$ over $$\mathbb{Z}_p$$ satisfying conditions (1) $$Y \otimes_{\mathbb{Z}_p}\mathbb{Q}_p\simeq X$$ (2) $$Y$$ is flat over $$\mathbb{Z}_p$$ and regular (3) the morphism $$Y\to \text{Spec}~\mathbb{Z}_p$$ is subjective ?

• "subjective" = "surjective"? :) Commented Sep 6, 2022 at 20:49

Yes, this follows from Nagata's compactification theorem in this generality as follows. If we consider the morphism $$X \to \operatorname{Spec} \mathbb{Z}_p$$, there exists a proper scheme $$\overline{Y} \to \operatorname{Spec} \mathbb{Z}_p$$ and an open immersion $$X \subset \overline{Y}$$ over $$\operatorname{Spec} \mathbb{Z}_p$$. When $$X$$ is quasi-projective, we can just take $$\overline{Y}$$ to be the closure inside $$\mathbb{P}^n_{\mathbb{Z}_p}$$. Up to replacing $$\overline{Y}$$ by its normalization, we can suppose that it is normal and thus regular in codimension $$1$$. Up to replacing $$\overline{Y}$$ with the closure of $$X$$, we can suppose that $$X$$ is a dense open. Thus $$\overline{Y} \to \operatorname{Spec} \mathbb{Z}_p$$ is proper, every component of $$\overline{Y}$$ dominates $$\operatorname{Spec} \mathbb{Z}_p$$ and $$\overline{Y}$$ is reduced so we conclude that $$\overline{Y} \to \operatorname{Spec} \mathbb{Z}_p$$ is flat and proper and in particular surjective.
By regularity in codimension $$1$$, we have a maximal open subset $$U \subset \overline{Y}$$ whose complement has codimension $$2$$ such that $$U$$ is regular and flat over $$\mathbb{Z}_p$$. By flatness, the fiber $$\overline{Y}|_0$$ over the closed point has codimension $$1$$ so $$U \cap \overline{Y}|_0 \neq \emptyset$$ and $$U \to \operatorname{Spec} \mathbb{Z}_p$$ is surjective. Since $$X$$ is smooth, $$X \subset U$$. Let $$Z = U_{\eta} \setminus X$$ where $$U_\eta$$ is the generic fiber of $$U$$ and let $$\overline{Z}$$ be the closure of $$Z$$ in $$U$$. Finally, let $$Y = U \setminus \overline{Z}$$. Then $$Y$$ does the job.
Pick a nonempty affine open $$U\subset X$$. Then $$U$$ is the generic fiber of some affine $$V$$, surjective and flat over $$\mathbb{Z}_p$$ (this is easy). We may assume $$V$$ normal, and then observe (as Dori does) that the regular locus $$W$$ of $$V$$ is surjective over $$\mathrm{Spec}(\mathbb{Z}_p)$$, and has $$U$$ as generic fiber. Since $$U$$ is open in both $$X$$ and $$W$$, we can glue $$X$$ and $$W$$ along $$U$$ to get a scheme $$Y$$ as desired. Note that by construction, the closed fibers of $$W$$ and $$Y$$ coincide.
I suspect that the OP has omitted the (reasonable) requirement that $$Y$$ be separated. Let me show that the above $$Y$$ is (equivalently, that $$Y\to\mathrm{Spec}(\mathbb{Z}_p)$$ is a separated morphism) , using the valuative criterion. So let $$S\to\mathrm{Spec}(\mathbb{Z}_p)$$ be the spectrum of a valuation ring, with generic point $$\eta$$ and closed point $$s$$. Let $$\alpha, \beta:S\rightrightarrows Y$$ be two $$\mathbb{Z}_p$$-morphisms with the same restriction to $$\eta$$. Now we have two cases:
• either $$S$$ sits above $$\mathrm{Spec}(\mathbb{Q}_p)$$ and then $$\alpha$$ and $$\beta$$ factor through $$U$$ which is separated, hence $$\alpha=\beta$$;
• or $$s$$ is above $$\mathrm{Spec}(\mathbb{F}_p)$$, and then $$\alpha(s)$$ and $$\beta(s)$$ are in the closed fiber of $$Y$$, wich is contained in $$W$$. This means that $$\alpha$$ and $$\beta$$ factor through $$W$$, because $$S$$ is local, and we conclude again that $$\alpha=\beta$$ because $$W$$ is separated.