In Cox, Little and Schenck's book **Toric Varieties** they show that a toric variety $ X\_{\Sigma} $ over a field of characteristic zero is complete if and only if the support is all of $ N\_{\mathbb{R}} $. This proof was very specific to varieties over fields of characteristic zero. When I looked at Oda and Fulton's books they seemed to work over characteristic zero as well.


Is a toric variety $ X\_{\Sigma} $ over an algebraically closed field $ k $ of characteristic $ p>0 $ complete if and only if the support of $ \Sigma $ is all of $ N\_{\mathbb{R}} $? I believe the answer should be yes for the following reason, but if anyone could give me a reference, then I would appreciate it.


Let $ R $ be a DVR of mixed characteristic with special fibre $ k $ and generic fibre $ L $. Also, let us define a scheme $ \mathcal{X}\_{\Sigma} $ over $ \operatorname{Spec}(R) $ to be
\begin{equation\*}
 \mathcal{X}\_{\Sigma} =\left(\coprod\_{\sigma \in \Sigma} \operatorname{Spec}(R[\sigma^{\vee} \cap M])\right)/\sim.
\end{equation\*}
We claim that the scheme $ \mathcal{X}\_{\Sigma} $ is faithfully flat over $ \operatorname{Spec}(R) $. The claim is local, so it suffices to prove that $ \operatorname{Spec}(R[\sigma^{\vee} \cap M]) $ is faithfully flat over $ \operatorname{Spec}(R) $. Since $ R[\sigma^{\vee} \cap M] $ is an $ R $-algebra, the claim follows.


If $ A $ is a DVR over $ k $, then let $ \operatorname{Spec}(B) $ be a faithfully flat, family of DVRs over $ \operatorname{Spec}(R) $, such that the special fibre is $ \operatorname{Spec}(A) $.


A normal, toric variety $ X\_{\Sigma} $ over a field $ L $ of characteristic zero is complete if and only if the support of $ \Sigma $ is all of $ N\_{\mathbb{R}} $. So since $ \operatorname{Spec}(B) $ is faithfully flat over $ \operatorname{Spec}(R) $, the valuative criterion of properness will hold for the morphism $ \mathcal{X}\_{\Sigma} \times\_{\operatorname{Spec}(R)} \operatorname{Spec}(L) \to \operatorname{Spec}(L) $ if and only if it holds for the morphism $ X\_{\Sigma} \to \operatorname{Spec}(k) $.



Is a toric variety over a field of positive characteristic complete if and only if the support is all of $ N_{\mathbb{R}} $?
The answer is yes. Fulton's proof is in fact valid positive characteristic as well. In the proof of "full support $\Rightarrow$ completeness" he uses the valuative criterion which, as you noticed, is true for all characteristic. For the opposite direction you only need to replace the zero characteristic language from Fulton's proof: if there is $v \in N$ which is not in any cone of $\Sigma$, then consider the corresponding one parameter subgroup $\lambda\_v$. If you choose a system of coordinates on $N$ so that $v = (v\_1, \ldots, v\_n)$, then $\lambda\_v$ is simply the map $k^\* \to X$ given by $t \mapsto (t^{v\_1}, \ldots, t^{v\_n})$. If $X$ were complete, then $\lambda\_v$ would extend to a map from $k \to X$. However, for every cone $\sigma \in \Sigma$, since $v \not\in \sigma$, there is $u\in \check{\sigma}$ such that $\langle u, v \rangle < 0$. Then $\chi^u$ is not defined at $0$, so that $0$ can not be mapped to anywhere on the open affine subset of $X$ corresponding to $\sigma$. Since $X$ is a union of these subsets, it follows that $X$ is not complete, as required.


