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) $.

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    $\begingroup$ Often it is good to read the masters. The treatise of Kempf-Mumford-Knudsen-Saint-Donat already treats toric varieties over algebraically closed fields of arbitrary characteristic. $\endgroup$ Commented May 2, 2023 at 5:34

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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.


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