Degenerations of toric varieties over valuation rings (Q2830656)

This paper investigates degenerations of toric varieties over valuation rings of finite rank generalizing the works of \textit{D. Mumford} [Compos. Math. 24, 239--272 (1972; Zbl 0241.14020)] and \textit{W. Gubler} [Contemp. Math. 589, 125--189 (2013; Zbl 1318.14061)].NEWLINENEWLINELet \(\nu : K^{\times} \to \mathbb{R}^{(k)}\) be a valuation on field \(K\) whose valuation ring is \(R\) and let \(\Gamma := \nu(K^{\times}) \subset \mathbb{R}^{(k)}\) be its value group. The authors explain how the induces order-preserving embedding \(\Gamma \hookrightarrow \mathbb{R}^{(k)}\) gives rise to a tower of definable subsets \(\mathcal{E}_0 \supset \mathcal{E}_1 \supset \dots \supset \mathcal{E}_n\) in \(\mathbb{R}^{(k)}\). Given a \(\mathbb{Z}\)-lattice \(M\), the main contribution of the paper is to define complete \(\Gamma\)-admissible fans \(\Sigma\) inside \(\text{Hom}_{\mathbb{Z}}(M, \mathbb{R}^{(k)}) \times \mathcal{E}_0\), provide a polyhedral framework for them, and then associate flat proper \(R\)-schemes \(\mathscr{Y}(\Sigma)\) to such fans. They also connect the geometric properties of \(\mathscr{Y}(\Sigma)\) to the combinatorial properties of \(\Sigma\), for instance, they show the generic fiber \(\mathscr{Y}(\Sigma)_K\) of \(\mathscr{Y}(\Sigma)\) is the toric \(K\)-variety associated to some complex in \(\Sigma\).