Completion of real fans and Zariski-Riemann spaces

Abstract

Given a real fan in a real space consisting of real convex polyhedral cones, we construct a complete real fan which contains the fan, by two completely different methods. The first one is purely combinatorial and a proof of a related version was sketched earlier by Ewald. The second one is based on Nagata's method of imbedding an abstract variety into a complete variety. For the second method, we introduce the theory of Zariski-Riemann space of a fan.

Support of Laurent series algebraic over the field of formal power series

“…Since τ 1 (ξ) is open and disjoint from τ 0 (ξ) then ω ∈ R >0 n \(τ 0 (ξ) ∪ τ 1 (ξ)). Since L is generated by a vector with Q-linearly independent coordinates we have that dim Q (Qω 1 + · · · + Qω n ) n − 1 ( 5 ) and this inequality is in fact an equality by Remark 5.14. By Theorem 5.13 there exists a Laurent polynomial p(x) ∈ K[x] x1···xn such that # (Supp(In ω (ξ + p))) = ∞.…”

“…By Corollary 5.15 there exist ω ∈ R >0 n \(τ 0 (ξ) ∪ τ 1 (ξ)), a Laurent polynomial p and a vector γ ∈ Z n such that Supp(ξ + p) ⊂ γ + ω ∨ and Supp(ξ + p) ∩ (γ + ω ∨ ) ⊂ γ + L is infinite where L is a half-line ending at the origin. Moreover we may assume that dim Q (Qω 1 + · · · + Qω n ) = n − 1 as shown in the proof of Corollary 5.15 (see (5)). We can also assume that none of the monomials of p lie on γ + L.…”

Completions of fans

“…Third, tight separability strengthens separability by ensuring that Σ is on one hand independent of Σ (that is, separable), but on the other hand not too much bigger than Σ (in some topological sense). It is at this point where in [4] the metric ε-arguments enter the scene.…”

“…Around the same time, Ewald stated in [2] the existence of completions of fans without proof, but in a way that suggests a direct and constructive proof. In 1996 he indeed sketched such a proof in his textbook on combinatorial convexity [3,III.2.8], and finally ten years later Ewald and Ishida published in [4] a refined version of this construction. (In loc.cit.…”

Nagata’s Compactification Theorem for Normal Toric Varieties over a Valuation Ring of Rank One

“…More precisely, it has been proved that every rational fan can be completed, see for instance [2, III. Theorem 2.8], [3] and [11]. This gives rise to an equivariant open embedding of the original variety into a complete normal toric variety.…”