Normal completions of toric varieties over rank one valuation rings and completions of $Γ$-admissible fans

高阶Gelfand-Kapranov-Zelevinsky扇形

Higher rank Gelfand-Kapranov-Zelevinsky fans

本文研究高阶GKZ扇形，通过点配置构造高阶拟估值，将相关紧致簇平坦退化为减少的紧致簇联合体，编码点配置的多面体细分。

Classification of normal toric varieties over a valuation ring of rank one

Normal toric varieties over a field or a discrete valuation ring are classified by rational polyhedral fans. We generalize this classification to normal toric varieties over an arbitrary valuation ring of rank one. The proof is based on a generalization of Sumihiro's theorem to this non-noetherian setting. These toric varieties play an important role for tropicalizations.

有理二重点奇点的Gröbner扇与最小嵌入环面消解

Gröbner Fans and Minimal Embedded Toric Resolutions of Rational Double Point Singularities

本文通过显式构造ADE奇点的Gröbner扇并利用profile概念进行加细，重构了有理二重点奇点的嵌入环面消解。

Toric flat families, valuations, and applications to projectivized toric vector bundles

Using the notion of a valuation into the semifield of piecewise linear functions, we give a classification of torus equivariant flat families of finite type over a toric variety base, by certain piecewise linear maps between fans. As a consequence we derive a classification of toric vector bundles phrased in terms of tropicalized linear spaces. We use these tools to give a characterization of the Mori dream space property for a projectivized toric vector bundle.

Nombre de composantes connexes d'une variété réelle et R-places

The purpose of this paper is to present results and open problems related to R-places. The first section recalls basic facts, the second introduces R-places and their relationship with orderings and valuations. The third part involves Real Algebraic Geometry and gives results proved using the space of R-places. Theorem 14 gives explicitly, in terms of the function field of the variety, the number of connected components of a non-empty smooth projective real variety. The fourth and fifth parts are devoted to the links with the real holomorphy rings and the valuation fans. Then we present an app…

Local Tropicalization

In this paper we propose a general functorial definition of the operation of \emph{local tropicalization} in commutative algebra. Let $R$ be a commutative ring, $Γ$ a finitely generated subsemigroup of a lattice, $γ: Γ\rightarrow R/ R^*$ a morphism of semigroups, and $\V(R)$ the topological space of valuations on $R$ taking values in $\R \cup \infty$. Then we may \emph{tropicalize} with respect to $γ$ any subset $\W$ of the space of valuations $\V(R)$. By definition, we get a subset of a rational polyhedral cone canonically associated to $Γ$, enriched with strata at infinity. In particular, wh…