相关论文: Valuations, Deformations, and Toric Geometry

Let $R$ be a complete equicharacteristic noetherian local domain and $\nu$ a valuation of its field of fractions whose valuation ring dominates $R$ with trivial residue field extension. The semigroup of values of $\nu$ on $R\setminus \{0\}$…

Let (R; m; k) be a local noetherian domain with field of fractions K and R_v a valuation ring, dominating R (not necessarily birationally). Let v|K be the restriction of v to K; by definition, v|K is centered at R. Let \hat{R} denote the…

We study Abhyankar valuations of excellent equicharacteristic local domains with an algebraically closed residue field. For zero dimensional valuations we prove that whenever the ring is complete and the semigroup of values taken by the…

Let $R$ be a complete equicharacteristic noetherian local domain with an algebraically closed residue field $k$. Let $\nu$ be a zero dimensional valuation of rank one centered in $R$ with value group $\Phi$. We show that there is a…

We show that there exists a strong connection between the generic formal neighborhood at a rational arc lying in the Nash set associated with a toric divisorial valuation on a toric variety and the formal neighborhood at the generic point…

Let k be a field. This paper investigates the embedding dimension and codimension of Noetherian local rings arising as localizations of tensor products of k-algebras. We use results and techniques from prime spectra and dimension theory to…

We examine local cohomology in the setting of valuation rings. The novelty of this investigation stems from the fact that valuation rings are usually non-Noetherian, whereas local cohomology has been extensively developed mostly in a…

In this thesis we study toric degenerations of projective varieties. We compare different constructions to understand how and why they are related as s first step towards developing a global framework. In focus are toric degenerations…

This note is supposed to be an introduction to those concepts of toric geometry that are necessary to understand applications in the context of string and F-theory dualities. The presentation is based on the definition of a toric variety in…

The main goal of this paper is to study the structure of the graded algebra associated to a valuation. More specifically, we prove that the associated graded algebra ${\rm gr}_v(R)$ of a subring $(R,\mathfrak{m})$ of a valuation ring…

Let R be a complete discrete valuation ring of mixed characteristics, with algebraically closed residue field k. We study the existence problem of equivariant liftings to R of Galois covers of nodal curves over k. Using formal geometry, we…

This note presents two observations which have in common that they lie at the boundary of toric geometry. The first one because it concerns the deformation of affine toric varieties into non toric germs in order to understand how to avoid…

We study deformations of affine toric varieties. The entire deformation theory of these singularities is encoded by the so-called versal deformation. The main goal of our paper is to construct the homogeneous part of some degree -R of this,…

In this paper, an algebraic theory for local rings of finite embedding dimension is developed. Several extensions of (Krull) dimension are proposed, which are then used to generalize singularity notions from commutative algebra. Finally,…

Let $R$ be the coordinate ring of an affine toric variety. We show that the endomorphism ring $End_R(\mathbb A),$ where $\mathbb A$ is the (finite) direct sum of all (isomorphism classes of) conic $R$-modules, has finite global dimension.…

We will demonstrate how calculations in toric geometry can be used to compute quantum corrections to the relations in the chiral ring for certain gauge theories. We focus on the gauge theory of the del Pezzo 2, and derive the chiral ring…

Let k be a field of characteristic zero, K an algebraic function field over k, and V a k-valuation ring of K. Zariski's theorem of local uniformization shows that there exist algebraic regular local rings R_i with quotient field K which are…

We develop the notion of deformations using a valuation ring as ring of coefficients. This permits to consider in particular the classical Gerstenhaber deformations of associative or Lie algebras as infinitesimal deformations and to solve…

In this paper we discuss stable forms of extensions of algebraic local rings along a valuation in all dimensions over a field k of characteristic zero, and generalize a formula of Ghezzi, H\`a and Kashcheyeva describing the extension of…

This text, which appeared in volume 2313 of the Springer Lecture Notes in Math. dedicated to Catriona Byrne, presents problems in two different areas of algebraic geometry. The first concerns the role of ``infinitely singular'' or ``non…