Related papers: Toric Geometry and the Semple-Nash modification
The higher Nash blowup of an algebraic variety replaces singular points with limits of certain spaces carrying higher order data associated to the variety at non-singular points. In the case of normal toric varieties we give a combinatorial…
We initiate the study of the resolution of singularities properties of Nash blowups over fields of prime characteristic. We prove that the iteration of normalized Nash blowups desingularizes normal toric surfaces. We also introduce a prime…
We show that the $ n $-th Nash blowup of the toric surface singularity of type $ A_3 $ is singular for any $ n > 0 $. It was known that the normalization of the $ n $-th Nash blowup of a toric variety is also a toric variety associated to…
We prove that the higher Nash blowup of a normal toric variety defined over a field of positive characteristic is an isomorphism if and only if it is non-singular. We also extend a result by R. Toh-Yama which shows that higher Nash blowups…
We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group ${\rm SL}_2({\mathbb Z})$ to its preimage in the universal cover of ${\rm SL}_2({\mathbb R})$. With this…
It has been recently shown that the iteration of Nash modification on not necessarily normal toric varieties corresponds to a purely combinatorial algorithm on the generators of the semigroup associated to the toric variety. We will show…
We show that iterating Nash blowups resolve the singularities of normal toric surfaces satisfying the following property: the minimal generating set of the corresponding semigroup is contained in one or two segments. We also provide…
The Nash blowing-up (or modification) of an algebraic variety $X$ is a canonical process that produces a proper, birational morphism $\pi : X' \to X$ of varieties. It is expected that the singularities of $X'$ will be better than those of…
We classify smooth toric Fano varieties of dimension $n\geq 3$ containing a toric divisor isomorphic to $\PP^{n-1}$. As a consequence of this classification, we show that any smooth complete toric variety $X$ of dimension $n\geq 3$ with a…
We are interested in two classes of varieties with group action, namely toric varieties and spherical embeddings. They are classified by combinatorial objects, called fans in the toric setting, and colored fans in the spherical setting. We…
It is a long-standing question whether an arbitrary variety is desingularized by finitely many normalized Nash blow-ups. We consider this question in the case of a toric variety. We interpret the normalized Nash blow-up in polyhedral terms,…
We construct and study noncommutative deformations of toric varieties by combining techniques from toric geometry, isospectral deformations, and noncommutative geometry in braided monoidal categories. Our approach utilizes the same fan…
In this paper we describe the implementation that led to the counterexamples to the Nash blowup conjectures recently discovered by the authors. We also provide new examples of toric varieties with prescribed singularities that are not…
We introduce conditions on cones of normal toric varieties under which the polyhedron defining the normalized Nash blowup does not depend on the characteristic of the base field. As a consequence, we deduce several results on the resolution…
We present a new method to achieve an embedded desingularization of a toric variety. Let $W$ be a regular toric variety defined by a fan $\Sigma$ and $X\subset W$ be a toric embedding. We construct a finite sequence of combinatorial…
In this paper, we provide a combinatorial description of seminormal toric varieties. The corresponding combinatorial object is a fan equipped with a collection of groups assigned to each cone. This framework introduces a more general class…
Classical toric varieties are among the simplest objects in algebraic geometry. They arise in an elementary fashion as varieties parametrized by monomials whose exponents are a finite subset $\mathcal{A}$ of $\mathbb{Z}^n$. They may also be…
Toric varieties are a special class of rational varieties defined by equations of the form {\it monomial = monomial}. For a good brief survey of the history and role of toric varieties see [10]. Any toric variety $X$ contains a cover by…
After surveying higher K-theory of toric varieties, we present Totaro's old (c. 1997) unpublished result on expressing the corresponding homotopy theory via singular cohomology. It is a higher analog of the rational Chern character…
This short note solves the following problem: Given a map of normal toric varieties corresponding to a coherent subdivision of a cone, find an ideal such that the given map is the blowup of that ideal.