Related papers: Desingularization by blowings-up avoiding simple n…
We show how the notion of fantastacks can be used to effectively desingularize binomial varieties defined over algebraically closed fields. In contrast to a desingularization via blow-ups in smooth centers, we drastically reduce the number…
We give a combinatorial algorithm for equivariant embedded resolution of singularities of a toric variety defined over a perfect field. The algorithm is realized by a finite succession of blowings-up with smooth invariant centres that…
Given a variety $X$ over a perfect field, we study the partition defined on $X$ by the multiplicity (into equimultiple points), and the effect of blowing up at smooth equimultiple centers. Over fields of characteristic zero we prove…
We provide a procedure for resolving, in characteristic 0, singularities of a variety $X$ embedded in a smooth variety $Y$ by repeatedly blowing up the worst singularities, in the sense of stack-theoretic weighted blowings up. No history,…
Solutions to scalar curvature equations have the property that all possible blow-up points are isolated, at least in low dimensions. This property is commonly used as the first step in the proofs of compactness. We show that this result…
We provide a complete classification of the singularities of cluster algebras of finite type with trivial coefficients. Alongside, we develop a constructive desingularization of these singularities via blowups in regular centers over fields…
We provide explicit criteria for blow-up solutions of autonomous ordinary differential equations. Ideas are based on the quasi-homogeneous desingularization (blowing-up) of singularities and compactifications of phase spaces, which suitably…
After briefly recalling some computational aspects of blowing up and of representation of resolution data common to a wide range of desingularization algorithms (in the general case as well as in special cases like surfaces or binomial…
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,…
In this paper, given a compact Kcsc orbifolds of any dimension and with nontrivial holomorphic vector fields, we find sufficient conditions on the position of singular points in order to admit a Kcsc desingularization, generalizing the…
We present a simple and fast embedded resolution of varieties and principalization of ideals using torus actions on ambient smooth varieties with simple normal crossings (SNC) divisors. The canonical functorial resolution in characteristic…
Geometric treatments of blow-up solutions for autonomous ordinary differential equations and their blow-up rates are concerned. Our approach focuses on the type of invariant sets at infinity via compactifications of phase spaces, and…
This article shall serve as a quick reference for somebody who needs precise information on concepts and results related to resolution of singularities. As such, it is more a technical manual than a bedtime story. Topics which are covered:…
In this paper we show a general method to compactify certain open varieties by adding normal crossing divisors. This is done by proving that {\it blowing up along an arrangement of subvarieties} can be carried out. Important examples such…
Our previous papers introduce topological notions of normal crossings symplectic divisor and variety, show that they are equivalent, in a suitable sense, to the corresponding geometric notions, and establish a topological smoothability…
Stack-theoretic blow-ups have proven to be efficient in resolving singularities over fields of characteristic zero. In this article, we move forward towards positive characteristic where new challenges arise. In particular, the dimension of…
In this paper, we attempt to resolve the singularities of the zero variety of a $C^{\infty}$ function of two variables as much as possible by using ordinary blowings up. As a result, we formulate an algorithm to locally express the zero…
We extend the classical formula of Porteous for blowing-up Chern classes to the case of blow-ups of possibly singular varieties along regularly embedded centers. The proof of this generalization is perhaps conceptually simpler than the…
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…
Fixed a point O on a non-singular surface S and a complete mO-primary ideal I in its local ring, the curves on the surface X obtained by blowing-up I are studied in terms of the base points of I. Criteria for the principality of these…