Related papers: Partial local resolution by characteristic zero me…
The objective of this paper is to discuss invariants of singularities of algebraic schemes over fields of positive characteristic, and to show how they yield the simplification of singularities. We focus here on invariants which arise in an…
An algorithm for resolution of singularities in characteristic zero is described. It is expressed in terms of multi-ideals, that essentially are defined as a finite sequence of pairs, each one consiting of a sheaf of ideals and a positive…
These lecture notes provide a unified overview of most known canonical desingularization methods in characteristic zero. It starts with discussing the classical method, and then proceeds with the recently discovered ones: logarithmic…
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…
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…
Effective methods are introduced for testing zero-dimensionality of varieties at a point. The motivation of this paper is to compute and analyze deformations of isolated hypersurface singularities. As an application, methods for computing…
We address the question of normal-crossings-preserving resolution of singularities (NC-preserving resolution), and compare the cases of characteristic 0 and characteristic 2. In characteristic 0, it is shown by Belotto da Silva and…
A new proof of equivariant resolution of singularities under a finite group action in characteristic 0 is provided. We assume we know how to resolve singularities without group action. We first prove equivariant resolution of toroidal…
I begin by explaining to non-specialists why resolution of singularities in characteristic 0 works. Then I go into some ideas telling how it actually works. I finish with a brief discussion of related results on foliations. I report on work…
We give an overview of invariants of algebraic singularities over perfect fields. We then show how they lead to a synthetic proof of embedded resolution of singularities of 2-dimensional schemes.
We address the following question of partial desingularization preserving normal crossings. Given an algebraic (or analytic) variety X in characteristic zero, can we find a finite sequence of blowings-up preserving the normal-crossings…
In characteristic zero, the residual order constitutes, after the local multiplicity, the second key invariant for the resolution of singularities. It is defined as the order of the coefficient ideal in a local hypersurface of maximal…
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:…
We first introduce and study the notion of multi-weighted blow-ups, which is later used to systematically construct an explicit yet efficient algorithm for functorial logarithmic resolution in characteristic zero, in the sense of Hironaka.…
A theory of simultaneous resolution of singularities for families of embedded varieties (over a field of characteristic zero) parametrized by the spectrum of a suitable artinian ring, and compatible with a given algorithm of resolution, is…
We investigate the qualitative properties of a critical Hartree equation defined on punctured domains. Our study has two main objectives: analyzing the asymptotic behavior near isolated singularities and establishing radial symmetry of…
We give an alternative proof of the theorem by Kuznetsov and Lunts, stating that any separated scheme of finite type over a field of characteristic zero admits a categorical resolution of singularities. Their construction makes use of the…
Categorical resolution of singularities has been constructed in arXiv:1212.6170. It proceeds by alternating two steps of seemingly different nature. We show how to use the formalism of filtered derived categories to combine the two steps…
The concept of the maximal contact is the key in Hironaka's resolution theory. It treats local theory, and it is not effective in positive characteristics. This is the essential reason why Hironaka's theory treats only the case of…
The article investigates the behaviour of the characteristic zero resolution invariant when transcribed suitably to the case of surfaces in positive characteristic. By Moh's jumping phenomenon -- or the occurrence of kangaroo singularities…