Related papers: Functorial embedded resolution via weighted blowin…
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.…
In this paper we describe a computer implementation of Abramovich, Temkin, and Wlodarczyk's algorithm for resolving singularities in characteristic zero. Their "weighted resolution" algorithm proceeds by repeatedly blowing up along centers…
In characteristic zero, we construct logarithmic resolution of singularities, with simple normal crossings exceptional divisor, using weighted blow-ups.
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…
Let $X$ be a fs logarithmic scheme that is generically logarithmically smooth, and that admits a strict closed embedding into a logarithmically smooth scheme $Y$ over a field $\kk$ of characteristic zero. We construct a simple and fast…
In characteristic zero, we construct a canonical, functorial resolution algorithm by weighted blow-ups that strictly preserves the normal crossings (nc) locus, effectively answering Kollar's problem. Operating in full generality, our…
We discuss to what extent the local techniques of resolution of singularities over fields of characteristic zero can be applied to improve singularities in general. For certain interesting classes of singularities, this leads to an embedded…
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…
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…
Given a singular hypersurface in a regular 2-dimensional scheme essentially of finite type over a field, we construct an embedded resolution of singularities by weighted blow-ups. This differs from our earlier work which required…
In 2019, Abramovich--Temkin--Wlodarczyk and McQuillan used weighted blow-ups to obtain very fast and functorial algorithms for resolution of singularities in characteristic zero. Recently, Abramovich--Quek--Schober simplified the…
We prove that for any singular integral affine variety $X$ of finite presentation over a perfect field defined over $\mathbb Z$, there exists a smooth morphism from $Y$ onto $X$ such that $Y$ admits a resolution. That is, there exists a…
We prove the existence of resolution of singularities for arbitrary (not necessarily reduced or irreducible) excellent two-dimensional schemes, via permissible blow-ups. The resolution is canonical, and functorial with respect to…
This article is an exposition of an elementary constructive proof of canonical resolution of singularities in characteristic zero, presented in detail in Invent. Math. 128 (1997), 207-302. We define a new local invariant and get an…
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…
We show that stack-theoretic resolution of singularities preserving normal crossings (partial desingularization) by weighted blowings-up, can be obtained in a simple direct way from a splitting theorem of the first and third authors, using…
Fulton and MacPherson famously constructed a configuration space that encodes infinitesimal collision data by blowing up the diagonals. We observe that when generalizing their approach to configuration spaces of filtered manifolds (e.g. jet…
We prove that any three dimensional terminal singularity $P \in X$ can be resolved by a sequence of divisorial contractions with minimal discrepancies which are weighted blowups over points.
It is shown that, for any reduced algebraic variety in characteristic zero, one can resolve all but simple normal crossings (snc) singularities by a finite sequence of blowings-up with smooth centres which, at every step, avoids points…
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…