Searcharxiv⌕ Search

arXiv subjects

Dan Abramovich

Publications and source records attributed to Dan Abramovich.

At least 19 recordsLinked to original sources
arXiv2026

Partial desingularization up to normal-crossings in characteristic 0 and 2

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 Bierstone arxiv:2602.09114 and Włodarczyk arxiv:2602.14266 that, if one allows to introduce stack theoretic weighted blowups, any variety over a field of characteristic 0 admits a normal crossings resolution. We provide a principle that makes such results possible, Theorem 3.1.4. We further show that the coarse moduli space can be restricted to have higher pinch points (Definition 1.3.2), see Theorem 1.3.3. In contrast, in characteristic 2 we classify pinch points, and show that weighted blowups cannot lead to NC-preserving resolution of pinch points, although a pinch point is always the coarse moduli space of a NC stack (Theorem 1.4.1).

math.AG↗
arXiv2026

Torus actions, weighted blow-ups, and desingularization of plane curves

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 multi-weighted blow-ups. We deduce an inductive argument, despite the fact that higher dimensional tangent spaces arise, by taking torus actions and equivariant centers into account. In addition, we do not have to restrict to perfect base fields.

math.AG↗
arXiv2025

Principalization on logarithmically foliated orbifolds

In characteristic zero, we construct principalization of ideals on smooth orbifolds endowed with a normal crossings divisor and a foliation. We then illustrate how the method can be used in the general study of foliations via two applications. First, we provide a resolution of singularities of Darboux totally integrable foliations in arbitrary dimensions -- including rational and meromorphic Darboux foliations. Second, we show how to transform a generically transverse section into a transverse section.

math.AG↗
arXiv2025

Resolution of singularities for the dynamical mathematician

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 with André Belotto da Silva, Michael Temkin, and Jarosław Włodarczyk; any claim to originality is joint with them and appears in the paper [AdSTW25].

math.AG↗
arXiv2025

Functorial monomialization and uniqueness of centers for relative principalization

Theorem 1.2.6 of [ATW20] provides a relatively functorial logarithmic principalization of ideals on relative logarithmic orbifolds $X\to B$ in characteristic 0, relying on a delicate monomialization theorem for Kummer ideals. The paper [AdSTW25] provides a parallel avenue through weighted blowings up. In this paper we show that, if $X\to B$ is proper, monomialization of both Kummer and weighted logarithmic centers can be carried out in a manner which is functorial for base change by regular morphisms. This implies in particular logarithmic relative principalization of ideals and logarithmically smooth reduction of proper families of varieties in characteristic 0 in a manner equivariant for group actions and compatible with localization on the base.

math.AG↗
arXiv2024

Resolving plane curves using stack-theoretic 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 the tangent space of the Artin stack created after a weighted blow-up may increase, which makes it hard to apply inductive arguments -- even if the maximal order decreases. We focus on the case of curve singularities embedded into a smooth surface defined over a perfect field. For this special situation we propose a solution to overcome the inductive challenge through canonically constructed multi-weighted blow-ups.

math.AG↗
arXiv2024

Punctured logarithmic maps

We introduce a variant of stable logarithmic maps, which we call punctured logarithmic maps. They allow an extension of logarithmic Gromov-Witten theory in which marked points have a negative order of tangency with boundary divisors. As a main application we develop a gluing formalism which reconstructs stable logarithmic maps and their virtual cycles without expansions of the target, with tropical geometry providing the underlying combinatorics. Punctured Gromov-Witten invariants also play a pivotal role in the intrinsic construction of mirror partners by the last two authors in arXiv:1909.07649, conjecturally relating to symplectic cohomology, and in the logarithmic gauged linear sigma model in upcoming work of the second author with Felix Janda and Yongbin Ruan.

math.AG↗
arXiv2024

Desingularization of binomial varieties using toric Artin stacks

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 of steps and the number of charts appearing along the process. Furthermore, we discuss how our considerations extend to a partial simultaneous normal crossings desingularization of finitely many binomial hypersurfaces.

math.AG↗
arXiv2023

The integral Chow ring of weighted blow-ups

We give a formula for the integral Chow rings of weighted blow-ups. Along the way, we also compute the integral Chow rings of weighted projective stack bundles, a formula for the Gysin homomorphism of a weighted blow-up, and a generalization of the splitting principle. In addition, in the appendix we compute the Chern class of a weighted blow-up.

math.AG↗
arXiv2021

Logarithmic resolution via multi-weighted blow-ups

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. Specifically, for a singular, reduced closed subscheme $X$ of a smooth scheme $Y$ over a field of characteristic zero, we resolve the singularities of $X$ by taking proper transforms $X_i \subset Y_i$ along a sequence of multi-weighted blow-ups $Y_N \to Y_{N-1} \to \dotsb \to Y_0 = Y$ which satisfies the following properties: (i) the $Y_i$ are smooth Artin stacks with simple normal crossing exceptional loci; (ii) at each step we always blow up the worst singular locus of $X_i$, and witness on $X_{i+1}$ an immediate improvement in singularities; (iii) and finally, the singular locus of $X$ is transformed into a simple normal crossing divisor on $X_N$.

math.AG↗
arXiv2021

Boundedness of the space of stable logarithmic maps

We prove that the moduli space of stable logarithmic maps with fixed numerical invariants, from logarithmic curves to a fixed projective target logarithmic scheme with fine and saturated logarithmic structure, is a proper algebraic stack. This was previously known only with further restrictions on the logarithmic structure of the target.

math.AG↗
arXiv2020

Relative desingularization and principalization of ideals

In characteristic zero, we construct relative principalization of ideals for logarithmically regular morphisms of logarithmic schemes, and use it to construct logarithmically regular desingularization of morphisms. These constructions are relatively canonical and even functorial with respect to logarithmically regular morphisms and arbitrary base changes. Relative canonicity means, that the principalization requires a fine enough non-canonical modification of the base, and once it is chosen the process is canonical. As a consequence we deduce the semistable reduction theorem over arbitrary valuation rings. In another our work in progress, the same problems will be solved canonically in the case of proper morphisms.

math.AG↗
arXiv2020

Decomposition of degenerate Gromov-Witten invariants

We prove a decomposition formula of logarithmic Gromov-Witten invariants in a degeneration setting. A one-parameter log smooth family X->B with singular fibre over b_0 \in B yields a family M(X/B,β) -> B of moduli stacks of stable logarithmic maps. We give a virtual decomposition of the fibre of this family over b_0 in terms of rigid tropical curves. This generalizes one aspect of known results in the case that the fibre X_{b_0} is a normal crossings union of two divisors. We exhibit our formulas in explicit examples.

math.AG↗
arXiv2020

Functorial embedded resolution via weighted blowings up

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, no exceptional divisors, and no logarithmic structures are necessary to carry this out; the steps are explicit geometric operations requiring no choices; and the resulting algorithm is efficient. A similar result was discovered independently by McQuillan in a manuscript posted on this archive simultaneously.

math.AG↗
arXiv2017

Resolution of singularities of complex algebraic varieties and their families

We discuss Hironaka's theorem on resolution of singularities in charactetistic 0 as well as more recent progress, both on simplifying and improving Hironaka's method of proof and on new results and directions on families of varieties, leading to joint work on toroidal orbifolds with Michael Temkin and Jarosław Włodarczyk.

math.AG↗
arXiv2017

Toroidal orbifolds, destackification, and Kummer blowings up

We show that any toroidal DM stack $X$ with finite diagonalizable inertia possesses a maximal toroidal coarsening $X_{tcs}$ such that the morphism $X\to X_{tcs}$ is logarithmically smooth. Further, we use torification results of [AT17] to construct a destackification functor, a variant of the main result of Bergh [Ber17], on the category of such toroidal stacks $X$. Namely, we associate to $X$ a sequence of blowings up of toroidal stacks $\widetilde{\mathcal{F}}_X\:Y\longrightarrow X$ such that $Y_{tc}$ coincides with the usual coarse moduli space $Y_{cs}$. In particular, this provides a toroidal resolution of the algebraic space $X_{cs}$. Both $X_{tcs}$ and $\widetilde{\mathcal{F}}_X$ are functorial with respect to strict inertia preserving morphisms $X'\to X$. Finally, we use coarsening morphisms to introduce a class of non-representable birational modifications of toroidal stacks called Kummer blowings up. These modifications, as well as our version of destackification, are used in our work on functorial toroidal resolution of singularities.

math.AG↗
arXiv2017

Principalization of ideals on toroidal orbifolds

Given an ideal $\mathcal I$ on a variety $X$ with toroidal singularities, we produce a modification $X' \to X$, functorial for toroidal morphisms, making the ideal monomial on a toroidal stack $X'$. We do this by adapting the methods of [Wło05], discarding steps which become redundant. We deduce functorial resolution of singularities for varieties with logarithmic structures. This is the first step in our program to apply logarithmic desingularization to a morphism $Z \to B$, aiming to prove functorial semistable reduction theorems.

math.AG↗