Searcharxiv⌕ Search

arXiv subjects

Michael Temkin

Publications and source records attributed to Michael Temkin.

At least 19 recordsLinked to original sources
arXiv2026

Geometrically multiplicative non-archimedean norms

Universally (or geometrically) multiplicative norms on Banach algebras over complete non-archimedean fields were used by Berkovich in his works on non-archimedean geometry, and later they were studied in some detail by Poineau. In this paper, we perform a more thorough study of the question when multiplicativity, spectrality and spectral multiplicativity of norms on algebras over real valued fields are preserved by ground field extensions. We obtain precise criteria quite analogous to the classical theory of geometric irreducibility and reducedness. In particular, we generalize the results of Poineau in a few aspects.

math.AG↗
arXiv2026

Torsion-stabilized modular curves of level $p$

This is the first paper of a project on new integral models $\mathcal{X}(N)$ of the modular curve $X(N)$. The final results for a general level $N$ will be obtained in the second paper, while this paper is devoted to giving all necessary background and definitions applicable to any $N$ and then working out the case of $\mathcal{X}(p)$ with all possible details. We define $\mathcal{X}(N)$ as the closure of $Y(N)$ in the space $\overline{\mathcal{M}}_{1,N^2}=\overline{\mathcal{M}}_{1,\Gamma}$, where $\Gamma=(\mathbb{Z}/N\mathbb{Z})^2$, and show that for $N=p$ it is the blowup of the Katz-Mazur model $\widetilde{\mathcal{X}}(p)$ at all supersingular points, and hence $(\mathcal{X}(p),Y(p))$ is the minimal toroidal resolution of $(\widetilde{\mathcal{X}}(p),Y(p))$. In fact, it is even log smooth over $(\mathbb{Z},\mathbb{Z}[1/p])$, but this is special for the case when $p=N$. One can tautologically view $\mathcal{X}(p)$ as the moduli space of $\Gamma$-stabilized genus-1 curves $(E,\Gamma)$ which can be smoothed to an elliptic curve labelled by its $N$-torsion, but our main results provide explicit criteria of the smoothability: $\mathcal{X}(p)$ parameterizes $\Gamma$-equivariant stable genus-1 curves $(E,\Gamma)$ such that the action satisfies two explicit conditions formulated in the paper.

math.AG↗
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 W{\l}odarczyk 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↗
arXiv2025

Dream resolution and principalization II: excellent schemes

This is the second paper in a project on dream (or memoryless) principalization and resolution methods. It extends this theory from the case of schemes with enough derivations, which was established in [Tem25], to general excellent schemes of characteristic zero. So, similarly to McQuillan's approach developed in [McQ20], the approach of [ATW24] is now extended to the generality of all excellent schemes of characteristic zero. In addition, we precisely describe the set of invariants of canonical centers and establish the resolution in the non-embedded form, where one applies simple (stack-theoretical) modifications along subschemes of a special form that we call tubes. In the regular case these are precisely the subschemes corresponding to canonical centers.

math.AG↗
arXiv2025

Dream resolution and principalization I: enough derivations

This is a first paper in a project on extending the dream principalization and resolution methods of [ATW24], [McQ20] and [Que22] to quasi-excellent, logarithmic and relative settings. We show that the main results of [ATW24] extend to regular schemes with enough derivations and are functorial with respect to all regular morphisms. This is already strong enough to formally imply that the same results hold in other categories, such as complex and p-adic analytic spaces. Our method has many common points with that of [ATW24], but the accent is now shifted towards the study of weighted centers and their coordinate presentations. Not only we hope that this is a bit simpler and more conceptual, this method will be easily applied in the logarithmic and relative settings in the sequel.

math.AG↗
arXiv2025

Small extensions of analytic fields

An extension $K/k$ of analytic (i.e. real valued complete) fields is called small if it is topologically-algebraically generated by finitely many elements. We prove that this property is inherited by subextensions and hence topological generating degree of such extensions is monotonic. Much more detailed results are obtained in the case of degree one. Let $k$ be an analytic algebraically closed field of positive residual characteristic $p$ and $K=\widehat{k(t)^a}$ with a non-trivial valuation. In a previous work it was shown that the set $I_{K/k}$ of intermediate complete algebraically closed subextensions $k\subseteq F\subseteq K$ is totally ordered by inclusion. In this paper we show that $I_{K/k}$ is an interval parameterized by the distance between $t$ and $F$. Moreover, logarithmic parameterizations induced by other generators differ by PL functions with slopes in $p^{\mathbb Z}$ and corners in $|K^\times|$, so $I_{K/k}$ acquires a natural PL structure.

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↗
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↗
arXiv2024

Adic curves: stable reduction, skeletons and metric structure

We study the structure of adic curves over an affinoid field of arbitrary rank. In particular, quite analogously to Berkovich geometry we classify points on curves, prove a semistable reduction theorem in the version of Ducros' triangulations, define associated curve skeletons and prove that they are deformational retracts in a suitable sense. An important new technical tool is an appropriate compactification of ordered groups that we call the ranger compactification. Intervals of rangers are then used to define metric structures and construct deformational retractions.

math.AG↗
arXiv2023

Relative and logarithmic resolution of singularities

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 desingularization of logarithmic schemes and morphisms, weighted methods: logarithmic and non-logarithmic ones, desingularization in wider contexts: quasi-excellent schemes and formal schemes, complex and non-archimedean spaces.

math.AG↗
arXiv2023

Height reduction for local uniformization of varieties and non-archimedean spaces

It is known since the works of Zariski that the essential difficulty in the local uniformization problem is met already in the case of valuations of height one. In this paper we prove that local uniformization of schemes and non-archimedean analytic spaces rigorously follows from the case of valuations of height one. For non-archimedean spaces this result reduces the problem to studying local structure of smooth Berkovich spaces.

math.AG↗
arXiv2022

Introduction to logarithmic geometry

These lecture notes provide an introduction to logarithmic geometry with a view towards recent applications in the desingularization theory.

math.AG↗
arXiv2022

Semistable reduction over thick log points

We establish a version of a semistable reduction theorem over a log point with a non-trivial nilpotent structure. In order to do this we extend the classical desingularization theories to non-reduced schemes with generically principal nilradical.

math.AG↗
arXiv2021

Non-archimedean pinchings

We develop the theory of pinchings for non-archimedean analytic spaces. In particular, we show that although pinchings of affinoid spaces do not have to be affinoid, pinchings of Hausdorff analytic spaces always exist in the category of analytic spaces.

math.AG↗
arXiv2020

Reduction and lifting problem for differential forms on Berkovich curves

Given a complete real-valued field $k$ of residue characteristic zero, we study properties of a differential form $\omega$ on a smooth proper $k$-analytic curve $X$. In particular, we associate to $(X,\omega)$ a natural tropical reduction datum combining tropical data of $(X,\omega)$ and algebra-geometric reduction data over the residue field $\widetilde{k}$. We show that this datum satisfies natural compatibility condition, and prove a lifting theorem asserting that any compatible tropical reduction datum lifts to an actual pair $(X,\omega)$. In particular, we obtain a short proof of the main result of a work [BCGGM20] by Bainbridge, Chen, Gendron, Grushevsky, and M\"oller.

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↗
arXiv2019

Descent for non-archimedean analytic spaces

In this paper we study two types of descent in the category of Berkovich analytic spaces: flat descent and descent with respect to an extension of the ground field. Quite surprisingly, the deepest results in this direction seem to be of the second type, including the descent of properties of being a good analytic space and being a morphism without boundary.

math.AG↗