Searcharxiv⌕ Search

arXiv subjects

Arnaud Mayeux

Publications and source records attributed to Arnaud Mayeux.

17 recordsLinked to original sources
arXiv2026

Dilatations of categories, via their lean formalization

Given a category $\calC$ and a center, that is a collection of pairs $(d_i, N_i)$ consisting of a morphism $d_i$ and a sieve $N_i$ over its codomain, the dilatation of $\calC$ is a new category $\calC'$ in which every $n \in N_i$ factors, uniquely and functorially, through $d_i$. This paper presents the theory of dilatations of categories through a full formalization of the construction and its main theorems in the Lean~4 proof assistant, on top of the Mathlib library. An appendix collects a systematic dictionary between the mathematical statements and the Lean declarations that formalize them.

cs.LO↗
arXiv2026

A polyptych of multi-centered deformation spaces

Extending Verdier's deformation space to the normal cone of a closed subscheme and Rost's double deformation space of a pair of nested closed subschemes, we introduce a notion of deformation spaces attached to chains of immersions of arbitrary lengths $n$. One main result, which builds on the formalism of multi-centered dilatations of schemes, is the existence of so-called panelization isomorphisms, which produce under suitable regularity conditions several canonical isomorphisms between a given deformation space of length $n$ and some deformation spaces of smaller lengths. Having these panelization isomorphisms also allows to give geometric descriptions of the strata -- certain restrictions of special interest -- of deformation spaces. \tableofcontents

math.AG↗
arXiv2025

Algebraic magnetism invariants of a double scalar action on the projective plane

This document is an expanded version of the notes from a talk at the \textit{Arithmetic and Algebraic Geometry Week} conference, which took place in Iasi in September 2025. In this note, we compute the pure magnets (certain semigroups) and the associated attractors for a double scalar action of $\mathbb{G}_m^2$ on $\mathbb{P}^2$. This is mostly expository and provides a non-affine example illustrating the invariants of Algebraic Magnetism in a simple and visual case. Nevertheless, we introduce the notion of lambdafiable magnets, in the general setting, to relate certain magnets to cocharacters. Finally we announce recent advanced results on Algebraic Magnetism obtained in \cite{BM} and solving positively some conjectures stated in \cite{Ma}.

math.AG↗
arXiv2025

Algebraic Magnetism

For a diagonalizable monoid scheme $A(M)_S$ acting on an algebraic space $X$, we introduce for any submonoid $N$ of $M$ an attractor space $X^N$. We then investigate and study various aspects of attractors associated to monoids.

math.AG↗
arXiv2025

Algebraic Magnetism Invariants of Self-Actions of Diagonalizable Monoid Schemes

We provide a method to compute the pure magnets of the action of a diagonalizable monoid scheme on itself. This is described in terms of minimal generators of the sharp monoid obtained quotienting by the face of invertible elements. In particular, in this example, algebraic magnetism detects sharpness and minimal generators of the monoid modulo invertible elements.

math.AG↗
arXiv2025

Conjecture: the set of prime numbers is supernatural

Prime numbers are fascinating by the way they appear in the set of natural numbers. Despite several results enlighting us about their repartition, the set of prime numbers is often informally qualified as misterious. In the present paper, we introduce a formalism allowing to state a formal conjecture: the set of prime numbers is supernatural. Our conjecture has no analog in the existing literature. This conjecture is expected to be a hard challenge for any kind of intelligence. We also find a Fermat-like function giving more prime numbers than Fermat's function.

math.GM↗
arXiv2025

On multi-graded Proj schemes

We review the construction (due to Brenner--Schröer) of the Proj scheme associated with a ring graded by a finitely generated abelian group. This construction generalizes the well-known Grothendieck Proj construction for $\mathbb{N}$-graded rings; we extend some classical results (in particular, regarding quasi-coherent sheaves on such schemes) from the $\mathbb{N}$-graded setting to this general setting, and prove new results that make sense only in the general setting of Brenner--Schröer. Finally, we show that flag varieties of reductive groups, as well as some vector bundles over such varieties attached to representations of a Borel subgroup, can be naturally interpreted in this formalism.

math.AG↗
arXiv2024

Dilatations of categories

Dilatations modify categories by imposing that some morphisms factorize through some others. This is formalized by a universal property. This text is devoted to introduce and study this construction. Examples of dilatations of categories include localizations of categories and dilatations of rings.

math.CT↗
arXiv2024

Multi-centered dilatations, congruent isomorphisms and Rost double deformation space

We introduce multi-centered dilatations of rings, schemes and algebraic spaces, a basic algebraic concept. Dilatations of schemes endowed with a structure (e.g. monoid, group or Lie algebra) are in favorable cases schemes endowed with the same structure. As applications, we use our new formalism to contribute to the understanding of mono-centered dilatations, to formulate and deduce some multi-centered congruent isomorphisms and to interpret Rost double deformation space as a "double-centered" dilatation.

math.AG↗
arXiv2024

A survey on algebraic dilatations

In this text, we wish to provide the reader with a short guide to recent works on the theory of dilatations in Commutative Algebra and Algebraic Geometry. These works fall naturally into two categories: one emphasises foundational and theoretical aspects and the other applications to existing theories.

math.AG↗
arXiv2024

Comparing Bushnell-Kutzko and Sécherre's constructions of types for $\mathrm{GL}_{N}$ and its inner forms with Yu's construction

Let $F$ be a non-archimedean local field, $A$ be a central simple $F$-algebra, and $G$ be the multiplicative group of $A$. To construct types for supercuspidal representations of $G$, simple types by Sécherre and Yu's construction are already known. In this paper, we compare these constructions. In particular, we show essentially tame supercuspidal representations of $G$ defined by Bushnell-Henniart are nothing but tame supercuspidal representations defined by Yu.

math.RT↗
arXiv2020

Néron blowups and low-degree cohomological applications

We define dilatations of general schemes and study their basic properties. Dilatations of group schemes are -- in favorable cases -- again group schemes, called Néron blowups. We give two applications to their cohomology in degree zero (integral points) and degree one (torsors): we prove a canonical Moy-Prasad isomorphism that identifies the graded pieces in the congruent filtration of $G$ with the graded pieces in its Lie algebra $\mathfrak g$, and we show that many level structures on moduli stacks of $G$-bundles are encoded in torsors under Néron blowups of $G$.

math.AG↗
arXiv2017

Représentations supercuspidales

This text is a response to the following question: What are the methods to build supercuspidal complex representations of p-adic reductive groups and are there ties between them ? We will give an overview of the Bushnell-Kutzko and Yu constructions. Then, given a simple maximum tame stratum, a simple character $θ$ associated to this stratum and a representation of the level zero of a subset of G, we associate a Yu data generic and therefore a representation constructed by Yu, this representation corresponds to a simple type given by a $β$ - extension of $θ$.

math.RT↗