Related papers: Two-dimensional Id\`eles with Cycle Module Coeffic…
In the present paper we discuss questions concerning the arithmetic resolution for etale cohomology. Namely, consider a smooth quasi-projective variety X over a field k together with the local scheme U at a point x. Let Y be a smooth proper…
For divisors over smooth projective varieties, we show that the volume can be characterized by the duality between pseudo-effective cone of divisors and movable cone of curves. Inspired by this result, we give and study a natural…
In this paper, we apply Clausen-Scholze's theory of solid modules to the existence of adelic decompositions for schemes of finite type over $\mathbb{Z}$. Specifically, we use the six-functor formalism for solid modules to define the…
We will establish a nearby and vanishing cycle formalism for the arithmetic $\mathscr{D}$-module theory following Beilinson's philosophy. As an application, we define smooth objects in the framework of arithmetic $\mathscr{D}$-modules whose…
We generalize Iskovskih's theorem about surfaces without irregularity and bigenus from the smooth case to regular surfaces over arbitrary fields, with special focus on the case of imperfect fields. This includes surfaces that are…
We construct a nontrivial cyclic cocycle on the Weyl algebra of a symplectic vector space. Using this cyclic cocycle we construct an explicit, local, quasi-isomorphism from the complex of differential forms on a symplectic manifold to the…
We use Donaldson invariants of regular surfaces with p_g >0 to make quantitative statements about modulispaces of stable rank 2 sheaves. We give two examples: a quantitative existence theorem for stable bundles, and a computation of the…
The ascent and descent of the Mittag-Leffler property were instrumental in proving Zariski locality of the notion of an (infinite dimensional) vector bundle by Raynaud and Gruson in \cite{RG}. More recently, relative Mittag-Leffler modules…
The purpose of this paper is twofold. First, we survey known results about theta dualities on moduli spaces of sheaves on curves and surfaces. Secondly, we establish new such dualities in the surface case. Among others, the case of elliptic…
We show, for all $n\ge 2$ even and $d\ge 2+\frac{4}{n}$, that the moduli of smooth degree $d$ hypersurfaces of $\mathbb{P}^{n+1}$ contains infinitely many different Hodge loci whose Zariski tangent space has the same codimension as the…
We show that on the $d$-dimensional cube $I^d\equiv [0,1]^d$ the discrete moduli of smoothness which use only the values of the function on a diadic mesh are sufficient to determine the moduli of smoothness of that function. As an important…
This a first step to develop a theory of smooth, etale and unramified morphisms between noetherian formal schemes. Our main tool is the complete module of differentials, that is a coherent sheaf whenever the map of formal schemes is of…
We show that there exists a fine moduli space for torsion-free sheaves on a projective surface, which have a "good framing" on a big and nef divisor. This moduli space is a quasi-projective scheme. This is accomplished by showing that such…
We show a conditional exactness statement for the Nisnevich Gersten complex associated to an $\mathbb{A}^1$-invariant cohomology theory with Nisnevich descent for smooth schemes over a Dedekind ring with only infinite residue fields. As an…
We consider a notion of stability for sheaves, which we call multi-Gieseker stability that depends on several ample polarisations $L_1, \dots, L_N$ and on an additional parameter $\sigma \in \mathbb{Q}_{\geq 0}^N\setminus\{0\}$. The set of…
A duality theorem for the stable module category of representations of a finite group scheme is proved. One of its consequences is an analogue of Serre duality, and the existence of Auslander-Reiten triangles for the $\mathfrak{p}$-local…
We prove two-sided inequalities between the integral moduli of smoothness of a function on $\mathbb{R}^d/\mathbb{T}^d$ and the weighted tail-type integrals of its Fourier transform/series. Sharpness of obtained results in particular is…
This article grew out of an effort to understand the smooth mapping class groups of certain 4-manifolds in a geometric manner. We prove a smooth analog of the Birman-Hilden theorem for manifolds that admit a hyperk\"ahler structure. This…
The acquisition of the defining equations of Rees algebras is a natural way to study these algebras and allows certain invariants and properties to be deduced. In this paper, we consider Rees algebras of codimension 2 perfect ideals of…
In this paper we explore conditions for a curve in a smooth projective surface to have a free product of cyclic groups as the fundamental group of its complement. It is known that if the surface is $\mathbb P^2$, then such curves must be of…