Related papers: Chatelet's Theorem in Synthetic Algebraic Geometry

Let $K$ be a number field or a function field in one variable over a finite field, and let $K^{sep}$ be a separable closure of $K$. Let $C/K$ be a smooth, complete, connected curve. We prove a strong theorem of Fekete-Szego type for adelic…

We introduce the N\'eron-Severi Lie algebra of a Soergel module and we determine it for a large class of Schubert varieties. This is achieved by investigating which Soergel modules admit a tensor decomposition. We also use the…

Let $k$ be a field that is finitely generated over the field of rational numbers and $Br(k)$ the Brauer group of $k$. Let $X$ be an absolutely irreducible smooth projective variety over $k$, let $Br(X)$ be the cohomological…

Schmidt's subspace theorem in terms of Seshadri constants for closed subschemes in subgeneral position has been already developed sharply. We derive our theorem for numerically equivalent ample divisors by dint of the above theory step by…

Let X be a smooth double cover of a geometrically ruled surface defined over a separably closed field of characteristic different from 2. The main result of this paper is a finite presentation of the 2-torsion in the Brauer group of X with…

We prove that the Hilbert scheme of points on a normal quasi-projective surface with at worst rational double point singularities is irreducible.

Let $\{P_i\}_{1 \leq i \leq r}$ and $\{Q_i\}_{1 \leq i \leq r}$ be two collections of Brauer Severi surfaces (resp. conics) over a field $k$. We show that the subgroup generated by the $P_i's$ in $Br(k)$ is the same as the subgroup…

We study the scheme of formal arcs on a singular algebraic variety and its images under truncations. We prove a rationality result for the Poincare series of these images which is an analogue of the rationality of the Poincare series…

We apply techniques from symplectic geometry to extend and give a new proof of the complex convexity theorem of Gindikin-Kroetz.

We construct a smooth and projective surface over an arbitrary number field that is a counterexample to the Hasse principle but has the infinite etale Brauer-Manin set. We also construct a surface with a unique rational point and the…

We describe the set of points of the trianguline variety over a given local Galois representation. Global analogues describing companion points in eigenvariety by [Bre14] and [HN17], can be thought of as a rational analogue to the weight…

We prove a conjecture stated in a previous paper by the author about the existence of canonical filtrations for a family of vertex operator algebras in rational levels.

We classify morphisms from proper varieties to Brauer-Severi varieties, which generalizes the classical correspondence between morphisms to projective space and globally generated invertible sheaves. As an application, we study del Pezzo…

In the recent paper arXiv:1807.02721, B. Lawrence and A. Venkatesh develop a method of proving finiteness theorems in arithmetic geometry by studying the geometry of families over a base variety. Their results include a new proof of both…

We consider a set-theoretic version of mereology based on the inclusion relation $\subseteq$ and analyze how well it might serve as a foundation of mathematics. After establishing the non-definability of $\in$ from $\subseteq$, we identify…

Let $\mathcal X$ be a regular variety, flat and proper over a complete regular curve over a finite field, such that the generic fiber $X$ is smooth and geometrically connected. We prove that the Brauer group of $\mathcal X$ is finite if and…

In this article, we study geometric aspects of semi-arithmetic Riemann surfaces by means of number theory and hyperbolic geometry. First, we show the existence of infinitely many semi-arithmetic Riemann surfaces of various shapes and prove…

We prove a quantitative version of the Polynomial Szemeredi Theorem for difference sets. This result is achieved by first establishing a higher dimensional analogue of a theorem of Sarkozy (the simplest non-trivial case of the Polynomial…

In a previous work we established a super Schur-Weyl-Brauer duality between the orthosymplectic supergroup of superdimension $(m|2n)$ and the Brauer algebra with parameter $m-2n$. This led to a proof of the first fundamental theorem of…

Rational quartic spectrahedra in 3-space are semialgebraic convex subsets in $\mathbb{R}^3$ of semidefinite, real symmetric $(4 \times 4)$-matrices, whose boundary admits a rational parameterization. The Zariski closure in…