Related papers: Restricted trichotomy in higher dimensions

Given a $K$-vector space $V$, let $\sigma(V,K)$ denote the covering number, i.e. the smallest (cardinal) number of proper subspaces whose union covers $V$. Analogously, define $\sigma(M,R)$ for a module $M$ over a unital commutative ring…

We explicitly describe a structure of a regular cell complex $K(L)$ on the moduli space $M(L)$ of a planar polygonal linkage $L$. The combinatorics is very much related (but not equal) to the combinatorics of the permutahedron. In…

Let $G$ be a connected reductive algebraic group over an algebraically closed field of positive characteristic, $\mathfrak{g}$ be its Lie algebra, and $B$ be a Borel subgroup. We prove a formula for the dimensions of extension groups, in…

Let $\mathcal{M}_{n,2n+2}$ be the coarse moduli space of CY manifolds arising from a crepant resolution of double covers of $\mathbb{P}^n$ branched along $2n+2$ hyperplanes in general position. We show that the monodromy group of a good…

We develop a microlocal theory, in the sense of Kashiwara-Schapira, for Zariski-constructible sheaves on rigid analytic varieties. We define and study monodromic sheaves, the monodromic Fourier transform, specialisation, microlocalisation,…

We demonstrate that for an arbitrary number of identical particles, each defined on a Hilbert-space of arbitrary dimension, there exists a whole ladder of relations of complementarity between local, and every conceivable kind of joint (or…

In this paper we describe the general theory of constructing toroidal compactifications of locally symmetric spaces and using these to compute dimension formulas for spaces of modular forms. We focus explicitly on the case of the orthogonal…

We show that if a complete, doubling metric space is annularly linearly connected then its conformal dimension is greater than one, quantitatively. As a consequence, we answer a question of Bonk and Kleiner: if the boundary of a one-ended…

We study the minimal dimension of maximal commutative subalgebras of the matrix algebra $M_n(k)$ over an algebraically closed field. While examples with dimension strictly smaller than n are known for $n \geq 14$, no such examples are known…

We discuss an orbifold of the toroidally compactified heterotic string which gives a global reduction of the dimension of the moduli space while preserving the supersymmetry. This construction yields the moduli space of the first of a…

Over a local ring $R$, the theory of cohomological support varieties attaches to any bounded complex $M$ of finitely generated $R$-modules an algebraic variety $V_R(M)$ that encodes homological properties of $M$. We give lower bounds for…

We prove that the classical algebraic varieties over algebraically closed fields can be defined over arbitrary fields $k.$ Then we prove that for associative algebras $A$, there exist local representing objects $A_M$ for simple modules $M.$…

We give a new axiomatic treatment of the Zilber trichotomy, and use it to complete the proof of the trichotomy for relics of algebraically closed fields, i.e., reducts of the ACF-induced structure on ACF-definable sets. More precisely, we…

This paper is concerned with the minimum distance computation for higher dimensional toric codes defined by lattice polytopes. We show that the minimum distance is multiplicative with respect to taking the product of polytopes, and behaves…

We consider the Zariski space of all places of an algebraic function field $F|K$ of arbitrary characteristic and investigate its structure by means of its patch topology. We show that certain sets of places with nice properties (e.g., prime…

We study the existence and properties of birationally equivalent models for elliptically fibered varieties. In particular these have either the structure of Mori fiber spaces or, assuming some standard conjectures, minimal models with a…

If in a given rank $r$, there is an irreducible complex local system with torsion determinant and quasi-unipotent monodromies at infinity on a smooth quasi-projective variety, then for every prime number $\ell$, there is an absolutely…

An algebraic variety is called $\mathbb{A}^{1}$-cylindrical if it contains an $\mathbb{A}^{1}$-cylinder, i.e. a Zariski open subset of the form $Z\times\mathbb{A}^{1}$ for some algebraic variety Z. We show that the generic fiber of a family…

Let (R,m) be a complete local ring, a an ideal of R and M a finitely generated R-module. The aim of this paper is to show that for any non-negative integer n, the least integer i such that the i-th local cohomology with respect to a is not…

It is well known that knots are countable in ordinary knot theory. Recently, knots {\it with intersections} have raised a certain interest, and have been found to have physical applications. We point out that such knots --equivalence…