Related papers: Highly Versal Torsors

A set $R \subseteq V(G)$ is a resolving set of a graph $G$ if for all distinct vertices $v,u \in V(G)$ there exists an element $r \in R$ such that $d(r,v) \neq d(r,u)$. The metric dimension $\dim(G)$ of the graph $G$ is the minimum…

For a graph $G$ and partition $\mathcal{U}$ of its vertex set, an independent transversal of $(G, \mathcal{U})$ is an independent set of $G$ that contains one vertex from each block of $\mathcal{U}$. Buys, Kang, and Ozeki studied when a…

Let $A$ be an abelian variety over the function field $K$ of a curve over a finite field. We describe several mild geometric conditions ensuring that the group $A(K^{\rm perf})$ is finitely generated and that the $p$-primary torsion…

A set $S$ of vertices in a hypergraph is \textit{strongly independent} if every hyperedge shares at most one vertex with $S$. We prove a sharp result for the number of maximal strongly independent sets in a $3$-uniform hypergraph analogous…

We construct finitely generated torsion-free solvable groups $G$ that have infinite rank, but such that all finitely generated torsion-free metabelian subquotients of $G$ are virtually abelian. In particular all finitely generated…

We show that every orbispace satisfying certain mild hypotheses has 'enough' vector bundles. It follows that the K-theory of finite rank vector bundles on such orbispaces is a cohomology theory. Global presentation results for smooth…

We define parahoric $\cG$--torsors for certain Bruhat--Tits group scheme $\cG$ on a smooth complex projective curve $X$ when the weights are real, and also define connections on them. We prove that a $\cG$--torsor is given by a homomorphism…

Let $G$ be a finite group and construct a graph $\Delta(G)$ by taking $G\setminus\{1\}$ as the vertex set of $\Delta(G)$ and by drawing an edge between two vertices $x$ and $y$ if $\langle x,y\rangle$ is cyclic. Let $K(G)$ be the set…

Notions of rank abound in the literature on tensor decomposition. We prove that strength, recently introduced for homogeneous polynomials by Ananyan-Hochster in their proof of Stillman's conjecture and generalised here to other tensors, is…

Given a graph $G$ and a family $\mathcal{G} = \{G_1,\ldots,G_n\}$ of subgraphs of $G$, a transversal of $\mathcal{G}$ is a pair $(T,\phi)$ such that $T \subseteq E(G)$ and $\phi: T \rightarrow [n]$ is a bijection satisfying $e \in…

A subset U of a group G is called k-universal if U contains a translate of every k-element subset of G. We give several nearly optimal constructions of small k-universal sets, and use them to resolve an old question of Erdos and Newman on…

The aim of this paper is to give a proof of the restriction theorems for principal bundles with a reductive algebraic group as structure group in arbitrary characteristic. Let $G$ be a reductive algebraic group over any field $k=\bar{k}$,…

Let $K$ be a differential field over $\C$ with derivation $D$, $G$ a finite linear automorphism group over $K$ which preserves $D$, and $K^G$ the fixed point subfield of $K$ under the action of $G$. We show that every finite-dimensional…

Restriction is a natural quasi-order on $d$-way tensors. We establish a remarkable aspect of this quasi-order in the case of tensors over a fixed finite field -- namely, that it is a well-quasi-order: it admits no infinite antichains and no…

Motivated by the study of decompositions of tensors as Hadamard products (i.e., coefficient-wise products) of low-rank tensors, we introduce the notion of Hadamard rank of a given point with respect to a projective variety: if it exists, it…

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $c,d,m$ be positive integers. Let $D$ be a $p$-divisible group of codimension $c$ and dimension $d$ over $k$. Let $\scrD$ be a versal deformation of $D$ over a smooth…

We analyse a class of quantum field theory models illustrating some of the possibilities that have emerged in the general study of the short distance properties of superselection sectors, performed in a previous paper (together with R.…

We introduce the notion of Universally Decodable Matrices of Genus g (UDMG), which for g=0 reduces to the notion of Universally Decodable Matrices (UDM) introduced in [8]. A UDMG is a set of L matrices over a finite field, each with K rows,…

We show that an equivariant vector bundle on a complete toric variety is nef or ample if and only if its restriction to every invariant curve is nef or ample, respectively. Furthermore, we show that nef toric vector bundles have a…

Let $U$ be a unipotent group which is graded in the sense that it has an extension $H$ by the multiplicative group of the complex numbers such that all the weights of the adjoint action on the Lie algebra of $U$ are strictly positive. We…