Related papers: High order free hyperplane arrangements in 3-dimen…
We establish the well-posedness in Gevrey function space with optimal class of regularity 2 for the three dimensional Prandtl system without any structural assumption. The proof combines in a novel way a new cancellation in the system with…
On a $3$D manifold, a Weyl geometry consists of pairs $(g, A) =$ (metric, $1$-form) modulo gauge $\widehat{g} = {\rm e}^{2\varphi} g$, $\widehat{A} = A + {\rm d}\varphi$. In 1943, Cartan showed that every solution to the Einstein-Weyl…
Let $\mathbb{F}_q^d$ be the $d$-dimensional vector space over the finite field with $q$ elements. For a subset $E\subseteq \mathbb{F}_q^d$ and a fixed nonzero $t\in \mathbb{F}_q$, let $\mathcal{H}_t(E)=\{h_y: y\in E\}$, where $h_y$ is the…
Ziegler showed that free arrangements have free restricted multiarrangements (multirestrictions). After Ziegler's work, several results concerning "reverse direction", namely characterizing freeness of an arrangement via that of…
In this note we give a negative answer to a question proposed by Almendra-Hern\'andez and Mart\'inez-Sandoval. Let $n\le m$ be positive integers and let $X$ and $Y$ be sets of sizes $n$ and $m$ in $\mathbb{R}^{n-1}$ such that every pair of…
In this paper we revisit and extend the work done by Chaturvedu et al, as well as Dabrowski and Parashar. The basic premise is to take a deformed coordinate system and give is a concrete realization. This realization is given by a parameter…
We give a complete classification of modular categories of dimension $p^3m$ where $p$ is prime and $m$ is a square-free integer. When $p$ is odd, all such categories are pointed. For $p=2$ one encounters modular categories with the same…
We compare each coefficient of the reduced characteristic polynomial of a simple arrangement and that of its Ziegler restriction. As a consequence we can show that the former is not less than the latter in the category of tame arrangements.…
We give a counterexample to a conjecture by Miasnikov, Ventura and Weil, stating that an extension of free groups is algebraic if and only if the corresponding morphism of their core graphs is onto, for every basis of the ambient group. In…
In this paper, we study the class of free multiarrangements of hyperplanes. Specifically, we investigate the relations between freeness over a field of finite characteristic and freeness over the rationals.
In this note we study the freeness of the module of derivations on all moduli of the $X_3$ arrangement with multiplicities. We use homological techniques stemming from work of Yuzvinsky, Brandt, and Terao which have recently been developed…
A graphic arrangement is a subarrangement of the braid arrangement whose set of hyperplanes is determined by an undirected graph. A classical result due to Stanley, Edelman and Reiner states that a graphic arrangement is free if and only if…
Given integers $a$ and $m\ge 2$, let $\Hm$ be the following set of integral points $$ \Hm= \{(x,y) \ : \ xy \equiv a \pmod m,\ 1\le x,y \le m-1\} $$ We improve several previously known upper bounds on $v_a(m)$, the number of vertices of the…
We consider the behaviour of logarithmic differential forms on arrangements and multiarrangements of hyperplanes under the operations of deletion and restriction, extending early work of G\"unter Ziegler. The restriction of logarithmic…
Let $k$ be a field, $m$ a positive integer, $\mathbb{V}$ an affine subvariety of $\mathbb{A}^{m+3}$ defined by a linear relation of the form $x_{1}^{r_{1}}\cdots x_{m}^{r_{m}}y=F(x_{1}, \ldots , x_{m},z,t)$, $A$ the coordinate ring of…
We show that the number of lines in an $m$--homogeneous supersolvable line arrangement is upper bounded by $3m-3$ and we classify the $m$--homogeneous supersolvable line arrangements with two modular points up-to lattice-isotopy. A lower…
We give a complete classification of free and non-free multiplicities on the $A_3$ braid arrangement. Namely, we show that all free multiplicities on $A_3$ fall into two families that have been identified by Abe-Terao-Wakefield (2007) and…
We show that if $A$ and $H$ are Hopf algebras that have equivalent tensor categories of comodules, then one can transport what we call a free Yetter-Drinfeld resolution of the counit of $A$ to the same kind of resolution for the counit of…
Let A be a (central) arrangement of hyperplanes in a finite dimension complex vector space V. Let M(A) be the dependence matroid determined by A. The Orlik-Solomon algebra OS(M) of a matroid M is the exterior algebra on the points modulo…
We study a class of first-order theories whose complete quantifier-free types with one free variable either have a trivial positive part or are isolated by a positive quantifier-free formula--plus a few other technical requirements. The…