Related papers: Arrangements of ideal type

Let $V(I)$ be a polarized projective variety or a subvariety of a product of projective spaces and let $A$ be its (multi-)homogeneous coordinate ring. Given a full-rank valuation $\mathfrak v$ on $A$ we associate weights to the coordinates…

For any ideal $I$ of finite projective dimension in a commutative noetherian local ring $R$, we prove that if the conormal module $I/I^2$ has finite projective dimension over $R/I$, then $I$ must be generated by a regular sequence. This…

We consider the polynomial ring in finitely many variables over an algebraically closed field of positive characteristic, and initiate the systematic study of ideals preserved by the action of the general linear group by changes of…

M.-P. Sch\"utzenberger asked to determine the support of the free Lie algebra ${\mathcal L}_{{\mathbb Z}_{m}}(A)$ on a finite alphabet $A$ over the ring ${\mathbb Z}_{m}$ of integers $\bmod m$ and all the corresponding pairs of twin and…

In a commutative ring $R$ with unity, given an ideal $I$ of $R$, Anderson and Badawi in 2011 introduced the invariant $\omega(I)$, which is the minimal integer $n$ for which $I$ is an $n$-absorbing ideal of $R$. In the specific case that $R…

We prove that the modules of differential operators of order 2 on the classical Coxeter arrangements are free by exhibiting bases. For this purpose, we use Cauchy-Sylvester's theorem on compound determinants and Saito-Holm's criterion. In…

We study the free (associative, non-commutative) Baxter algebra on one generator. The first explicit description of this object is due to Ebrahimi-Fard and Guo. We provide an alternative description in terms of a certain class of trees,…

Let $A_1$ be the (first) Weyl algebra, and let $G$ be its automorphism group. We study the natural action of $G$ on the space of isomorphism classes of right ideals of $A_1$ (equivalently, of finitely generated rank 1 torsion-free right…

Let $K$ be a unital associative and commutative ring and let $K \langle X \rangle$ be the free unital associative $K$-algebra on a non-empty set $X$ of free generators. Define a left-normed commutator $[a_1, a_2, \dots , a_n]$ inductively…

We study the combinatorics of pseudoline arrangements in the real projective plane. Our focus lies on two classes of arrangements: simplicial arrangements and arrangements whose characteristic polynomials have only real roots. We derive…

With each of the usual tree forcings I (e.g., I = Sacks forcing S, Laver forcing L, Miller forcing M, Mathias forcing R, etc.) we associate a sigma--ideal i^0 on the reals as follows: A \in i^0 iff for all T \in I there is S \leq T (i.e. S…

We study a generalized version of Terao's famous addition theorem for free arrangements to the category of those with projective dimension one. Namely, we give a criterion to determine the algebraic structure of logarithmic derivation…

We consider a pair of semigroups associated to a signed poset, called the root semigroup and the weight semigroup, and their semigroup rings, $R_P^\mathrm{rt}$ and $R_P^\mathrm{wt}$, respectively. Theorem 4.1.5 gives generators for the…

The aim of this note is to discuss the following quite queer Problem: \noindent GIVEN \noindent i) the free non-commutative polynomial ring, ${\Cal P} := {\Bbb F}\langle X_1,\ldots,X_n\rangle$ {\em (public)}, \noindent ii) a bilateral ideal…

Many ${\mathcal W}$-(super)algebras which are defined by the generalized Drinfeld-Sokolov reduction are also known or expected to have coset realizations. For example, it was conjectured by Ito that the principal ${\mathcal W}$-superalgebra…

In classical and real algebraic geometry there are several notions of the radical of an ideal I. There is the vanishing radical defined as the set of all real polynomials vanishing on the real zero set of I, and the real radical defined as…

The aim of this thesis is to investigate the Betti diagrams of squarefree monomial ideals in polynomial rings. We use two key tools to help us study these diagrams. The first is the Stanley-Reisner Correspondence, which assigns a unique…

Using the recent results on square-free Gr\"obner degenerations by Conca and Varbaro, we proved that if a homogeneous ideal $I$ of a polynomial ring is such that its initial ideal $\mathrm{in}_<(I)$ is square-free and $\beta_0(I) =…

We extend the relation between random matrices and free probability theory from the level of expectations to the level of all correlation functions (which are classical cumulants of traces of products of the matrices). We introduce the…

Let $I_1,\dots,I_n$ be ideals generated by linear forms in a polynomial ring over an infinite field and let $J = I_1 \cdots I_n$. We describe a minimal free resolution of $J$ and show that it is supported on a polymatroid obtained from the…