Related papers: Steiner Configurations ideals: containment and col…

We investigate symbolic and regular powers of monomial ideals. For a square-free monomial ideal $I$ in $k[x_0, \ldots, x_n]$ we show $I^{t(m+e-1)-e+r)}$ is a subset of $M^{(t-1)(e-1)+r-1}(I^{(m)})^t$ for all positive integers $m$, $t$ and…

In the paper we prove the containment $I^{(nm)}\subset M^{(n-1)m}I^m$, for a radical ideal $I$ of $s$ general points in $\mathbb{P}^n$, where $s\geq 2^n$. As a corollary we get that the Chudnovsky Conjecture holds for a very general set of…

By an easy application of Skoda's theorem on ideal generation, a non-local version of the Briancon-Skoda theorem is obtained. In particular, the symbolic powers $I^{(p)}$ of a zero dimensional radical ideal $I$ generated by $r$ holomorphic…

In this paper, we initiate the study of discrepancy questions for combinatorial designs. Specifically, we show that, for every fixed $r\ge 3$ and $n\equiv 1,3 \pmod{6}$, any $r$-colouring of the triples on $[n]$ admits a Steiner triple…

Hochster and Huneke proved in \cite{HH5} fine behaviors of symbolic powers of ideals in regular rings, using the theory of tight closure. In this paper, we use generalized test ideals, which are a characteristic $p$ analogue of multiplier…

Let $K$ be a field and $I$ a monomial ideal of the polynomial ring $S=K[x_1,..., x_n]$ generated by monomials $u_1,u_2,..., u_t$. We show that $S/I$ is pretty clean if either: 1) $u_1,u_2,..., u_t$ is a filter-regular sequence, 2)…

Let $R$ be a (not necessarily commutative) ring whose additive group is finitely generated and let $U_n(R) \subset GL_n(R)$ be the group of upper-triangular unipotent matrices over $R$. We study how the homology groups of $U_n(R)$ vary with…

The purpose of this note is twofold. We present first a vanishing theorem for families of linear series with base ideal being a fat points ideal. We apply then this result in order to give a partial proof of a conjecture raised by Bocci,…

Given a polynomial ring $C$ over a field and proper ideals $I$ and $J$ whose generating sets involve disjoint variables, we determine how to embed the associated primes of each power of $I+J$ into a collection of primes described in terms…

We systematically extend the statement that the configurational entropy provides an alternative approach to studying gravitational stability of compact objects, carried out in the previous work of M. Gleiser and N. Jiang, Phys. Rev. D {\bf…

In this survey article we give a brief history of symbolic powers and its connection with the interesting problem of set-theoretic complete intersection. We also state a few problems and conjectures. Recently, in connection to symbolic…

Let $G$ be a graph and suppose we are given, for each $v \in V(G)$, a strict ordering of the neighbors of $v$. A set of matchings ${\cal M}$ of $G$ is called internally stable if there are no matchings $M,M' \in {\cal M}$ such that an edge…

This paper is concerned with the tight closure of an ideal $I$ in a commutative Noetherian ring $R$ of prime characteristic $p$. The formal definition requires, on the face of things, an infinite number of checks to determine whether or not…

We develop a notion of containment for independent sets in hypergraphs. For every $r$-uniform hypergraph $G$, we find a relatively small collection $C$ of vertex subsets, such that every independent set of $G$ is contained within a member…

We consider for two based graphs $G$ and $H$ the sequence of graphs $G_k$ given by the wedge sum of $G$ and $k$ copies of $H$. These graphs have an action of the symmetric group $\Sigma_k$ by permuting the $H$-summands. We show that the…

The aim of this paper is to study mixed multiplier ideals associated to a tuple of ideals in a two-dimensional local ring with a rational singularity. We are interested in the partition of the real positive orthant given by the regions…

Let S=K[x_1,x_2,...,x_n] be a polynomial ring in n variables over a field K. Stanley's conjecture holds for the modules I and S/I, when I is a critical monomial ideal. We calculate the Stanley depth of S/I when I is a canonical critical…

Let $n,s,k$ be three positive integers such that $1\leq s\leq(n-k+1)/k$ and let $[n]=\{1,\ldots,n\}$. Let $H$ be a $k$-graph with vertex set $\{1,\ldots,n\}$, and let $e(H)$ denote the number of edges of $H$. Let $\nu(H)$ and $\tau(H)$…

Let $\mathcal{H}_d^{(t)}$ ($t \geq -d$, $t>-3$) be the reproducing kernel Hilbert space on the unit ball $\mathbb{B}_d$ with kernel \[ k(z,w) = \frac{1}{(1-\langle z, w \rangle)^{d+t+1}} . \] We prove that if an ideal $I \triangleleft…

Let $V_1, V_2, V_3, \dots $ be a sequence of $\mathbb{Q}$-vector spaces where $V_n$ carries an action of $\mathfrak{S}_n$ for each $n$. {\em Representation stability} and {\em multiplicity stability} are two related notions of when the…