A short note on containment of cores

An Elementary Approach to Containment Relations Between Symbolic and Ordinary Powers of Certain Monomial Ideals

The purpose of this note is to find an elemenary explanation of a surprising result of Ein--Lazarsfeld--Smith \cite{ELS} and Hochster--Huneke \cite{HH} on the containment between symbolic and ordinary powers of ideals in simple cases. This line of research has been very active ever since, see for instance \cites{BC,HaH,DST} and the references therein, by now the literature on this topic is quite extensive. By `elementary' we refer to arguments that among others do not make use of resolution of singularities and multiplier ideals nor tight closure methods. Let us quickly recall the statement \c…

A vanishing theorem and symbolic powers of planar point ideals

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, Harbourne and Huneke concerning containment relations between ordinary and symbolic powers of planar point ideals.

On codimension two flats in Fermat-type arrangements

In the present note we study certain arrangements of codimension $2$ flats in projective spaces, we call them "Fermat arrangements". We describe algebraic properties of their defining ideals. In particular, we show that they provide counterexamples to an expected containment relation between ordinary and symbolic powers of homogeneous ideals.

Cartier 代数的 Cartier 核心映射

The Cartier core map for Cartier algebras

本文为 Cartier 代数定义 Cartier 核心映射，证明其连续性、保包含性及对相容理想的固定作用，并推广至任意理想，还刻画了 Stanley-Reisner 环的素一致 F-相容理想。

The $*$-core of the graded maximal ideal in a Stanley-Reisner ring

We consider ideals $I$ in a Stanley-Reisner ring $k[Δ]$ over the simplical complex $Δ$, such that the tight closure of $I$, $I^*$, is equal to $\mathfrak{m}$, the standard graded maximal ideal of $k[Δ]$. We determine the minimal number of generators of $I$ to be the $\dim Δ+1$ and note the important role this value plays in bounding the intersection of all such ideals $I$. We make mention of this intersection in special cases of Stanley-Reisner rings. We conclude with a description of how this work relates to integral closure.

The $cl-core$ of an ideal

We expand the notion of core to $cl$-core for Nakayama closures $cl$. In the characteristic $p>0$ setting, when $cl$ is the tight closure, denoted by *, we give some examples of ideals when the core and the *-core differ. We note that *-core$(I)=$ core$(I)$, if $I$ is an ideal in a one-dimensional domain with infinite residue field or if $I$ is an ideal generated by a system of parameters in any Noetherian ring. More generally, we show the same result in a Cohen--Macaulay normal local domain with infinite perfect residue field, if the analytic spread, $\ell$, is equal to the *-spread and $I$ i…

Efficient generation of ideals in core subalgebras of the polynomial ring k[t] over a field k

This note aims at finding explicit and efficient generation of ideals in subalgebras $R$ of the polynomial ring $S=k[t]$ ($k$ a field) such that $t^{c_0}S \subseteq R$ for some integer $c_0 > 0$. The class of these subalgebras which we call cores of $S$ includes the semigroup rings $k[H]$ of numerical semigroups $H$, but much larger than the class of numerical semigroup rings. For $R=k[H]$ and $M \in \operatorname{Max}R$, our result eventually shows that $μ_{R}(M) \in \{1,2,μ(H)\}$ where $μ_{R}(M)$ (resp. $μ(H)$) stands for the minimal number of generators of $M$ (resp. $H$), which covers in t…