Fermat-like平面和线排列的符号幂的初始次数

Fermat-type configurations of lines in $\mathbb P^3$ and the containment problem

The purpose of this note is to show a new series of examples of homogeneous ideals $I$ in ${\mathbb K}[x,y,z,w]$ for which the containment $I^{(3)}\subset I^2$ fails. These ideals are supported on certain arrangements of lines in ${\mathbb P}^3$, which resemble Fermat configurations of points in ${\mathbb P}^2$, see \cite{NagSec16}. All examples exhibiting the failure of the containment $I^{(3)}\subseteq I^2$ constructed so far have been supported on points or cones over configurations of points. Apart of providing new counterexamples, these ideals seem quite interesting on their own.

A homological criterion for the containment between symbolic and ordinary powers for some ideals of points in $\mathbb{P}^2$

We establish a criterion for the (failure of) the containment $I^{(m)}\subset I^r$ for 3-generated ideals $I$ defining reduced sets of points in $\mathbb{P}^2$. Our criterion arises from studying the minimal free resolutions of the powers of $I$, specifically the minimal free resolutions for $I^m$ and $I^r$. We apply this criterion to two point configurations that have recently arisen as counterexamples to a question of B. Harbourne and C. Huneke: the Fermat configuration and the Klein configuration.

Symbolic Rees algebras and set-theoretic complete intersections

In this paper we extend a result of Cowsik on set-theoretic complete intersection and a result Huneke, Morales and Goto and Nishida about Noetherian symbolic Rees algebras of ideals. As applications, we show that the symbolic Rees algebras of the following ideals are Noetherian and the ideals are set-theoretic complete intersections: (a) the edge ideal of a complete graph, (b) the Fermat ideal and (c) the Jacobian ideal of a certain hyperplane arrangement.

Configurations of Linear Spaces of Codimension Two and the Containment Problem

We show that the union of certain codimension two flats in Fermat hyperplane arrangements does not have the symbolic cube of the defining ideal contained in the square of the ideal.

Singular loci of reflection arrangements and the containment problem

This paper provides insights into the role of symmetry in studying polynomial functions vanishing to high order on an algebraic variety. The varieties we study are singular loci of hyperplane arrangements in projective space, with emphasis on arrangements arising from complex reflection groups. We provide minimal sets of equations for the radical ideals defining these singular loci and study containments between the ordinary and symbolic powers of these ideals.

不可约二次曲线上的符号一般初始点系

The Symbolic Generic Initial System of Points on an Irreducible Conic

本文研究了K[x，y，z]中理想I的符号一般初始系对应于P2的r个点在不可约二次曲线上的排列的极限行为。特别地，当r大于或等于4时，我们证明了该系统的极限形状是由（2，0）和（0，r/2）直线上的所有点组成的R2的子集.

Ideals generated by $a$-fold products of linear forms have linear graded free resolution

Given $Σ\subset R:=\mathbb K[x_1,\ldots,x_k]$, where $\mathbb K$ is a field of characteristic 0, any finite collection of linear forms, some possibly proportional, and any $1\leq a\leq |Σ|$, we prove that $I_a(Σ)$, the ideal generated by all $a$-fold products of $Σ$, has linear graded free resolution. This allows us to determine a generating set for the defining ideal of the Orlik-Terao algebra of the second order of a line arrangement in $\mathbb P_{\mathbb{K}}^2$, and to conclude that for the case $k=3$, and $Σ$ defining such a line arrangement, the ideal $I_{|Σ|-2}(Σ)$ is of fiber type. We …

On the containment problem for fat points ideals

In this note we show that Harbourne's conjecture is true for symbolic powers of ideals of points, we check that the stable version of this conjecture is valid for ideals of very general points (resp. generic points) in $\mathbb P_{\mathbb K}^N$ (resp. $\mathbb P_{\mathbb K(\underline{z})}^N$). We also show that this conjecture and the Harbourne-Huneke conjecture are true for a class of ideals $I$ defining fat points obtained from line arrangements in $\mathbb P_{\mathbb K}^2$.