Chudnovsky 的猜想与稳定的 Harbourne-Huneke 包含

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 $\mathbb P_{\mathbb K}^2$.

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.

Containments of symbolic powers of ideals of generic points in $\PP^3$

We show that the Conjecture of Harbourne and Huneke, $I^{(Nr-(N-1))} \subset M^{(r-1)(N-1)}I^{r}$ holds for ideals of generic (simple) points in $\PP^3$. As a result, for such ideals we prove the following bounds, which can be recognized as generalizations of Chudnovsky bounds: $α(I^{(3m-k)}) \geq mα(I)+2m-k$, for any $m \geq 1$ and $k=0,1,2$. Moreover, we obtain lower bounds for the Waldshmidt constant for such ideals.

On the containment $I^{(3)} \subseteq I^2$ and configurations of triple points in Böröczky line arrangements

We study sets of triple points of Böröczky's arrangements of lines in the context of the containment problem proposed by Harbourne and Huneke. We show that in the class of those arrangements, the smallest counterexample to the containment $I^{(3)} \subseteq I^2$ is obtained when the number of lines is equal to 12.

Symbolic powers of ideals of generic points in P^3

B. Harbourne and C. Huneke conjectured that for any ideal $I$ of fat points in $P^N$ its $r$-th symbolic power $I^{(r)}$ should be contained in $M^{(N-1)r}I^r$, where $M$ denotes the homogeneous maximal ideal in the ring of coordinates of $P^N$. We show that this conjecture holds for the ideal of any number of simple (not fat) points in general position in $P^3$ and for at most $N+1$ simple points in general position in $P^N$. As a corollary we give a positive answer to Chudnovsky Conjecture in the case of generic points in $P^3$.

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.

Waldschmidt常数的下界及Demailly猜想对于一般点和非常一般点的研究

Lower bounds for Waldschmidt constants and Demailly's Conjecture for general and very general points

本文证明了关于Waldschmidt常数与第二符号幂初始度下界的关系的Demailly猜想，并讨论了Harbourne-Huneke包含关系及Demailly猜想在一般点和低维射影空间中的结果。

德马依猜想与包含问题

Demailly's Conjecture and the Containment Problem

本文研究了德马依猜想在一般点集中的情况，并证明了关于符号幂与普通幂之间包含关系的猜想，该猜想特别推导出德马依界，适用于通用行列式理想和星形配置定义理想。

Symbolic powers of ideals defining F-pure and strongly F-regular rings

Given a radical ideal $I$ in a regular ring $R$, the Containment Problem of symbolic and ordinary powers of $I$ consists of determining when the containment $I^{(a)} \subseteq I^b$ holds. By work of Ein-Lazersfeld-Smith, Hochster-Huneke and Ma-Schwede, there is a uniform answer to this question, but the resulting containments are not necessarily best possible. We show that a conjecture of Harbourne holds when $R/I$ is F-pure, and prove tighter containments in the case when $R/I$ is strongly F-regular.

Interpolation and The Weak Lefschetz Property

Our starting point is a basic problem in Hermite interpolation theory, namely determining the least degree of a homogeneous polynomial that vanishes to some specified order at every point of a given finite set. We solve this problem if the number of points is small compared to the dimension of their linear span. This also allows us to establish results on the Hilbert function of ideals generated by powers of linear forms. The Verlinde formula determines such a Hilbert function in a specific instance. We complement this result and also determine the Castelnuovo-Mumford regularity of the corresp…

Steiner Configurations ideals: containment and colouring

Given a homogeneous ideal $I \subseteq k[x_0,\dots,x_n]$, the Containment problem studies the relation between symbolic and regular powers of $I$, that is, it asks for which pair $m, r \in \mathbb{N}$, $I^{(m)} \subseteq I^r$ holds. In the last years, several conjectures have been posed on this problem, creating an active area of current interests and ongoing investigations. In this paper, we investigated the Stable Harbourne Conjecture and the Stable Harbourne -- Huneke Conjecture and we show that they hold for the defining ideal of a Complement of a Steiner configuration of points in $\mathb…

The Alpha Problem & Line Count Configurations

Motivated by the work of Chudnovsky and the Eisenbud-Mazur Conjecture on evolutions, Harbourne and Huneke give a series of conjectures that relate symbolic and regular powers of ideals of fat points in $\mathbb P^n$. The conjectures involve both containment statements and bounds for the initial degree in which there is a non-zero form in an ideal. Working with initial degrees, we verify two of these conjectures for special line count configurations in projective 2-space over an algebraically closed field of characteristic 0.

符号幂的费马点配置理想初始次数

The Initial Degree of Symbolic Powers of Ideals of Fermat Configuration of Points

本文计算了费马点配置理想符号幂的最小生成次数，验证了Chudnovsky猜想、Demailly猜想和Harbourne-Huneke包容问题，并计算了Waldschmidt常数和渐进 resurgence 数。

Symbolic powers versus regular powers of ideals of general points in P^1 x P^1

Recent work of Ein-Lazarsfeld-Smith and Hochster-Huneke raised the problem of which symbolic powers of an ideal are contained in a given ordinary power of the ideal. Bocci-Harbourne developed methods to address this problem, which involve asymptotic numerical characters of symbolic powers of the ideals. Most of the work done up to now has been done for ideals defining 0-dimensional subschemes of projective space. Here we focus on certain subschemes given by a union of lines in ${\bf P}^3$ which can also be viewed as points in ${\bf P}^1\times {\bf P}^1$. We also obtain results on the closely r…