In the present paper, the authors deliver an interesting overview of the so-called containment problem for ordinary and symbolic powers of homogeneous ideals. Let us briefly present the main idea standing behind this subject. Let \(I \subset R=\mathbb{K}[x_{0},\dots, x_{N}]\) be a homogeneous ideal. If \(I = \langle f_{1},\dots, f_{n}\rangle\) is given explicitly in terms of generators, then the \(r\)-th ordinary power of \(I\), denote by \(I^{r}\), is generated by products \(f_{i_{1}}\cdots f_{i_{r}}\) of length \(r\). Now we can define the \(m\)-th symbolic power of \(I\), denoted by \(I^{(m)}\), as \[ I^{(m)} = \bigcap_{P \in \text{Ass}(I)} (I^{m}R_{P} \cap R), \] where the intesection is taken in the field of fractions and \(\text{Ass}(I)\) denotes is the set of associated primes of \(I\). The cornerstone of this subject is a merger of the following two natural problems.
Problem 1. Let \(I \subset R\) be a homogeneous ideal. Determine generators of \(I^{(m)}\) for a certain \(m\geq 2\).
Problem 2. Decide for which \(m\) and \(r\) there is the containment \(I^{(m)} \subset I^{r}\).
The groundbreaking and elegant result due to Ein, Lazarsfeld and Smith in characteristic \(0\), and due to Hochster and Huneke in positive characteristic, provides a fundamental relation between ordinary and symbolic powers of homogeneous ideals.
Theorem. Let \(I \subset R\) be a homogeneous ideal such that every component of its zero locus \(V(I)\) has codimension at most \(e\). Then the containment \[ I^{(m)} \subset I^r \] holds for all \(m\geq er\).
It is natural to ask whether the above result is sharp. In particular, the very first attempt towards this question was indicated by C. Huneke [“Open problems on powers of ideals”, Notes from a workshop on Integral Closure, Multiplier Ideals and Cores, AIM, December 2006, http://www.aimath.org/WWN/integralclosure/Huneke.pdf].
Question (Huneke). Let \(I\) be a saturated ideal of a reduced finite set of points in \(\mathbb{P}^{2}\). Does the containment \[ I^{(3)} \subset I^{2} \] hold?
This problem was further generalized by Harbourne and Bocci jointly with Huneke to any arbitrary projective space \(\mathbb{P}^{N}\).
Question (Bocci-Harbourne-Huneke). Let \(I\) be a saturated ideal of a finite set of reduced points in \(\mathbb{P}^{N}\). Does the containment \[ I^{(m)} \subset I^{r} \] hold for \(m \geq Nr - (N-1)\)?
The main aim of this overview is to deliver recent developments on the containment problem and to present some counterexamples to the above questions that have been discovered recently. The first counterexample to Huneke’s question was found by Dumnicki, Szemberg and Tutaj-Gasińska [M. Dumnicki et al., J. Algebra 393, 24–29 (2013; Zbl 1297.14008)] – their ideal \(I\) is given by \(12\) triple points determined by the dual Hesse configuration of \(9\) lines in the complex projective plane. Surprisingly, almost all known counterexamples to Huneke’s questions are determined by very specific arrangements of lines possessing some extremal properties – these are coming from classical reflection groups and some of them play an important role in the so-called orchard problem. It would be really nice to know whether we can find some natural numerical criteria which could possibly allow to determine new non-trivial counterexamples to the questions. At the end of the overview, the authors present a list of open problems to think about, some of them are really challenging.