On the containment problem

Abstract

The purpose of this note is to provide an overview of the containment problem for symbolic and ordinary powers of homogeneous ideals, related conjectures and examples. We focus here on ideals with zero dimensional support. This is an area of ongoing active research. We conclude the note with a list of potential promising paths of further research.

arXiv comment13 pages, 1 figure

References in corpus (13)

• Uniform Bounds and Symbolic Powers on Smooth Varieties
• Comparison of symbolic and ordinary powers of ideals
• Are symbolic powers highly evolved?
• Counterexamples to the $I^{(3)}\subset I^2$ containment
• Resurgences for ideals of special point configurations in ${\mathbf{P}}^N$ coming from hyperplane arrangements
• Bounded Negativity and Arrangements of Lines
• Asymptotic regularity of powers of ideals of points in a weighted projective plane
• A counterexample to the containment $I^{(3)}\subset I^2$ over the reals
• Variations on Nagata's Conjecture
• The Hesse pencil of plane cubic curves
• Star configurations in ${\mathbb{P}}^n$
• A vanishing theorem and symbolic powers of planar point ideals
• An Elementary Approach to Containment Relations Between Symbolic and Ordinary Powers of Certain Monomial Ideals

Cited by in corpus (14)

• Fermat-type configurations of lines in ${\mathbb{P}}^3$ and the containment problem
• Computations involving symbolic powers
• The 21 reducible polars of Klein's quartic
• From Pappus Theorem to parameter spaces of some extremal line point configurations and applications
• Algebraic stories from one and from the other pockets
• Uniform Harbourne-Huneke Bounds via Flat Extensions
• On the Sylvester-Gallai and the orchard problem for pseudoline arrangements
• The containment problem and a rational simplicial arrangement
• On the containment problem for fat points ideals
• Lower bounds for Waldschmidt constants of generic lines in ${\mathbb{P}}^3$ and a Chudnovsky-type theorem
• Containment problem and combinatorics
• Containment problem for the quasi star configurations of points in ${\mathbb{P}}^2$
• Comparison of symbolic and ordinary powers of parity binomial edge ideals
• Containment problems for fat point schemes via asymptotic multiplier ideals