Steiner Configurations ideals: containment and colouring

TL;DR

This paper proves several conjectures related to the containment problem for ideals associated with Steiner configurations and explores the connection between hypergraph colorability and ideal containments.

Contribution

It establishes the validity of the Stable Harbourne and Harbourne--Huneke Conjectures for Steiner configuration ideals and links hypergraph colorability to containment failures.

Findings

Stable Harbourne and Harbourne--Huneke Conjectures hold for Steiner configuration ideals.

The ideal of a Steiner configuration complement has expected resurgence.

Hypergraph colorability relates to containment issues in cover ideals.

Abstract

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 $\mathbb{P}^{n}_{k}$. We can also show that the ideal of a Complement of a Steiner Configuration of points has expected resurgence, that is, its resurgence is strictly less than its big height, and it also satisfies Chudnovsky and Demailly's Conjectures. Moreover, given a hypergraph $H$, we also study the relation between its colourability and the failure of the containment problem for the cover ideal associated to $H$. We apply these results in the case that $H$ is a Steiner System.