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 pairs $m,r\in \mathbb{N}$, $I^{\left(m\right)}\subseteq I^r$ 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 ${\mathbb{P}}_k^n$. 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.

中文翻译：

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Steiner配置的理想选择：密闭性和着色

给定一个均匀的理想 $\mathrm{一世}\subseteq ķ[X_0，\dots ，X_{ñ}]$，遏制问题研究了I的符号幂与规则幂之间的关系，即它要求哪些对$米，\mathrm{[R}\in \mathbb{ñ}$， ${\mathrm{一世}}^{（米）}\subseteq {\mathrm{一世}}^{\mathrm{[R}}$持有。在过去的几年中，对此问题提出了一些推测，这创造了当前兴趣和正在进行的调查的活跃领域。在本文中，我们研究了稳定的Harbourne猜想和稳定的Harbourne-Huneke猜想，并表明它们对点的Steiner构型的补码的定义理想成立。${\mathbb{P}}_{ķ}^{ñ}$。我们还可以证明，点的Steiner构型的补充的理想具有预期的回潮，即其回潮严格小于其高度，并且还满足Chudnovsky和Demailly的猜想。此外，给定超图H，我们还研究了与H有关的覆盖理想状态下，其可着色性与收容问题的失败之间的关系。我们在H是Steiner系统的情况下应用这些结果。