4 problems

Let $I\subseteq k[x_0,\dots,x_n]$ be a homogeneous radical ideal of big height $h$, and let $\mathcal{M}=(x_0,\dots,x_n)$ be the graded maximal ideal. Stable Harbourne–Huneke conje…

Let $\mathbb K$ be a field of characteristic $0$, let $R=\mathbb K[x_0,\ldots,x_N]$, and let $I$ be the defining ideal of a finite set of points in $\mathbb P_{\mathbb K}^N$. Stabl…

Let $I$ be a homogeneous ideal, and let $e$ be the maximum height of an associated prime of $I$. Associated-prime-height containment conjecture. One has … The known containment…

Let $K[\mathbb{P}^{N}]$ be the homogeneous coordinate ring, let $M$ be its homogeneous maximal ideal, and let $I$ be the ideal of a finite set of points $p_i\in\mathbb{P}^{N}$. Evo…