13 problems

Let $I$ be the defining ideal of $s$ reduced points in ${\mathbb P}^N$, let $\mathfrak m$ denote the homogeneous maximal ideal, and let $h$ be the parameter appearing in the contai…

Eisenbud–Mazur conjecture. The containment

Let $I$ be a radical ideal of big height $c$ in a regular ring. For each positive integer $n$, let $I^{(n)}$ denote the $n$th symbolic power of $I$. Stable Harbourne conjecture. On…

Stable Harbourne conjecture. For all $n \gg 0$,

Beheshti–Eisenbud conjecture. For $q\gg0$,

Stable Harbourne–Huneke containment conjecture. Both containments hold for all sufficiently large $r$ as specified above. The stable containment is presented as implying Chudnovsky…

Let $Z\subset {\mathbb{P}}^N$ be a fat point scheme. For a homogeneous ideal $J$, write $\alpha(J)$ for its least nonzero degree, and write $\widehat{\alpha}(I(Z))$ for the Waldsch…

Let $I\subseteq {\mathbb{K}}[{\mathbb{P}}^N]$ be a radical homogeneous ideal, and let $h_I$ denote the maximum height of an associated prime of $I$. Grifo's conjecture. The contain…

Let $\mathbbm k$ be a field, let $\mathbb P^N_{\mathbbm k}$ be projective space, and let $I\subseteq\mathbbm k[\mathbb P^N_{\mathbbm k}]$ be a homogeneous radical ideal of big heig…

Bocci–Cooper–Harbourne conjecture. For all $m\geq1$,

Let $K$ be an algebraically closed field, let $I$ be the radical ideal of a finite set of points in $\mathbb{P}^{N}$, and let $M=(x_0,\ldots,x_N)$ be the maximal homogeneous ideal…

Let $K$ be an algebraically closed field, let $M=(x_0,\ldots,x_N)$ be the maximal homogeneous ideal of $K[\mathbb{P}^{N}]$, and let $I$ be the radical ideal of a finite set of poin…

Let $K$ be an algebraically closed field, let $R=K[\mathbb{P}^{N}]$, and let $I\subseteq R$ be a homogeneous ideal. Bocci-Harbourne-Essential conjecture. For every $r$, … The state…