13 problems

Let $\mathcal{A}$ be a central arrangement in $\mathbb{K}^{n+1}$. Write $\operatorname{rgin}(J(\mathcal{A}))$ for the generic initial ideal of its Jacobian ideal, and set … A minim…

Moreno's conjecture. The ideal $J$ is almost reverse lexicographic. The conjecture predicts a precise form for the initial ideal of a generic complete intersection. The source late…

Let $S_d(U)$ be a squeezed sphere, or an S-squeezed sphere, constructed from a shifted order ideal $U$. A simplicial sphere has the strong Lefschetz property when its Lefschetz map…

Kalai's conjecture. For every such shifted order ideal,

Let $S$ be a graded polynomial ring and let $I\subset S$ be a homogeneous ideal. Write $\operatorname{gin}(I)$ for the generic initial ideal of $I$. The homological degree comparis…

Let $R=k[x_1,x_2,\ldots,x_r]$ be a polynomial ring over a field. An ideal $I\subseteq R$ has the saturated chain property if, for every non-minimal associated prime ideal $Q$ of…

Let $I$ be a monomial ideal, let $D_{\mathcal L}(I)$ be its distraction with respect to the matrix $\mathcal L$, let $\operatorname{drl}$ denote degree reverse lexicographic order,…

Let $P=K[x_1,dots,x_n]$ be a polynomial ring, let $I$ be an ideal of distractions stemming from a strongly stable ideal, and let $x_n$ not divide zero modulo $I$. Write $I_{x_n}$ f…

Let $A$ be a polynomial ring, let $I\subseteq A$ be a $\mathbb{Z}^m$-graded Cartwright–Strumfels ideal, let $\operatorname{gin}(I)$ be its $\mathbb{Z}^m$-graded generic initial ide…

Let $A=\mathbb{K}[x_1,\dots,x_n]$ be a polynomial ring over a field $\mathbb{K}$, let $I\subseteq A$ be a $\mathbb{Z}^m$-graded Cartwright–Sturmfels ideal, and let…

Let $S$ be the polynomial ring with its ${\mathbb Z}^n$-grading, let $\mathfrak m$ be its graded maximal ideal, let $I$ be a Cartwright–Sturmfels (CS) ideal, and let $J$ be its…

Let $S$ be the polynomial ring with its ${\mathbb Z}^n$-grading, let $I$ be a Cartwright–Sturmfels (CS) ideal, and let $J$ be its ${\mathbb Z}^n$-graded generic initial ideal. The…

Let $R$ be a polynomial ring over an infinite field, and let $I$ be a graded ideal of $R$. The completely $\mathfrak m$-full/componentwise linearity conjecture. The ideal $I$ is co…