7 problems

Initial-ideal containment conjecture. Then

Equivalent formulation. The map $f\cdot:R_w\to M_{w+3m}$ is an isomorphism for every $w\bsin\mathbb{N}$. This is presented as an equivalent linear-algebraic formulation of the weig…

Let $n_1$ be a positive integer, and let $R_{n_1,1}$ be the ring and $I_{L_2(n_1,1)}$ the ideal appearing in the paper, with $\mathrm{in}_{\prec}(I_{L_2(n_1,1)})$ denoting its init…

Let $K$ be a field, let $R=K[x_1,\ldots,x_n]$, and let $I=(f_1,\ldots,f_n)$ be a generic homogeneous ideal of type $(n;d_1,\ldots,d_n)$. Let $\operatorname{in}(I)$ be the initial i…

Let $H$ be a numerical semigroup minimally generated by $a_1<\dots<a_n$. Let $\operatorname{width}(H)=a_n-a_1$, let $I_H^*$ denote the ideal of initial forms associated to the defi…

The cluster-count criterion. One has

Sullivant's conjecture. For every $r$, one has