16 problems

Moreno-Socías's conjecture. The ideal $I$ is weakly reverse lexicographical.

Let $G$ be a closed graph, let $J_G$ be its binomial edge ideal, and let $<$ be a diagonal term order. Ene–Herzog–Hibi conjecture. The graded Betti numbers of $J_G$ agree with thos…

Initial-ideal containment conjecture. Then

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 $S=K[x_1,\ldots,x_n]$ and let $f_1,f_2,\ldots,f_n\in S$ be a generic sequence of homogeneous polynomials of degrees $d_1,d_2,\ldots,d_n$. Set $I=(f_1,\ldots,f_n)$, let $J$ be t…

Let $S=K[x_1,\ldots,x_n]$ and let $I\subset S$ be an $\mathfrak m$-primary homogeneous ideal generated in degree $d$, where $\mathfrak m=(x_1,\ldots,x_n)$. Suppose that the minimal…

Let $G$ be a closed graph, let $J_G$ be its binomial edge ideal, and let $\operatorname{in}_{\operatorname{lex}}(J_G^m)$ denote the lexicographic initial ideal of $J_G^m$. Write…

Let $R=K[X]$ and let $\frac{}{}\prec$ be the monomial order defined by Soll and Welker. Let $I_t$ be the determinantal ideal and let $\Delta$ be the simplicial complex defined by t…

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 $P=\mathbb{C}[x,y,z]$ be a polynomial ring with weights $\operatorname{wt}(x)=1$, $\operatorname{wt}(y)=2$, and $\operatorname{wt}(z)=3$. Let $f\bsin P$ be a general…

Let $G$ be a graph, let $J_G$ be its binomial edge ideal in a polynomial ring $S$, and let $\operatorname{in}_<(J_G)$ be its initial ideal with respect to a term order $<$. Regular…

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 $I=(f_1,\ldots,f_r)$ be a complete intersection ideal in a convergent power series ring, and let $I'=(f_1',\ldots,f_r')$, where each $f_i'$ is the $n$-jet of $f_i$ for some…

The cluster-count criterion. One has

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

Let $S$ be the homogeneous coordinate ring, let $Y$ be a $\mathrm{GL}$-stable subset of the Hilbert scheme $\mathbf{Hilb}_{p(t)}^{n}$, and let $\mathbf{G}_Y$ denote its double-gene…