9 problems

Let $G$ be a simple graph on vertex set $V(G)=[n]$, let $R=\Bbbk[x_1,\ldots,x_n]$, and let $I(G^d)$ be the edge ideal of the $d$-th power of $G$, where $d$ is a positive integer. T…

Let $X\subseteq\mathbb P^r$ be a nondegenerate projective variety. Its Castelnuovo–Mumford regularity is denoted by $\operatorname{reg} X$, its degree by $\deg X$, and its codimens…

For each $n$, define … where $R_{v,w}$ is the associated graded ring of the local ring of the Schubert variety $X_w$ at the torus-fixed point $e_v$. Maximal-regularity conjecture.…

Let $GL_n/B$ be the complete flag variety, with Schubert varieties $X_w$ indexed by permutations $w\in\mathfrak S_n$. For $u\leq v\leq w$ in Bruhat order, let $R_{u,w}$ and…

Regularity conjecture. The Castelnuovo–Mumford regularity satisfies

Let $G$ be a simple graph, and let $I(G)$ be its edge ideal. For each integer $s\geq 1$, the symbolic power $I(G)^{(s)}$ and ordinary power $I(G)^s$ are ideals in the associated po…

Let $B\subseteq GL_n(\mathbb{C})$ be the Borel subgroup of upper triangular matrices, and let $w\in S_n$ be a Grassmannian permutation with descent at position $k$. Let $Y$ be the…

Let $R$ be a polynomial ring over a field, and let $I\subseteq R$ be an ideal with minimal homogeneous generators whose number and degrees are fixed. The regularity-bound conjectur…

Let $R=k[x_0,\ldots,x_n]$, and let $I=(f_1,\ldots,f_t)$ be a homogeneous ideal generated in degrees $d_1\geq d_2\geq \cdots\geq d_t\geq 1$, of codimension $r$. Set…