8 problems

Global generation conjecture. One can choose $S$ and $M_S\in{\cal M}_S$ such that $X\subset S$ is the zero-locus of a regular section

Let $r\ge1$ and $c\ge2-r$. Let $A=R/I_A$ (and, when $c>2-r$, let $B=R/I_B$) be defined by the $(t-r+1)\times(t-r+1)$ minors of a general homogeneous $t\times(t+c-1)$ matrix…

Let $r\ge1$, $c>2-r$, and let $\mathcal{A}$ be a homogeneous $t\times(t+c-1)$ matrix with entries of degree $a_j-b_i$. Set $A=R/I_{t-r+1}(\mathcal{A})$ and suppose…

Fix integers $r\ge1$, $c>2-r$, $t\ge2$, and $d\ge1$. Let $\mathcal{A}$ be a homogeneous $t\times(t+c-1)$ matrix with entries of degree $e$, and set $A=R/I_{t-r+1}(\mathcal{A})$. Su…

The dimension conjecture. Under these hypotheses,

Let $A$ be the degree matrix of a standard determinantal scheme, and let $h^A$ denote the associated $h$-vector. Assume that $A$ has no zero entries. Converse conjecture. The vecto…

Let $W(\underline{b};\underline{a})$ be the family of determinantal schemes associated with integers $a_0\le a_1\le\cdots\le a_{t+c-2}$ and $b_1\le\cdots\le b_t$, and let…

Let $W(\underline{b};\underline{a})$ be the family of determinantal schemes associated with integer sequences $a_0\le a_1\le\cdots\le a_{t+c-2}$ and $b_1\le\cdots\le b_t$, and let…