6 problems

Let $A$ be a $d \times n$ matrix with $n-d=3$, and let $\mathcal{G}_A$ denote its Gröbner fan. Sturmfels–Thomas conjecture. Every maximal cone in $\mathcal{G}_A$ has three or four…

Let $I$ be a toric ideal of codimension three, and let $C$ be a Gröbner cone of $I$. Four-facet conjecture. The cone $C$ has at most four facets. The example immediately preceding…

Let $A\in{\bf Z}^{d\times n}$ be a unimodular matrix of rank $d$, and let $B$ be a Gale transform of $A$. The fan $\Gamma(B)$ is the fan associated with the Gale transform $B$. Spe…

Let $N$ be a numerical semigroup with associated face $F_N$, and suppose that $N$ is staircase and Kunz. When $k=3$, the four-facet conjecture. $F_N$ has at most $4$ facets. This c…

Maximal-set realization conjecture. For every set of ideals $J_1,\ldots,J_s\in\textnormal{\textsf{S}}_{p(t)}^{n}$ such that

The Gröbner-fan maximality conjecture. For every cone of maximal dimension $\mathcal{C} \in \textnormal{GF}(\textnormal{\bf Hilb}_{p(t)}^{n})$, there exists a term order $\Omega$ s…