71 problems

Exceptional-set characterization. The exceptional set $\mathcal{E}(A)$ is empty if and only if the toric ideal $I_A$ is Cohen–Macaulay.

Finite abelian group bound. For any finite abelian group $G$ and any positive integer $n$,

Let $G$ be a graph and let $I_G$ be its cut ideal. A graph is $K_5$-minor-free when it has no minor isomorphic to $K_5$. Quartic generation conjecture. … This extends the proposed…

Let $G$ be a graph, let $I_G$ be its cut ideal, and let $\mu(I_G)$ denote the maximal degree of a binomial in a minimal generating set of $I_G$. A graph $H$ is a minor of $G$ if it…

Let $M$ be a matroid, with toric ideal $I_M$. For bases $B_1,B_2,D_1,D_2$, call $y_{B_1}y_{B_2}-y_{D_1}y_{D_2}$ a double-swap binomial when the pair $D_1,D_2$ is obtained from…

Let $M$ be a matroid and let $I_M$ be its toric ideal. A quadratic binomial in $I_M$ is a binomial homogeneous of degree two in the base variables. The quadratic-generation conject…

Let $M$ be a matroid and let $I_M$ be its toric ideal, the kernel of the map sending each base variable $y_B$ to the corresponding squarefree monomial. The matroid quadratic Gröbne…

Let $I$ be a toric ideal defining a projectively normal $r$-dimensional toric variety. The normality Gröbner-basis conjecture. The ideal $I$ has a Gröbner basis consisting of binom…

Let $M$ be a matroid on the ground set $\{1,2,\ldots,n\}$. Fix a field $k$, let $S_M=k[y_B:B$ a base of $M]$, and let $I_M$ be the kernel of the homomorphism…

Let $T$ be a rooted tree, let ${\mathcal A}_T$ be the configuration of exponent vectors from the toric parametrization, and let $P_T$ be the convex hull of the columns of…

Let $T$ be a path of length $n$, and let $I_T$ be its toric ideal of phylogenetic invariants. The ideal $I_T$ should be generated in degree $3$, with $2n-4$ generators of degree…

Let $T$ be a rooted tree and let $I_T$ be the toric ideal associated to the homogeneous binary Markov model. Suppose either that $T$ is binary, or more generally that every non-lea…

Complete bipartite graph ideal degree conjecture. The graph ideal for $G=K_{m,n}$ is generated in degree at most

Let ${\mathcal A}\subset\mathbb Z^d$ be a finite graded, normal configuration, and let $I_{\mathcal A}$ be its toric ideal. Write $\dim({\mathcal A})=d$ for the Krull dimension of…

Sturmfels's conjecture. The exceptional set is empty if and only if $I_A$ is Cohen–Macaulay. For Cohen–Macaulay $I_A$, the rank equals $\operatorname{vol}(A)$ for every parameter,…

Let $A\in{\bf Z}^{d\times n}$ be a matrix of rank $d$. The Gröbner fan of $A$ is the fan whose cones encode the reduced Gröbner bases of the toric ideal associated to $A$. Sturmfel…

Perfect-contractility conjecture. The following conditions are equivalent:

Matroid-base quadraticity conjecture. The toric ideal $I_{P_B(\mathcal{M})}$ is quadratic.

Bøgvad's conjecture. The toric ideal $I_P$ is quadratic.

Let $G$ be a simple graph, let $A_G$ be its incidence matrix, let $I_G$ be the corresponding toric ideal, and let $M$ be a minimal Markov basis for $I_G$. A Markov basis is distanc…

Koszul filtration conjecture. The quotient $S/(G)$ has a Koszul filtration. Moreover, if $(G)$ is toric, then $S/(G)$ has a monomial Koszul filtration.

Hibi–Qureshi conjecture. The ideal $I_{\mathcal{P}}$ arises as the toric ideal of the edge ring of a finite simple graph if and only if $\mathcal{P}$ is simple.

Let $L$ be a linear subspace of dimension $r$ whose matroid $\mathrm{M}(L)$ is connected. Set $\mathcal E_L=\mathcal S_L\oplus\mathcal Q_L$. Vanishing conjecture for dual matroid c…

Quadratic Parke–Taylor generation conjecture. The ideal $\tilde{\mathcal{I}}(\mathrm{PT}_n^\circ)\subseteq\mathbb{C}[z_\sigma^{\pm}]$ is generated by the quadratics in $\calF$ and…

Quadratic kernel-generation conjecture. For $n\geq 7$, the set $\calB_n$ generates $\ker_\mathbb{Z}(A_n)$. In particular,…