6 problems

Let $S$ be a polynomial ring, let $I_{\mathcal L}$ be a lattice ideal in $S$, and let $M$ be its initial ideal with respect to some term order. For a prime ideal $P$ in $S$, write…

Pure-resolution conjecture. This stronger conjecture would imply a finite classification, up to inclusion of nonzero Gale vectors and permutation of coordinates, of the relevant hi…

Let $\Bbbk$ be a field of characteristic $0$, and let $I_\mathcal L$ be a nondegenerate Cohen–Macaulay lattice ideal such that … If $\operatorname{codim} I_\mathcal L\geq 2$, then…

Let ${\mathcal P}$ be a polyomino, and let $[a,b]$ be an interval containing ${\mathcal P}$. The polyomino ${\mathcal P}$ is simple if every cell $C$ not belonging to…

Width-and-degree generation conjecture. From the computation, one conjectures that $\ker(\pi)$ is generated, up to symmetry, by moves of width and degree at most three.

Engström's conjecture. For any lattice ideal with a Gröbner basis, there is an $N$ such that the connectivity of every $N$-large fiber graph satisfies