6 problems

Eisenbud–Huneke–Ulrich conjecture. If $I$ is $\mathfrak{m}$-primary and the resolution of $I$ is linear for $p$ steps, where $p\geq 1$, then

Gross–Popescu's conjecture. The homogeneous ideal of $A$ in this embedding is generated by quadrics and cubics.

Harbourne–Huneke degree-bound conjecture. For every $r>0$,

Let $S=K[x_1,\dots,x_n]$ be a polynomial ring over a field $K$, let $I\subset S$ be a suitable homogeneous ideal, and let $f_I(k)=\operatorname{depth}(S/I^k)$. A function…

Unbounded resurgence-gap conjecture. There exists a family of homogeneous polynomial ideals $\{I_k\}_{k\in\mathbb N}$, possibly in different polynomial rings, such that

Sufficiently general ideal conjecture. A sufficiently general homogeneous ideal should have a standard basis consisting entirely of essential forms.