4 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

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

Let $S$ be a standard graded algebra over a field $k$. For a homogeneous ideal $Q\subseteq S$, its depth function is the function $n\mapsto\operatorname{depth} S/Q^n$ for $n\geq 1$…

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