4 problems

Let $n=3$, and let $Z\subset\mathbf{P}^2$ be a rational cuspidal curve, meaning that $Z$ is homeomorphic to $\mathbf{P}^1$. Let $M'_k$ denote the degree-$k$ graded piece of the mod…

Let $C:f=0$ be a plane curve of degree $d$. Let $ct(f)$ and $st(f)$ denote the coincidence and stability thresholds of its Milnor algebra, and let $T=3(d-2)$. Freeness criterion co…

Let $S$ be the graded polynomial ring defining a projective hypersurface $V(f):f=0$, and let $M(f)=S/J_f$ be its Milnor algebra, where $J_f$ is the Jacobian ideal. Let…

Unimodality conjecture. The sequence of dimensions $\dim SD(J_f)_k$ is unimodal, namely