4 problems

Link-existence conjecture. For any $i$, $3\leq i\leq d-2$, if the $i$-th link does not exist, so that there is a common factor in the initial degree of the ideal of $Y_{i-1}$, then…

Let $C$ be a set of two skew lines in $\mathbb{P}^{4}$, spanning a hyperplane $H$, and let $C'$ be another set of two skew lines in $\mathbb{P}^{4}$, spanning a hyperplane $H'$. Tw…

Let $S={\Bbbk}[x,y,z]$, let $d$ be a positive integer, and let $[S/I]\in\mathrm{Hilb}^d(\mathbb{A}^3)$ be a smoothable point. Broken Gorenstein characterization conjecture. The fol…

Let $I$ be a grade $c$ perfect ideal in a local Noetherian ring $R$. Let $d$ denote the deviation of $I$ and $t$ the minimal number of generators of $\operatorname{Ext}^c(R/I,R)$.…