Let $Q$ be a regular local ring. An ideal $I$ of $Q$ is said to be licci if it is linked to a complete intersection. It is known that all perfect ideals of grade $2$ are licci, while most those of grade $4$ are not licci. The grade $3$ case is particularly interesting. It is conjectured by Christensen, Veliche and Weyman that a perfect ideal of grade $3$ in $Q$ is licci if its free resolution is of Dynkin formats. Their approach to proving the conjecture involves the classification of ideals by the corresponding Tor algebra structure.
In this talk, I will briefly talk about how they narrow down the candidates to those of Dynkin formats. Then I want to share with you the classification they used to attack the conjecture and how far they've got with this topic recently.