Classify non-Gorenstein finite Cohen–Macaulay type rings in dimension at least three

Classify, up to isomorphism, all d-dimensional (d ≥ 3) Cohen–Macaulay complete local rings over an algebraically closed field of characteristic zero that are of finite Cohen–Macaulay type and are not Gorenstein.

Background

The paper summarizes what is known about the classification of Cohen–Macaulay complete local rings (R, m, k) of finite Cohen–Macaulay type when the residue field k is algebraically closed of characteristic zero. In dimensions 0, 1, and 2, complete classifications are available: Gorenstein cases are simple singularities and, more generally in dimension 2, finite CM type rings are quotient singularities of the form k[[x,y]]^G.

For dimensions d ≥ 3, the situation diverges sharply. While Gorenstein cases have a clearer structure, the non-Gorenstein case lacks a full classification. The authors explicitly note that the classification problem in this regime remains open and only two examples are currently known, highlighting a significant gap compared with the comprehensive results in lower dimensions.