Cohen-Macaulay modules over non-isolated surface singularities

This talk is based on my joint work with Yuriy Drozd, arXiv:1002.3042.
I am going to explain the classification of indecomposable Cohen-Macaulay modules over a certain class of non-isolated Gorenstein surface singularities called degenerate cusps. The core of our approach is a categorical construction, allowing to reduce this classification problem to a certain matrix problem of tame representation type.
I am going to illustrate our method on the case of the rings $k[[x,y,z]]/(x^3+y^2-xyz)$, $k[[x,y,z]]/(xyz)$ and $k[[x,y,u,v]]/(xy,uv)$. In the case of degenerate cusps, which are hypersurface singularities, our technique leads to a description of all indecomposable matrix factorizations of the underlying polynomials.