# Cluster tilting for one-dimensional hypersurface singularities

## TL;DR

This paper explores the structure of Cohen-Macaulay modules over one-dimensional hypersurface singularities, linking them to cluster tilting theory and classifying associated 2-Calabi-Yau tilted algebras.

## Contribution

It provides criteria for the existence of cluster tilting objects and describes them using homological methods, including applications to specific curve singularities.

## Key findings

- Classified 2-CY tilted algebras for simple singularities
- Identified symmetric 2-CY tilted algebras with $ au^2=\id$
- Connected Cohen-Macaulay modules with cluster tilting theory

## Abstract

In this article we study Cohen-Macaulay modules over one-dimensional hypersurface singularities and the relationship with the representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological methods, using higher almost split sequences and results from birational geometry. We obtain a large class of 2-CY tilted algebras which are finite dimensional symmetric and satisfy $\tau^2=\id$. In particular, we compute 2-CY tilted algebras for simple and minimally elliptic curve singularities.