Stability of local rings of dimension 2

The notion of stability of a local ring arises naturally in investigating completed moduli spaces via geometric invariant theory. A scheme corresponding to a boundary point of a moduli space of nonsingular varieties cannot have unstable singular points. This paper reports some results concerning the stability of surface singularities. If a two-dimensional Cohen-Macaulay local ring R is semistable, its multiplicity must be less than seven and either equal to or 1 less than its embedding dimension. If the multiplicity of R is equal to its embedding dimension, the singularity of R must be a simple or cyclic elliptic singularity or the nonnormal limit of such a singularity. The results for the case when the multiplicity of R is less than its embedding dimension are still incomplete; the possible singularities that may arise when the multiplicity of R is equal to 2 or 3 are described.