Open AccessStability of local rings of dimension 2
Author(s) -
Jayant Shah
Publication year - 1978
Publication title -
proceedings of the national academy of sciences of the united states of america
Language(s) - English
Resource type - Journals
eISSN - 1091-6490
pISSN - 0027-8424
DOI - 10.1073/pnas.75.9.4085
Subject(s) - multiplicity (mathematics) , singularity , gravitational singularity , mathematics , moduli space , embedding , pure mathematics , local ring , invertible matrix , singularity theory , invariant (physics) , isolated singularity , moduli , mathematical analysis , ring (chemistry) , physics , mathematical physics , computer science , quantum mechanics , chemistry , organic chemistry , artificial intelligence
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.