On Classification of Toric Singularities

In 1988 S. Mori, D. Morrison, and I. Morrison published a paper [4] in which they gave a computer-based conjectural classification of four-dimensional terminal cyclic quotient singularities of prime index. This classification was partially proven in 1990 by G. Sankaran (cf. [7]). Basically, the conjecture of [4] says that these singularities form a finite number of series (cf. definition in section 2 below) with the precise list of series given. While a precise list of series in higher dimensions would be way too long, the qualitative version of this conjecture makes sense for toric singularities of any dimension. It turned out, to my great surprise, that this and much more was basically proven in 1991 in a beautiful paper by Jim Lawrence (cf. [3]). His proof and motivation came from the geometry of numbers and he was obviously unaware of the papers of Mori-Morrison-Morrison and Sankaran. So in this short note I just bring this all together. In section 2 the necessary definitions are given and the main theorem is proven. In section 3 some interesting related open problems are discussed. Acknowledgments. I can claim only little credit for the proof of the main theorem, as this is just an algebro-geometric corollary of the result of Jim Lawrence. I discovered the paper of Lawrence using MathSciNet.