On numerically Gorenstein quasi-simple elliptic singularities with $\textbf {C}^ *$-action

Abstract:

Let $(X,x)$ be a numerically Gorenstein elliptic singularity with ${\mathbb {C}^{\ast }}$-action and $\pi :(\tilde X,A) \to (X,x)$ the minimal good resolution. Assume that the central curve of $\pi$ is an elliptic curve. We will determine the configuration of the w.d. graph (weighted dual graph) of $A$ and obtain a condition for $(X,x)$ to be a maximally elliptic singularity in the sense of Yau.