Abstract

Gorenstein quasi-homogeneous surface singularities, studied by Dolgachev, Neumann and others, correspond to lifts of Fuchsian groups into the universal covering of PSL . I will show that the space of Gorenstein quasi-homogeneous surface singularities corresponding to a certain Fuchsian group is a finite affine space of Z/mZ-valued functions on the Fuchsian group, called m-Arf functions. Using m-Arf functions, I will count connected components of the space of Gorenstein quasi-homogeneous surface singularities and prove that any connected component is homeomorphic to a quotient of R^d by a discrete group. This work is connected to the earlier results of Atiyah and Mumford on spin structures on compact Riemann surfaces and of Jarvis, Kimura and Vaintrob on moduli spaces of higher spin curves. This is joint work with Sergey Natanzon.