Wild quotient singularities of arithmetic surfaces and their regular models

Abstract

This thesis adresses the problem of tame and wild cyclic quotient singularities of local Noetherian rings and arithmetic surfaces. In chapter 2, we study the ring of invariants of a local Noetherian ring by a tame cyclic action. We collect and generalize classic results on tame cyclic quotient singularities in the context of toric geometry. In chapter 3, we prove algebraic results on the invariant ring of a local Noetherian ring by a possibly wild cyclic action of prime order. The central result is a characterization of monogenous extensions which can be read as a regularity criterion for the invariant ring generalizing a criterion of Serre. In chapter 4, we relate the structure of a quotient singularity of a regular arithmetic surface to the action of the group on its models. In particular, we prove results relating the minimal normal crossings desingularization of the quotient to certain models of the original surface.