A study on tame and wild quotient singularities in characteristic p

A closed point $x$ on a normal scheme $X$ is a quotient singularity if the corresponding local ring $R=\mathcal{O}_{X,x}$ arises as the ring of invariants by the action of a finite group $G$ on a regular local ring $A$. If the characteristic of the residue field of $R$ does not divide the order of $G$, it is called a tame quotient singularity; it is wild otherwise. The aim of this thesis is to discuss the resolution of quotient singularities in three distinct settings: complex toric surfaces, tame cyclic quotient singularities on a normal curve over a Dedekind scheme, wild quotient singularities on a $2$-dimensional scheme. While the former two settings can be solved algorithmically and the data of the resolution can be encoded in a so-called Hirzebruch-Jung continued fraction, the latter can be much more unpredictable and difficult to compute explicitly. We will study the dual graph attached to the minimal regular resolution in each case, and see that in the first two cases it is a Dynkin graph of type $A_n$, while in the wild case it always contains a vertex of valency at least $3$.