Discriminant groups of wild cyclic quotient singularities

TL;DR

This paper explicitly describes the resolution of certain wild cyclic quotient singularities in dimension two across various primes, revealing new possibilities for intersection matrix determinants and supporting conjectures about ramification behavior.

Contribution

It provides explicit resolutions for wild Z/pZ-quotient singularities and shows that any power of p can be the determinant of their intersection matrices, extending known classifications.

Findings

Explicit resolutions for wild cyclic quotient singularities in dimension two.

Any power of p can appear as the determinant of the intersection matrix.

Evidence supporting the conjecture about ramification at the origin.

Abstract

Let p be prime. We describe explicitly the resolution of singularities of several families of wild Z/pZ-quotient singularities in dimension two, including families that generalize the quotient singularities of type E_6, E_7, and E_8 from p=2 to arbitrary characteristics. We prove that for odd primes, any power of p can appear as the determinant of the intersection matrix of a wild Z/pZ-quotient singularity. We also provide evidence towards the conjecture that in this situation one may choose the wild action to be ramified precisely at the origin.