Discriminant groups of wild cyclic quotient singularities

Lorenzini, Dino; Schröer, Stefan

doi:10.2140/ant.2023.17.1017

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Discriminant groups of wild cyclic quotient singularities

Dino Lorenzini and Stefan Schröer

Vol. 17 (2023), No. 5, 1017–1068

DOI: 10.2140/ant.2023.17.1017

Abstract

Let $p$ be prime. We describe explicitly the resolution of singularities of several families of wild $ℤ ∕ p ℤ$ -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 $p$ odd, any power of $p$ can appear as the determinant of the intersection matrix of a wild $ℤ ∕ p ℤ$ -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.

Keywords

quotient singularity, cyclic, wild, discriminant group, resolution of singularities, Brieskorn singularity, Dynkin diagram

Mathematical Subject Classification

Primary: 13A50, 14B05, 14E15, 14J17

Milestones

Received: 19 February 2021

Revised: 11 March 2022

Accepted: 6 July 2022

Published: 9 May 2023

Authors

Dino Lorenzini
Department of Mathematics University of Georgia Athens, GA United States

Stefan Schröer
Mathematisches Institut Heinrich-Heine-Universität Düsseldorf Düsseldorf Germany

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Vol. 17, No. 5, 2023