The McKay correspondence and local heights for wild-by-tame split metacyclic groups

TL;DR

This paper explores the McKay correspondence for certain wild-by-tame split metacyclic groups, calculating stringy motives and Euler numbers, revealing dependence on representation choices and group structure.

Contribution

It provides new formulas for stringy motives and Euler numbers for quotient varieties of wild-by-tame groups, highlighting differences from classical cases.

Findings

Stringy Euler number depends on the representation and group choice.

Crepant resolutions may not have Euler characteristic equal to the number of conjugacy classes.

Computed the v-function related to stacky local height functions.

Abstract

We study the McKay correspondence for the representations of certain wild-by-tame split metacyclic groups whose order is divisible by the characteristic of the base field. We calculate the stringy motive of the quotient variety and find a formula for its stringy Euler number. As a consequence, we prove that a crepant resolution of the quotient variety (provided one exists) does not in general have Euler characteristic equal to the number of conjugacy classes in $G$, in contrast to the classical case. In particular, we show it depends on the choice of representation as well as the group. As part of this, we compute the v-function associated to a $G$-representation, corresponding to a stacky local height function.