On linear $\alpha_p$-quotients

TL;DR

This paper investigates the properties of quotient singularities arising from linear p-actions on affine spaces, providing explicit resolutions, classifying singularities, and computing motivic invariants, while testing a conjecture through computational methods.

Contribution

It offers a detailed analysis of p-quotients, reduces a conjecture to an explicit combinatorial problem, and proves the equality of invariants for many primes.

Findings

Quotient singularities can be classified as log canonical, canonical, or terminal.

The stringy motivic invariants of p-quotients are computed explicitly.

The conjecture relating these invariants to p-quotients is verified for many primes.

Abstract

We study linear $\alpha_p$-actions on affine spaces and the associated quotient singularities, using explicit stacky resolutions. We describe when the quotient singularities are log canonical, canonical or terminal, and we compute their stringy motivic invariants. The second author and Fabio Tonini conjectured that these invariants coincide with those of linear $\mathbb{Z}/p$-quotients: our approach reduces this conjecture to an equality of explicit multi-sets, which we check for a large number of primes using a computer software. A general proof of the equality of multi-sets is given in the appendix written by Linus R\"osler.