Pathological MMP singularities as $\alpha_p$-quotients

TL;DR

This paper constructs examples of pathological MMP singularities in positive characteristic using $ ext{alpha}_p$-quotients, revealing new types of singularities and families with unusual fiber properties.

Contribution

It introduces a method to produce pathological MMP singularities in all positive characteristics via $ ext{alpha}_p$-quotients, expanding the understanding of singularity behavior.

Findings

Existence of non-$S_3$ terminal singularities in positive characteristic.

Construction of families with smooth or klt general fibers and non-$S_2$ special fibers.

Bound on the dimension of examples proportional to the characteristic.

Abstract

We construct pathological examples of MMP singularities in every positive characteristic using quotients by $\alpha_p$-actions. In particular, we obtain non-$S_3$ terminal singularities, as well as locally stable (respectively stable) families whose general fibers are smooth (respectively klt, Cohen--Macaulay and $F$-injective) and whose special fibers are non-$S_2$. The dimensions of these examples are bounded below by a linear function of the characteristic.