Deformations of isolated cyclic quotient singularities in arbitrary characteristic

TL;DR

This paper proves that toric surface singularities and cyclic quotient singularities in any dimension can deform into similar singularities, extending known results to arbitrary characteristic and confirming Riemenschneider's conjecture.

Contribution

It demonstrates that isolated cyclic quotient singularities deform to similar singularities in all characteristics, including mixed characteristic, and confirms Riemenschneider's conjecture.

Findings

Toric surface singularities deform to toric surface singularities in arbitrary characteristic.

Isolated cyclic quotient singularities deform to similar singularities in all characteristics.

Riemenschneider's conjecture is established for all dimensions and characteristics.

Abstract

We show that toric surface singularities deform to toric surface singularities - both in equal and mixed characteristic. As an application, we establish Riemenschneiders conjecture that isolated cyclic quotient singularities of any dimension deform to isolated cyclic quotient singularities in equal and mixed characteristic.