Elliptic singularities and threefold flops in positive characteristic

TL;DR

This paper proves that any flop of a smooth threefold over an algebraically closed field of positive characteristic is smooth, by examining elliptic singularities and Gorenstein curves in this setting.

Contribution

It establishes the smoothness of flops in positive characteristic and analyzes elliptic singularities over imperfect fields, which was previously not well-understood.

Findings

Any flop of a smooth threefold in positive characteristic is smooth.

Elliptic singularities over imperfect fields are characterized and studied.

Gorenstein curves of genus one are analyzed in the context of threefold flops.

Abstract

Let $X$ be a smooth threefold over an algebraically closed field of positive characteristic. We prove that an arbitrary flop of $X$ is smooth. To this end, we study Gorenstein curves of genus one and two-dimensional elliptic singularities defined over imperfect fields.