Smoothness and jet schemes

TL;DR

This paper establishes criteria linking the smoothness of schemes over algebraically closed fields to properties of their jet schemes, providing characterizations in both zero and positive characteristic cases.

Contribution

It introduces new criteria for scheme smoothness based on jet scheme properties, including flatness of truncation morphisms and reducedness conditions.

Findings

In characteristic zero, non-singularity iff a truncation morphism of jet schemes is flat.

In positive characteristic, non-singularity characterized by reducedness and jet scheme properties.

A simple criterion that a scheme is non-singular if and only if its jet scheme is non-singular.

Abstract

This paper shows some criteria for a scheme of finite type over an algebraically closed field to be non-singular in terms of jet schemes. For the base field of characteristic zero, the scheme is non-singular if and only if one of the truncation morphisms of its jet schemes is flat. For the positive characteristic case, we obtain a similar characterization under the reducedness condition on the scheme. We also obtain by a simple discussion that the scheme is non-singular if and only if one of its jet schemes is non-singular.