Isomorphisms of jet schemes

TL;DR

This paper investigates the relationship between schemes and their jet schemes, showing conditions under which isomorphisms of jet schemes imply isomorphisms of the original schemes, and highlighting limitations of this implication.

Contribution

It establishes that isomorphisms of jet schemes induced by base scheme morphisms imply base scheme isomorphisms, but general jet scheme isomorphisms do not guarantee base scheme isomorphisms.

Findings

Isomorphism of jet schemes induced by base scheme isomorphism implies the base schemes are isomorphic.

Existence of jet scheme isomorphisms alone does not ensure base scheme isomorphism.

The paper clarifies the conditions under which jet scheme isomorphisms reflect base scheme isomorphisms.

Abstract

If two schemes are isomorphic, then their $m$-jet schemes are isomorphic for all $m$. In this paper we consider the converse problem. We prove that if an isomorphism of the $m$-jet schemes is induced from a morphism of the base schemes, then the morphism of the base schemes is an isomorphism. But we also prove that just the existence of isomorphisms between $m$-jet schemes does not yield the existence of an isomorphism between the base schemes.