On jet closures of singularities

TL;DR

This paper introduces new algebraic invariants for singularities based on jet closures and support closures, explores their properties, and applies them to specific ideal cases, advancing the understanding of singularity classification.

Contribution

It defines two local algebras associated with jet closures, proves their invariance, and introduces a jet index to analyze jet schemes' information recovery.

Findings

The two algebras are invariants of singularities.

Explicit computations for monomial and homogeneous ideals.

Introduction of a new jet index and filtration.

Abstract

The jet closure and jet support closure were first introduced by de Fernex, Ein and Ishii to solve the local isomorphism problem. In this paper, we introduce two local algebras associated to jet closure and jet support closure respectively. We show that these two algebras are invariants of the singularities. We compute and investigate these invariants for some interesting cases, such as the cases of monomial ideals and homogeneous ideals. We also introduce a new filtration and jet index to jet closures. The jet index describes which jet scheme recover the information of base scheme. Moreover, we obtain some properties of the jet index. Keywords:jet closure, jet support closure and filtration.