Jet schemes and invariant theory

TL;DR

This paper investigates the properties of jet schemes under group actions, providing criteria for invariant ring isomorphisms and classifying specific cases, with implications for invariant theory and algebraic geometry.

Contribution

It establishes criteria for when the invariant ring map is an isomorphism across all jet levels, especially for classical groups and certain representations, and classifies cases for ^*_.

Findings

Criteria for isomorphism of invariant rings using Luna's slice theorem.

Complete classification of ^*_ for ^*_ representations.

Examples of invariant ring maps that are surjective but not injective.

Abstract

Let $G$ be a complex reductive group and $V$ a $G$-module. Then the $m$th jet scheme $G_m$ acts on the $m$th jet scheme $V_m$ for all $m\geq 0$. We are interested in the invariant ring $\mathcal{O}(V_m)^{G_m}$ and whether the map $p_m^*\colon\mathcal{O}((V//G)_m) \rightarrow \mathcal{O}(V_m)^{G_m}$ induced by the categorical quotient map $p\colon V\rightarrow V//G$ is an isomorphism, surjective, or neither. Using Luna's slice theorem, we give criteria for $p_m^*$ to be an isomorphism for all $m$, and we prove this when $G=SL_n$, $GL_n$, $SO_n$, or $Sp_{2n}$ and $V$ is a sum of copies of the standard representation and its dual, such that $V//G$ is smooth or a complete intersection. We classify all representations of $\mathbb{C}^*$ for which $p^*_{\infty}$ is surjective or an isomorphism. Finally, we give examples where $p^*_m$ is surjective for $m=\infty$ but not for finite $m$, and where it is surjective but not injective.