Number of components of the nullcone

TL;DR

This paper determines the number of irreducible components of the nullcone of unstable vectors for certain algebraic group representations, answering a specific open question in invariant theory.

Contribution

It provides a complete answer to Joseph's question for all pairs where the invariants form a free algebra, identifying the nullcone's component count.

Findings

Number of components explicitly calculated for all relevant pairs

Clarifies the structure of nullcones in these algebraic group actions

Advances understanding of invariant theory and nullcone geometry

Abstract

Answering the question of A. Joseph, for every pair $(G, V)$ where $G$ is a connected simple linear algebraic group and $V$ is a simple algebraic $G$-module with a free algebra of invariants, the number of irreducible components of the nullcone of unstable vectors in $V$ is found.