The separating variety for the basic representations of the additive group

TL;DR

This paper studies the structure of the separating variety for basic additive group representations, revealing when the variety is irreducible and implications for polynomial separating algebras.

Contribution

It characterizes the irreducible components of the separating variety for indecomposable rational linear representations of the additive group, identifying cases with unique or multiple components.

Findings

For certain dimensions, the separating variety is irreducible.

In other cases, the separating variety has two irreducible components.

No polynomial separating algebras exist in cases with multiple components.

Abstract

For a group $G$ acting on an affine variety $X$, the separating variety is the closed subvariety of $X\times X$ encoding which points of $X$ are separated by invariants. We concentrate on the indecomposable rational linear representations $V_n$ of dimension $n+1$ of the additive group of a field of characteristic zero, and decompose the separating variety into the union of irreducible components. We show that if $n$ is odd, divisible by four, or equal to two, the closure of the graph of the action, which has dimension $n+2$, is the only component of the separating variety. In the remaining cases, there is a second irreducible component of dimension $n+1$. We conclude that in these cases, there are no polynomial separating algebras.