Cylinders and the zero locus of the plinth ideal

TL;DR

This paper establishes a precise relationship between invariant cylinders under a group action on an affine variety and the zero locus of the plinth ideal associated with a locally nilpotent derivation.

Contribution

It proves that the complement of principal invariant cylinders coincides with the zero locus of the plinth ideal, clarifying the geometric structure of these invariants.

Findings

The complement of all principal invariant cylinders equals the zero locus of the plinth ideal.

Provides a characterization of invariant cylinders in affine varieties.

Connects geometric invariants with algebraic ideals in the context of group actions.

Abstract

Given a $\mathbb{G}_\mathrm{a}$-action on an affine variety $X$, we show that the complement of the union of all principal invariant cylinders in $X$ is equal to the zero locus of the plinth ideal of the corresponding locally nilpotent derivation.