Rank and rigidity of locally nilpotent derivations of affine fibrations

TL;DR

This paper introduces a new framework for understanding the rank and rigidity of locally nilpotent derivations in affine fibrations, providing classifications for specific derivations in three-dimensional cases.

Contribution

It defines notions of rank and rigidity for locally nilpotent derivations of affine fibrations and classifies fixed point free derivations in three-dimensional cases.

Findings

Characterization of locally nilpotent derivations with slices in affine fibrations.

Classification of fixed point free derivations in $ ext{A}^3$-fibrations.

Analogy established between derivations of polynomial algebras and affine fibrations.

Abstract

In this exposition, we propose a notion of rank and rigidity of locally nilpotent derivations of affine fibrations. We show that the concept is analogous to the perception of rank and rigidity of locally nilpotent derivations of polynomial algebras. Our results characterize locally nilpotent derivations of $\mathbb{A}^3$-fibrations having slice by classifying the fixed point free locally nilpotent derivations in terms of their ranks.