A note on homogeneous rank $2$ locally nilpotent derivations on $k[X,Y,Z]$

TL;DR

This paper classifies and describes the structure of homogeneous locally nilpotent derivations of rank 2 on polynomial rings, focusing on their triangularizability and explicit generators, especially for degrees related to prime numbers.

Contribution

It provides a classification of irreducible homogeneous locally nilpotent derivations of rank 2, including conditions for triangularizability and explicit generator descriptions.

Findings

Derivations of degree p-2 are triangularizable for prime p.

Non-triangularizable derivations of degree pq-2 are characterized for primes p,q.

Explicit generators of image ideals are described for certain derivations.

Abstract

In this article we show that for every prime number $p$, any irreducible homogeneous locally nilpotent derivations of rank $2$ and degree $p-2$ are triangularizable. Further, we describe the structure of irreducible non-triangularizable homogeneous locally nilpotent derivations of rank $2$ and degree $pq-2$, where $p,q$ are prime numbers. Consequently, we give explicit descriptions of the generators of the image ideals of certain homogeneous locally nilpotent derivations of rank $2$.