$\mathbb T$-homogeneous locally nilpotent derivations of trinomial algebras

TL;DR

This paper characterizes homogeneous locally nilpotent derivations of trinomial algebras, which are key in understanding Cox rings of certain varieties with torus actions.

Contribution

It provides a detailed description of these derivations, advancing the understanding of the algebraic structure of trinomial algebras and their geometric applications.

Findings

Classified homogeneous locally nilpotent derivations of trinomial algebras.

Connected derivations to Cox rings of complexity one torus action varieties.

Enhanced understanding of algebraic structures related to torus actions.

Abstract

A trinomial algebra is a commutative finitely generated algebra given by a system of compatible relations each of which is a polynomial with three terms. Such algebras arise as the Cox rings of varieties admitting a complexity one torus action. We describe locally nilpotent derivations of a trinomial algebra that are homogeneous under a natural torus action of complexity one.