On conjugacy of additive actions in the affine Cremona group

TL;DR

This paper investigates the conjugacy of additive actions on algebraic varieties, showing that for projective space, the conjugating automorphism can be chosen within the affine Cremona group using basic polynomials.

Contribution

It proves that additive actions on projective space are conjugate via elements in the affine Cremona group, characterized by basic polynomials of local algebras.

Findings

Additive actions on projective space are conjugate via affine Cremona group elements.

Conjugating elements can be explicitly described using basic polynomials.

The results connect additive actions with the structure of the affine Cremona group.

Abstract

An additive action on an irreducible algebraic variety $X$ is an effective action $\mathbb{G}_a^n\times X\to X$ with an open orbit of the vector group $\mathbb{G}_a^n$. Any two additive actions on $X$ are conjugate by a birational automorphism of $X$. We prove that, if $X$ is the projective space, the conjugating element can be chosen in the affine Cremona group and it is given by so-called basic polynomials of the corresponding local algebra.