On triviality of $\mathbb{A}^2$-forms admitting a nontrivial $\mathbb{G}_a$-action

TL;DR

This paper provides a structure theorem for $A^2$-forms with a nontrivial $G_a$-action over arbitrary fields, leading to conditions for triviality and a generalization of the Zariski Cancellation Theorem.

Contribution

It introduces a structure theorem for $A^2$-forms with $G_a$-actions over any field, extending known results and providing new triviality criteria.

Findings

Factorial $A^2$-forms with a $k$-rational point and $G_a$-action are trivial.

Generalization of the Zariski Cancellation Theorem for the affine plane over any field.

Examples show hypotheses are necessary for triviality.

Abstract

T. Kambayashi had shown that $\mathbb{A}^2$-forms over separable field extensions are necessarily polynomial rings. However, there exist inseparable $\mathbb{A}^2$-forms which are not necessarily polynomial rings. In this paper, we give a structure theorem for $\mathbb{A}^2$-forms over arbitrary field extensions admitting a nontrivial $\mathbb{G}_a$-action. From this structure theorem we derive some conditions under which an $\mathbb{A}^2$-form becomes trivial. In particular, we prove that over a field $k$, a factorial $\mathbb{A}^2$-form having a $k$-rational point and a non-trivial $\mathbb{G}_a$-action is trivial and we also give examples demonstrating that none of these hypotheses can be discarded. As a consequence of the structure theorem, we obtain a generalization of the Zariski Cancellation Theorem for the affine plane over an arbitrary field.