Additive actions on hyperquadrics of corank two

TL;DR

This paper classifies additive group actions on hyperquadrics of corank two, extending previous classifications for corank zero and one, especially focusing on cases where singularities are not fixed by the action.

Contribution

It provides a new classification of additive actions on hyperquadrics of corank two, addressing cases with non-fixed singularities, which was not previously understood.

Findings

Classified additive actions on hyperquadrics of corank two.

Extended understanding of group actions on singular hyperquadrics.

Identified cases where singularities are not fixed by the additive action.

Abstract

For a projective variety $X$ in $\mathbb{P}^{m}$ of dimension $n$, an additive action on $X$ is an effective action of $\mathbb{G}_{a}^{n}$ on $\mathbb{P}^{m}$ such that $X$ is $\mathbb{G}_{a}^{n}$-invariant and the induced action on $X$ has an open orbit. Arzhantsev and Popovskiy have classified additive actions on hyperquadrics of corank 0 or 1. In this paper, we give the classification of additive actions on hyperquadrics of corank 2 whose singularities are not fixed by the $\mathbb{G}_{a}^{n}$-action.