Commutative actions on smooth projective quadrics

TL;DR

This paper classifies all commutative algebraic group actions with an open orbit on smooth projective quadrics, showing uniqueness for higher dimensions and explicitly counting actions in lower dimensions.

Contribution

It proves that for dimensions n ≥ 3, there is a unique commutative action on Q_n, and explicitly classifies actions for lower dimensions.

Findings

Unique commutative action on Q_n for n ≥ 3

Three actions on Q_2 up to equivalence

Two actions on Q_1 up to equivalence

Abstract

By a commutative action on a smooth quadric $Q_n$ in $P^{n+1}$ we mean an effective action of a commutative connected algebraic group on $Q_n$ with an open orbit. We show that for $n \geq 3$ all commutative actions on $Q_n$ are additive actions described by Sharoiko in 2009. So there is a unique commutative action on $Q_n$ up to equivalence. For $n = 2$ there are three commutative actions on $Q_2$ up to equivalence, for $n = 1$ there are two commutative actions on $Q_1$ up to equivalence.